# Corrected inference about the extreme Expected Shortfall in the general Max-Domain of Attraction

Abdelaati Daouia, Gilles Stupfler, Antoine Usseglio-Carleve

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Abdelaati Daouia, Gilles Stupfler, Antoine Usseglio-Carleve. Corrected inference about the extreme Expected Shortfall in the general Max-Domain of Attraction. 2025. hal-04648262v3

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# Corrected inference about the extreme Expected Shortfall in the general Max-Domain of Attraction

Abdelaati Daouia<sup>a</sup>, Gilles Stupfler<sup>b</sup> & Antoine Usseglio-Carleve<sup>c</sup>

<sup>a</sup> Toulouse School of Economics, University of Toulouse Capitole, France
ORCID: 0000-0003-2621-8860

<sup>b</sup> Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
ORCID: 0000-0003-2497-9412

<sup>c</sup> Avignon Université, LMA UPR 2151, 84000 Avignon, France
ORCID: 0000-0002-8148-3758

**Abstract.** The use of the Expected Shortfall as a solution for various deficiencies of quantiles has gained substantial traction in the field of risk assessment over the last 20 years. Existing approaches to its inference at extreme levels remain limited to distributions that are both heavy-tailed and have a finite second tail moment. This constitutes a strong restriction in areas like finance and environmental science, where the random variable of interest may have a much heavier tail or, at the opposite, may be light-tailed or short-tailed. Under a wider semiparametric extreme value framework, we develop comprehensive asymptotic theory for Expected Shortfall estimation above extreme quantiles in the class of distributions with finite first tail moment, regardless of whether the underlying extreme value index is positive, negative, or zero. By relying on the moment estimators of the scale and shape extreme value parameters, we construct corrected asymptotic confidence intervals whose finite-sample coverage is found to be close to the nominal level on simulated data. We illustrate the usefulness of our construction on two sets of financial loss returns and flood insurance claims data.

**AMS Subject Classifications:** 62G05, 62G20, 62G30, 62G32.

**Keywords:** Expected Shortfall, extreme value moment estimator, inference, second-order extended regular variation, semiparametric extrapolation.
# 1 Introduction

Tail risk assessment is concerned with the analysis of rare events that carry potential serious impacts on healthcare systems, the environment or the economy. The most common univariate risk measure, referred to as Value-at-Risk, is the quantile $q(\tau)$ of a random risk variable $X$, for a suitable $\tau \in (0, 1)$. Unfortunately, this risk measure is unable to give any idea of the shape of the distribution of $X$ beyond $q(\tau)$, and does not define a coherent risk measure in a financial and actuarial sense, see Artzner et al. (1999) and Acerbi (2002).

A better alternative to $q(\tau)$ in these respects is the Expected Shortfall at level $\tau$, namely

$$ \mathrm{ES}(\tau) := \frac{1}{1-\tau} \int_{\tau}^{1} q(t)dt. $$

When the distribution of $X$ is continuous, this is the $\tau$-Conditional Value-at-Risk or Conditional Tail Expectation CTE($\tau$) = $\mathbb{E}(X | X > q(\tau))$ (Rockafellar and Uryasev, 2002). The Expected Shortfall risk measure takes all values of the variable $X$ beyond $q(\tau)$ into account. Being a spectral risk measure with positive and nonincreasing risk spectrum, it is coherent and comonotonically additive, see Theorem 4.47 p.180 and Remark 4.85 p.199 in Föllmer and Schied (2004). It is preferred by practitioners concerned with exposure to catastrophic financial events (Acerbi and Tasche, 2002a,b). It is also favored by major regulators, including the EU, UK, Australia and Canada, which will require the use of ES(97.5%), rather than VaR(99%), in alternative internal models from 1 January 2025. In the EU, this is codified by Article 325ba(1) of Regulation (EU) No 2019/876, updating the Capital Requirements Regulation (EU) No 575/2013, implementing the Basel Committee on Banking Supervision rules.

The estimation of the Expected Shortfall has so far mostly focused on its empirical counterpart. To simplify the presentation, we consider in this introduction the setting when $n(1-\tau)$ is an integer and $F$ is continuous. Then the empirical estimator of $\mathrm{ES}(\tau)$ is

$$ \frac{1}{1-\tau} \int_{\tau}^{1} \hat{q}_{n}(t)dt \quad \text{or equivalently} \quad \frac{1}{n(1-\tau)} \sum_{i=1}^{n(1-\tau)} X_{n-i+1:n}, $$

where $\hat{q}_n(t) = X_{\lfloor nt \rfloor:n}$ is the empirical quantile function of a sample $(X_1, \dots, X_n)$ from $X$, with $X_{1:n} \le X_{2:n} \le \dots \le X_{n:n}$ being the order statistics in ascending order. This is an $L$-statistic, whose asymptotic behavior has been considered by Brazauskas et al. (2008) for fixed $\tau$, and by El Methni et al. (2014, 2018), El Methni and Stupfler (2017) and Goegebeur et al. (2022) at levels $\tau_n \uparrow 1$ as $n \to \infty$, under the assumption that $X_+ = \max(X, 0)$ has a finite variance. This restricts appreciably the range of potential applications; for example, the analysis of the French commercial losses fire insurance dataset of El Methni and Stupfler (2018) suggests that the underlying distribution there is integrable but has an infinite second
moment. Apart from El Methni et al. (2018), who consider a general extreme value setting
in nonparametric regression, existing results also assume that the distribution of X is heavy-
tailed. This may constitute a substantial restriction for environmental applications (Beirlant
et al., 2004) but also in finance (Daouia et al., 2024a). The work of El Methni et al. (2018) is,
however, restricted to intermediate risk measures ES(τ_n) that are only moderately extreme
and stay within the sample.

Developing inference procedures for the empirical Expected Shortfall at extreme levels under minimal restrictions on the tail of X is the original motivation for this article. We work under the second-order version of the general extreme value assumption about the quantile function q(·), which essentially conveys that the right tail of the distribution of X can be approximated by that of a Generalized Pareto distribution. This makes it possible to construct extrapolated estimators of the Expected Shortfall using extreme value estimators of the scale and shape parameters of the approximating Generalized Pareto distribution. Depending on the class of estimators that is used, the extreme value assumption about q(·) may not be sufficient; for instance, if the moment estimators of Dekkers et al. (1989) are employed, then an analogous extreme value assumption about log q(·) should hold, see pp.103-104 in de Haan and Ferreira (2006). This requires extra technical assumptions about the extreme value parameters of X, whose necessity for practical purposes is unclear.

Lifting these restrictions is the second motivation for this paper. We first treat the case
of an intermediate, in-sample level τ_n ↑ 1 by examining the asymptotic behavior of two dif-
ferent classes of estimators: the empirical version and a semiparametric estimator based on
an asymptotic equivalent of ES(τ) − q(τ) as τ ↑ 1. In particular, we obtain the convergence
of the centered and renormalized empirical Expected Shortfall to a stable distribution when
the right tail of X has an infinite second moment. We then get the asymptotic distributions
of extrapolated versions of these Expected Shortfall estimators using the moment estimators
of Dekkers et al. (1989); a useful tool for that purpose, which is of independent interest, is
a new uniform second-order inequality on the log-quantile function of X that holds under
no restrictions whatsoever on the extreme value parameters of X. In particular, we obtain
in passing the asymptotic normality of the moment extreme value index estimator under
no restriction on the second-order extreme value parameter, contrary to the classical result
of Dekkers et al. (1989) which is stated in the reference monograph by de Haan and Fer-
reira (2006, Theorem 3.5.4 on p.104), see Lemma A.10 in the Appendix. We find that the
quality of the asymptotic approximations to the finite-sample distributions of the extrapo-
lated Expected Shortfall estimators can be disappointing, which motivates our construction
of corrected asymptotic confidence intervals by building upon a fine understanding of the
asymptotic behavior of the estimators and the uncertainty introduced by plugging parame-
ter estimates into asymptotic variances and correlations. Our intervals are found to perform
reasonably well.

This article is organized as follows. We introduce our extreme value framework in Section 2. The asymptotic properties of our extreme Expected Shortfall estimators are examined in Section 3. Inference procedures are considered in Section 4. A finite-sample simulation study and two real data analyses are reported in Section 5. The Appendix contains all necessary mathematical proofs, further technical details especially about the construction of our inference procedures, and additional finite-sample results.

## 2 Statistical framework and motivation

Let $F: x \mapsto \mathbb{P}(X \le x)$ denote the distribution function of the random variable of interest $X$. The associated quantile function is $q: \tau \mapsto \inf\{x \in \mathbb{R} | F(x) \ge \tau\}$, and the tail quantile function is $U: t \mapsto q(1 - 1/t)$, for $t > 1$. We require the following second-order (extended) regular variation condition on $U$, which imposes that the right tail of the distribution of $X$ can be approximated by a Generalized Pareto distribution tail at a known rate:

$C_2(\gamma, a, \rho, A)$ There are $\gamma \in \mathbb{R}$, a scale function $a(\cdot) > 0$, a second-order parameter $\rho \le 0$ and an auxiliary function $A(\cdot)$ having constant sign and converging to 0 at infinity such that

$$ \forall x > 0, \lim_{t \to \infty} \frac{1}{A(t)} \left( \frac{U(tx) - U(t)}{a(t)} - \int_{1}^{x} s^{\gamma-1} ds \right) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds. $$

This is a standard condition in extreme value theory, which plays a crucial role in obtaining refined results on tail estimation, such as central limit theorems and bias reduction procedures, as pointed out in the discussion pp.302-303 in Fraga Alves et al. (2007). On the theoretical side, de Haan and Resnick (1996) explores its links with so-called von Mises conditions and its implications on convergence rates of linearly normalized sample maxima to their natural Generalized Extreme Value limits in several metrics. This kind of second-order condition cannot be avoided if no precise model structure is assumed on $F$ and a semiparametric procedure to recover the right tail behavior of $X$ is sought: see, for example, Sections 2.3 and 2.4 and Chapter 3 in de Haan and Ferreira (2006), Section 5.6 in Beirlant et al. (2004), and Chapter 2 in Falk et al. (2011) from Section 2.2 onwards. It should, however, be noted that its verification in practice is notoriously challenging, as no formal tests or diagnostic checks currently exist to assess its validity in the case of a general Max-Domain of Attraction (MDA). Estimators for second-order parameters have been proposed in Gomes and Martins (2002), Fraga Alves et al. (2003) and Caeiro et al. (2005) to assess the strength of the second-order behavior only in the heavy-tailed case, with later work by Cai et al. (2013) providing consistent second-order parameter estimators in the setting $\gamma \in (-\infty, 1/2)$. Let us also highlight that several simple sufficient criteria have been recently developed in Stupfler and Usseglio-Carleve (2025)
to verify second-order regular variation in the light-tailed case given a parametric statistical model. With these criteria one may show that the second-order condition is satisfied by most, if not all, of the popular light-tailed distributions, such as the Gaussian, log-normal, Generalized Gamma, Variance-Gamma and Gamma-Gompertz distributions. Table 1 gives a non-exhaustive list of distributions satisfying this mild assumption, with corresponding values of $\gamma$, $a(\cdot)$, $\rho$ and $A(\cdot)$, found by applying Lemmas A.1–A.3 in the Appendix.

It follows from assumption $C_2(\gamma, a, \rho, A)$ that for any $x > 0$, $U(tx) \approx U(t) + a(t) \int_1^x s^{\gamma-1}ds$ as $t \to \infty$. To put it differently, the distribution of $X-u$ given that $X > u$ is, as $u$ tends to $U(\infty) = q(1) \le \infty$, approximately Generalized Pareto with scale parameter $\sigma(u) = a(1/(1-F(u)))$, that in general depends on $u$ in a non-trivial way, and shape parameter $\gamma$, see Theorem 1.1.6.4 on p.10 in de Haan and Ferreira (2006). This implies that extreme quantiles of $X$ can be recovered from so-called intermediate quantiles (colloquially speaking, “extreme, but not too much”) given estimators of the scale function $a(\cdot)$ and shape parameter $\gamma$ of the Generalized Pareto model. A popular solution is the pair of moment estimators developed by Dekkers et al. (1989), which we introduce now. Let $X_1, \dots, X_n$ have the same distribution as $X$, and $X_{1:n} \le X_{2:n} \le \dots \le X_{n:n}$ be the associated order statistics, and assume that $U(\infty) = q(1) > 0$. Let also $(\tau_n)$ be an intermediate sequence of probabilities, that is, $\tau_n \uparrow 1$ and $n(1-\tau_n) \to \infty$. Set

$$M_n^{(1)} = \frac{1}{\lfloor n(1-\tau_n) \rfloor} \sum_{i=1}^{\lfloor n(1-\tau_n) \rfloor} \log \frac{X_{n-i+1:n}}{X_{\lfloor n\tau_n \rfloor:n}} \quad \text{and} \quad M_n^{(2)} = \frac{1}{\lfloor n(1-\tau_n) \rfloor} \sum_{i=1}^{\lfloor n(1-\tau_n) \rfloor} \log^2 \frac{X_{n-i+1:n}}{X_{\lfloor n\tau_n \rfloor:n}}.$$

Then the moment estimators of the scale parameter $a(t)$ at $t = (1 - \tau_n)^{-1}$ and of the shape parameter $\gamma$ are respectively

$$\hat{a}_n^{\text{Mom}}((1 - \tau_n)^{-1}) = X_{\lfloor n\tau_n \rfloor:n} M_n^{(1)}(1 - \hat{\gamma}_{n,-}^{\text{Mom}}) \\ \text{and } \hat{\gamma}_n^{\text{Mom}} = M_n^{(1)} + \hat{\gamma}_{n,-}^{\text{Mom}}, \text{ with } \hat{\gamma}_{n,-}^{\text{Mom}} = 1 - \frac{1}{2} \left( 1 - \frac{(M_n^{(1)})^2}{M_n^{(2)}} \right)^{-1}.$$

A semiparametric estimator of an extreme quantile $q(\tau'_n)$, where $(\tau'_n)$ is another sequence of probabilities such that $(1 - \tau'_n)/(1 - \tau_n) \to 0$, follows as

$$\hat{q}_n^*(\tau'_n) = X_{\lfloor n\tau_n \rfloor:n} + \hat{a}_n^{\text{Mom}}((1 - \tau_n)^{-1}) \int_1^{\{(1-\tau'_n)/(1-\tau_n)\}^{-1}} s^{\hat{\gamma}_n^{\text{Mom}}-1} ds.$$

See Theorem 4.3.1 on p.134 in de Haan and Ferreira (2006) and a discussion of the asymptotic properties of this estimator in Section 4.3.2 on p.140 therein.

When attempting to transpose that construction to the estimation of the Expected Short-
fall at extreme levels, it is natural to write first

$$ \frac{\mathrm{ES}(\tau) - q(\tau)}{a((1-\tau)^{-1})} = \int_{1}^{\infty} \frac{U((1-\tau)^{-1}x) - U((1-\tau)^{-1})}{a((1-\tau)^{-1})} \frac{dx}{x^2} \\ \approx \int_{1}^{\infty} \left( \int_{1}^{x} s^{\gamma-1}ds \right) \frac{dx}{x^2} = \frac{1}{1-\gamma} \quad \text{as} \quad \tau \uparrow 1 \quad (1) $$

when $\gamma < 1$. Besides, by Theorem 2.3.3 on p.44 in de Haan and Ferreira (2006), the function $a(\cdot)$ is regularly varying with index $\gamma$, so that

$$ \frac{\mathrm{ES}(\tau'_n) - q(\tau'_n)}{a((1-\tau_n)^{-1})} \approx \frac{1}{1-\gamma} \times \frac{a((1-\tau'_n)^{-1})}{a((1-\tau_n)^{-1})} \approx \frac{1}{1-\gamma} \left(\frac{1-\tau'_n}{1-\tau_n}\right)^{-\gamma}, \\ \frac{\mathrm{ES}(\tau_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} \approx \frac{1}{1-\gamma} \quad \text{and} \quad \frac{q(\tau'_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} \approx \int_1^{\{(1-\tau'_n)/(1-\tau_n)\}^{-1}} s^{\gamma-1}ds. $$

Combining these three approximations provides

$$ \frac{\mathrm{ES}(\tau'_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} \approx \frac{1}{1-\gamma} \int_1^{\{(1-\tau'_n)/(1-\tau_n)\}^{-1}} s^{\gamma-1}ds. \quad (2) $$

Equations (1) and (2) suggest the following class of extrapolated estimators of $\mathrm{ES}(\tau'_n)$:

$$ \overline{\mathrm{ES}}_n(\tau_n) + \hat{a}_n^{\mathrm{Mom}}((1-\tau_n)^{-1}) \times \frac{1}{1-\hat{\gamma}_n^{\mathrm{Mom}}} \int_1^{(1-\tau_n)/(1-\tau'_n)} s^{\hat{\gamma}_n^{\mathrm{Mom}}-1}ds, $$

$$ \text{with } \overline{\mathrm{ES}}_n(\tau_n) = \widehat{\mathrm{ES}}_n(\tau_n) = \frac{1}{\lfloor n(1-\tau_n) \rfloor} \sum_{i=1}^{\lfloor n(1-\tau_n) \rfloor} X_{n-i+1:n} \quad (3) $$

$$ \text{or } \overline{\mathrm{ES}}_n(\tau_n) = \widehat{\mathrm{ES}}_n(\tau_n) = X_{\lfloor n\tau_n \rfloor:n} + \hat{a}_n^{\mathrm{Mom}}((1-\tau_n)^{-1}) \frac{1}{1-\hat{\gamma}_n^{\mathrm{Mom}}}. \quad (4) $$

There are, in our view, two main difficulties related to the asymptotic behavior of these extrapolated estimators. The first one is that the asymptotic theory about $\widehat{\mathrm{ES}}_n(\tau_n)$ has so far been restricted to heavy-tailed distributions and, within this class, to the range $\gamma \in (0, 1/2)$, which may be a strong limitation depending on the application at hand; as we shall illustrate on several samples of real data, there are environmental and financial applications where the validity of the heavy right tail assumption is dubious, and when it is satisfied, the underlying distribution may have an infinite variance. This calls for the development of asymptotic theory for the extrapolated version of $\widehat{\mathrm{ES}}_n(\tau_n)$ in the general MDA, where $\gamma < 1$ can be negative, which would naturally require joint asymptotic theory between $\widehat{\mathrm{ES}}_n(\tau_n)$ and, say, the moment estimators of $a((1-\tau_n)^{-1})$ and $\gamma$. The second difficulty, which is actually linked to this point, is that while $U$ satisfies condition $C_2(\gamma, a, \rho, A)$, the currently available asymptotic theory for these moment estimators, and the extrapolated estimators derived from their use, requires the somewhat unnatural assumption $\gamma \neq \rho$ so that a similar condition holds on $\log U(\cdot)$, see for instance the discussions on pp.103 and 140 in de Haan and Ferreira (2006) and Theorem 3.5.4
on p.104 therein. We shall provide solutions to these two difficulties in the next section, by studying the asymptotic distribution of $\widehat{\mathrm{ES}}_n(\tau_n)$ in the general setting $\gamma < 1$, and by obtaining directly a slightly weaker, but sufficient, uniform second-order inequality on $\log U(\cdot)$ under no restriction on the pair $(\gamma, \rho)$.

### 3 Theory of extreme Expected Shortfall estimation

Let $X_1, \dots, X_n$ be independent and identically distributed copies of $X$, whose tail quantile function $U$ satisfies condition $C_2(\gamma, a, \rho, A)$. The objective is to estimate an Expected Shortfall $\mathrm{ES}(\tau)$ of $X$ above an extreme level $q(\tau)$, where $\tau = \tau_n \to 1$ as $n \to \infty$. As is typical in extreme value statistics, we start by the case of an intermediate level $\tau_n$. In such a case the target quantity can be estimated nonparametrically. Intermediate quantile estimators are then extrapolated to properly extreme levels $\tau'_n$, typically satisfying $n(1 - \tau'_n) = O(1)$, using a semiparametric extrapolation formula warranted by condition $C_2(\gamma, a, \rho, A)$ along with estimators of the scale function $a(\cdot)$ and the shape parameter $\gamma$. An important tool will be the following asymptotic expansion of $\mathrm{ES}(\tau)$ at high levels $\tau$; this shall especially be useful in controlling the bias of extreme value estimators of the Expected Shortfall. Throughout we let, to be consistent with Section 3.5 in de Haan and Ferreira (2006), $x_+ = \max(x, 0)$ and $x_- = \min(x, 0)$ for any $x \in \mathbb{R}$ so that $x = x_+ + x_-$ and $|x| = x_+ - x_-$; with this notation, $x_-$ is the opposite of what is traditionally considered the negative part of $x$ in real analysis.

**Proposition 1 (Second-order expansion of $\mathrm{ES}(\tau)$).** Assume that condition $C_2(\gamma, a, \rho, A)$ holds.

(i) Then we have, for any $\delta, \varepsilon > 0$ sufficiently small, that for $t$ large enough and all $x > 1$,

$$ \left| \frac{U(tx) - U(t)}{a(t)} - \int_{1}^{x} s^{\gamma-1} \mathrm{d}s - A(t) \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} \mathrm{d}u \right) \mathrm{d}s \right| \le \varepsilon |A(t)| x^{\gamma_{+} + \delta}. $$

(ii) If moreover $\gamma < 1$, we have, as $\tau \uparrow 1$,

$$ \begin{aligned} \frac{\mathrm{ES}(\tau) - q(\tau)}{a((1 - \tau)^{-1})} &= \frac{\mathrm{CTE}(\tau) - q(\tau)}{a((1 - \tau)^{-1})} + o(|A((1 - \tau)^{-1})|) \\ &= \frac{1}{1 - \gamma} + A((1 - \tau)^{-1}) \left( \frac{1}{(1 - \gamma)(1 - \gamma - \rho)} + o(1) \right). \end{aligned} $$

(iii) If moreover $\gamma \in (0, 1)$, then, as $\tau \uparrow 1$,

$$ \begin{aligned} \frac{\mathrm{ES}(\tau) - q(\tau)}{q(\tau)} &= \frac{\gamma}{1 - \gamma} + \left( \frac{a((1 - \tau)^{-1})}{q(\tau)} - \gamma \right) \left( \frac{1}{1 - \gamma} + o(1) \right) \\ &\quad + A((1 - \tau)^{-1}) \left( \frac{\gamma}{(1 - \gamma)(1 - \gamma - \rho)} + o(1) \right). \end{aligned} $$
Proposition 1(i) is a convenient complement to the second-order inequality in Theorem 2.3.6 on p.46 in de Haan and Ferreira (2006) with a weaker upper bound that is sufficient for our purposes. Proposition 1(ii) is an extension of Lemma 3(ii) in El Methni and Stupfler (2017), which does not apply under condition $C_2(\gamma, a, \rho, A)$ for $\gamma \le 0$, to the general extreme value framework we consider.

## 3.1 At intermediate levels

Let $\tau \in (0, 1)$ and recall the notation $\hat{q}_n(t) = X_{\lfloor nt \rfloor:n}$ for the empirical quantile function of the sample $(X_1, \dots, X_n)$. When there are no ties in the sample (this is true with probability 1 if $F$ is continuous) and if moreover $n(1 - \tau)$ is an integer, the empirical estimator of $\mathrm{ES}(\tau)$ satisfies

$$ \frac{1}{1-\tau} \int_{\tau}^{1} \hat{q}_{n}(t)dt = \frac{1}{\lfloor n(1-\tau) \rfloor} \sum_{i=1}^{\lfloor n(1-\tau) \rfloor} X_{n-i+1:n} = \frac{\sum_{i=1}^{n} X_i \mathbb{1}_{\{X_i > X_{\lfloor n\tau \rfloor:n}\}}}{\sum_{i=1}^{n} \mathbb{1}_{\{X_i > X_{\lfloor n\tau \rfloor:n}\}}}. $$

The latter estimator is the obvious empirical counterpart of the Conditional Tail Expectation $\mathrm{CTE}(\tau) = \mathbb{E}(X | X > q(\tau))$. It turns out that in general, in the case when $\tau = \tau_n \to 1$ and $n(1 - \tau_n) \to \infty$, as $n \to \infty$, corresponding to an Expected Shortfall above an extreme but in-sample level, the three estimators in the above equation have the same asymptotic behavior (see Proposition A.1 for a rigorous statement), and we therefore work throughout with the estimator $\widehat{\mathrm{ES}}_n(\tau_n)$ defined in (3). Our first result in this section shows that at such an intermediate level $\tau_n$, the estimator $\widehat{\mathrm{ES}}_n(\tau_n)$ is consistent, and provides its asymptotic behavior for any $\gamma < 1$. This is not a simple corollary of existing results, as Gaussian approximations to the tail empirical quantile process from Theorem 2.4.2 on p.51 in de Haan and Ferreira (2006) are not sufficient in order to handle the case $1/2 \le \gamma < 1$. To the best of our knowledge, no result on $\widehat{\mathrm{ES}}_n(\tau_n)$ has been shown up to now in this challenging case.

**Theorem 1** (Weak convergence of the empirical intermediate Expected Shortfall). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $\gamma < 1$. Let $\tau_n \uparrow 1$ be such that $n(1 - \tau_n) \to \infty$.

* Assume $\gamma < 1/2$. Then, if $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1}) = O(1)$, one has

$$ \sqrt{n(1 - \tau_n)} \frac{\widehat{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1 - \tau_n)^{-1})} \xrightarrow{d} \mathcal{N} \left( 0, \frac{2}{(1 - \gamma)(1 - 2\gamma)} \right). $$

* If $\gamma = 1/2$, we have, provided $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1})/\sqrt{\log(n(1 - \tau_n))} = O(1)$,

$$ \frac{\sqrt{n(1 - \tau_n)}}{\sqrt{\log(n(1 - \tau_n))}} \frac{\widehat{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1 - \tau_n)^{-1})} \xrightarrow{d} \mathcal{N}(0, 4). $$
• Assume $1/2 < \gamma < 1$. Then, if $(n(1 - \tau_n))^{1-\gamma} A((1 - \tau_n)^{-1}) = O(1)$, one has

$$ (n(1 - \tau_n))^{1-\gamma} \frac{\widehat{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1 - \tau_n)^{-1})} \xrightarrow{d} \frac{1}{\gamma} \left\{ -\frac{\Gamma(2 - 1/\gamma)}{1/\gamma - 1} \cos\left(\frac{\pi}{2\gamma}\right) \right\}^{\gamma} Z_{1/\gamma}, $$

where, for any $\alpha \in (1, 2)$, $Z_\alpha$ has a unit right-skewed stable distribution with Fourier transform

$$ \mathbb{E}(e^{itZ_{\alpha}}) = \exp\left(-|t|^{\alpha}\left\{1 - i \tan\left(\frac{\pi\alpha}{2}\right) \mathrm{sign}(t)\right\}\right). $$

It follows from Theorem 1 that the rate of convergence of $\widehat{\mathrm{ES}}_n(\tau_n)$ varies with the shape parameter $\gamma$: this rate is $1/(n(1 - \tau_n))^{\min(1/2, 1-\gamma)}$ for $\gamma \in (-\infty, 1/2) \cup (1/2, 1)$, with a phase transition at $\gamma = 1/2$ for which the rate of convergence is $\sqrt{\log(n(1 - \tau_n))}/\sqrt{n(1 - \tau_n)}$. The limiting distribution, meanwhile, is Gaussian for $\gamma \le 1/2$, and is a non-Gaussian stable distribution for $\gamma \in (1/2, 1)$. The intuition behind this finding is that, since $\widehat{\mathrm{ES}}_n(\tau_n)$ is a mean of high order statistics from $X$, its limiting behavior should be Gaussian when $X$ has a finite second moment, and non-Gaussian stable otherwise. The proof formalizes this intuition by connecting the stochastic behavior of $\widehat{\mathrm{ES}}_n(\tau_n)$ to that of a sample mean of i.i.d. Pareto random variables having extreme value index $\gamma$ using the R´enyi representation. A fundamental tool in making this connection is Proposition 1(i), whose use in the proof makes a bias term proportional to $A((1 - \tau_n)^{-1})$ appear; interestingly, the corresponding bias condition in Theorem 1, which is $(n(1 - \tau_n))^{\min(1/2, 1-\gamma)} A((1 - \tau_n)^{-1}) = O(1)$ for $\gamma \in (-\infty, 1/2) \cup (1/2, 1)$ and $\sqrt{n(1 - \tau_n)} A((1 - \tau_n)^{-1})/\sqrt{\log(n(1 - \tau_n))} = O(1)$ when $\gamma = 1/2$, gets weaker as $\gamma$ increases.

**Remark 1** (Compatibility with existing results in the Fr´echet MDA with finite second moment). When $0 < \gamma < 1$, $a(t)/U(t) \to \gamma$ as $t \to \infty$ and $\mathrm{ES}(\tau)/q(\tau) \to 1/(1 - \gamma)$ as $\tau \uparrow 1$, see Proposition 1(iii). It follows from Theorem 1 that when $0 < \gamma < 1/2$ and $\sqrt{n(1 - \tau_n)} A((1 - \tau_n)^{-1}) = O(1)$,

$$ \sqrt{n(1 - \tau_n)} \left( \frac{\widehat{\mathrm{ES}}_n(\tau_n)}{\mathrm{ES}(\tau_n)} - 1 \right) \xrightarrow{d} \mathcal{N} \left( 0, \frac{2\gamma^2(1 - \gamma)}{1 - 2\gamma} \right). $$

This is Theorem 2 in El Methni and Stupfler (2017) for the distortion function being the identity function; see also Theorem 1 in El Methni et al. (2014) in the regression setting.

An alternative class of estimators of $\mathrm{ES}(\tau_n)$, motivated by Equation (4), is the quantile-based semiparametric version

$$ \widehat{\mathrm{ES}}_n(\tau_n) = X_{[n\tau_n]:n} + \hat{a}_n((1 - \tau_n)^{-1}) \frac{1}{1 - \hat{\gamma}_n} $$

where $\hat{a}_n((1 - \tau_n)^{-1})$ is a relatively consistent estimator of $a((1 - \tau_n)^{-1})$, in the sense that $\hat{a}_n((1 - \tau_n)^{-1})/a((1 - \tau_n)^{-1})$ should converge in probability to 1, and $\hat{\gamma}_n$ is a consistent
estimator of the shape parameter $\gamma$. Examples include the semiparametric class of moment estimators (Dekkers et al., 1989), Generalized Pareto maximum likelihood estimators (Smith, 1987; Drees et al., 2004) and probability-weighted moment estimators (Hosking et al., 1985; Diebolt et al., 2007), all of which naturally relate to the Peaks-over-Threshold approach to extreme value theory. An alternative method is the Block Maxima approach, in which it has been shown that the Generalized Extreme Value maximum likelihood estimators satisfy the above convergence as well (Dombry and Ferreira, 2019, Theorem 2.2). A comparison of the Peaks-over-Threshold and Block Maxima approaches to extreme value analysis is provided in Bücher and Zhou (2021). These estimators, along with the intermediate empirical quantile $X_{[n\tau_n]:n}$, typically converge at the rate $1/\sqrt{n(1-\tau_n)}$ to a joint Gaussian limit, see Chapter 3 in de Haan and Ferreira (2006). The next result discusses the asymptotic properties of $\tilde{\text{ES}}_n(\tau_n)$ depending on those of the random vector $(X_{[n\tau_n]:n}, \hat{a}_n((1-\tau_n)^{-1}), \hat{\gamma}_n)$.

**Theorem 2** (Weak convergence of $\tilde{\text{ES}}_n(\tau_n)$ at intermediate levels). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $\gamma < 1$. Let $\tau_n \uparrow 1$ be such that $n(1-\tau_n) \to \infty$ and $\sqrt{n(1-\tau_n)}A((1-\tau_n)^{-1}) \to \lambda \in \mathbb{R}$. Assume that

$$ \sqrt{n(1-\tau_n)} \left( \frac{X_{[n\tau_n]:n} - q(\tau_n)}{a((1-\tau_n)^{-1})}, \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1, \hat{\gamma}_n - \gamma \right) \xrightarrow{d} (N_{\text{loc}}, N_{\text{scale}}, N_{\text{shape}}) $$

where the random vector $(N_{\text{loc}}, N_{\text{scale}}, N_{\text{shape}})$ has a nondegenerate distribution. Then

$$ \sqrt{n(1-\tau_n)} \left( \frac{\tilde{\text{ES}}_n(\tau_n) - \text{ES}(\tau_n)}{a((1-\tau_n)^{-1})}, \frac{X_{[n\tau_n]:n} - q(\tau_n)}{a((1-\tau_n)^{-1})}, \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1, \hat{\gamma}_n - \gamma \right) \\ \xrightarrow{d} \left( N_{\text{loc}} + \frac{1}{1-\gamma} N_{\text{scale}} + \frac{1}{(1-\gamma)^2} N_{\text{shape}} - \frac{\lambda}{(1-\gamma)(1-\gamma-\rho)}, N_{\text{loc}}, N_{\text{scale}}, N_{\text{shape}} \right). $$

As a matter of fact, Theorem 2 is a high-level result that does not require independent data; all that is required for its use is to be able to show the convergence of semiparametric extreme value estimators, which is known to hold under, for example, mixing assumptions on the data for the moment estimators and the Generalized Pareto maximum likelihood estimators, as can be seen from Theorem 6.2 in Drees (2003) in view of the discussion on p.625 therein. More generally, since the moment estimators of Dekkers et al. (1989) are smooth, Hadamard differentiable functionals of the tail empirical process $(X_{n-|ks|:n})_{0<s\le 1}$, the aforementioned results will hold as soon as a Gaussian approximation to the tail empirical process is available.

Inspecting the proof of Theorem 1(i) reveals that this result likewise holds under mixing settings (with an enlarged asymptotic variance), because its main ingredient is a Gaussian approximation to the tail empirical process. However, the proofs of Theorem 1(ii)-(iii) crucially rely on the Rényi representation of order statistics from an independent unit exponential sample. As pointed out in the introduction of Hsing (1991), which analyzes the asymptotic
properties of the Hill estimator Hill (1975) under α-mixing, the Rényi representation unfortunately does not work when dependence is present. The arguably much easier problem of extending the Rényi representation for independent but non identically distributed data is already highly involved, as explained in Nagaraja (2006).

Theorems 1 and 2 will be crucial components of the proofs of our asymptotic results about the extrapolated versions of $\widehat{\mathrm{ES}}_n$ and $\widehat{\mathrm{ES}}_n$ which, unlike $\widehat{\mathrm{ES}}_n(\tau_n)$ and $\widehat{\mathrm{ES}}_n(\tau_n)$, target the Expected Shortfall calculated above an extreme quantile level that can converge to 1 at any rate. This is the focus of our next section.

## 3.2 At extreme levels

Let $(\tau'_n)$ be another sequence of probabilities such that $(1-\tau'_n)/(1-\tau_n) \to 0$. Typical examples include $\tau'_n = 1-1/n$, with the associated extreme quantile $q(\tau'_n) \equiv q(1-1/n)$ having the order of magnitude of the sample maximum $X_{n:n}$ in the data. Equation (2) suggests the following, broad class of extrapolated estimators of $\mathrm{ES}(\tau'_n)$:

$$ \overline{\mathrm{ES}}_n^*(\tau'_n) = \overline{\mathrm{ES}}_n(\tau_n) + \hat{a}_n((1-\tau_n)^{-1}) \times \frac{1}{1-\hat{\gamma}_n} \int_1^{(1-\tau_n)/(1-\tau'_n)} s^{\hat{\gamma}_n-1} ds. $$

Here $\overline{\mathrm{ES}}_n(\tau_n)$ is a relatively consistent estimator of $\mathrm{ES}(\tau_n)$, such as $\widehat{\mathrm{ES}}_n(\tau_n)$ or $\widehat{\mathrm{ES}}_n(\tau_n)$, $\hat{a}_n((1-\tau_n)^{-1})$ is a relatively consistent estimator of $a((1-\tau_n)^{-1})$ and $\hat{\gamma}_n$ is a consistent estimator of $\gamma$. The objective of the next result in this section is to give a high-level result about the convergence of $\overline{\mathrm{ES}}_n^*(\tau'_n)$, along with its corollary when $\overline{\mathrm{ES}}_n(\tau_n) = \widehat{\mathrm{ES}}_n(\tau_n)$, resulting in an estimator $\overline{\mathrm{ES}}_n^*(\tau'_n) = \widehat{\mathrm{ES}}_n^*(\tau'_n)$, and its corollary when $\overline{\mathrm{ES}}_n(\tau_n) = \widehat{\mathrm{ES}}_n(\tau_n)$, resulting in an estimator $\overline{\mathrm{ES}}_n^*(\tau'_n) = \widehat{\mathrm{ES}}_n^*(\tau'_n)$.

**Theorem 3** (Asymptotic expansion of the extrapolated estimator $\overline{\mathrm{ES}}_n^*(\tau'_n)$). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $\gamma < 1$, and $\rho < 0$ if $\gamma \ge 0$. Let $\tau_n, \tau'_n \uparrow 1$ be such that $n(1-\tau_n) \to \infty$ and $1/d_n := (1-\tau'_n)/(1-\tau_n) \to 0$. Assume that

$$ \sqrt{n(1-\tau_n)} \left( \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) = O_P(1) \quad \text{and} \quad \sqrt{n(1-\tau_n)}(\hat{\gamma}_n - \gamma) = O_P(1). $$

Suppose finally that

* $\sqrt{n(1-\tau_n)}A((1-\tau_n)^{-1}) \to \lambda \in \mathbb{R}$,

* $\log(d_n)/\sqrt{n(1-\tau_n)} \to 0$.
Then

$$
\frac{\overline{\mathrm{ES}}_n^*(\tau_n') - \mathrm{ES}(\tau_n')}{a((1-\tau_n)^{-1})} \stackrel{\mathrm{d}}{=} \frac{\overline{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{\gamma_-}{1-\gamma_-} \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s \left( \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) \\
+ \left( \frac{1}{1-\gamma} - \frac{\gamma_-}{(1-\gamma_-)^2} \right) \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s (\hat{\gamma}_n - \gamma) \\
- \frac{\int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s}{\sqrt{n(1-\tau_n)}} \left( \lambda \frac{\gamma_- (1-2\gamma_- - \rho)}{(1-\gamma_-)(1-\gamma_- - \rho)(\gamma_- + \rho)} + \mathrm{o}_P(1) \right).
$$

An inspection of the proof reveals that the assumption that $\rho < 0$ when $\gamma \ge 0$ is necessary in order to control the bias term incurred in the approximation $a((1 - \tau_n')^{-1})/a((1 - \tau_n)^{-1}) \approx ((1 - \tau_n')/(1 - \tau_n))^{-\gamma}$ made when constructing the extrapolated estimator $\overline{\mathrm{ES}}_n^*(\tau_n')$. Condition $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1}) \to \lambda \in \mathbb{R}$ is a condition required due to having to control the bias incurred by using the semiparametric extrapolation formula (2). Condition $\log((1 - \tau_n)/(1 - \tau_n'))/\sqrt{n(1 - \tau_n)} \to 0$, meanwhile, is standard, see for example Theorem 4.3.1 on p.134 in de Haan and Ferreira (2006) in the context of extreme quantile estimation.

When using $\overline{\mathrm{ES}}_n(\tau_n) = \widehat{\mathrm{ES}}_n(\tau_n)$ for $\gamma < 1/2$, or $\overline{\mathrm{ES}}_n(\tau_n) = \widetilde{\mathrm{ES}}_n(\tau_n)$ when $\gamma < 1$, it follows from Theorems 1 and 2 that $(\overline{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n))/a((1-\tau_n)^{-1}) = \mathbb{O}_P(1/\sqrt{n(1-\tau_n)})$. We may then draw the following useful corollary.

**Corollary 1** (Weak convergence of the extrapolated estimator $\overline{\mathrm{ES}}_n^*(\tau_n')$). With the notation and under the conditions of Theorem 3, if actually

$$
\sqrt{n(1 - \tau_n)} \left( \frac{\overline{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1 - \tau_n)^{-1})}, \frac{\hat{a}_n((1 - \tau_n)^{-1})}{a((1 - \tau_n)^{-1})} - 1, \hat{\gamma}_n - \gamma \right) \xrightarrow{d} (Z_{\mathrm{loc}}, Z_{\mathrm{scale}}, Z_{\mathrm{shape}})
$$

where the trivariate random vector $(Z_{\mathrm{loc}}, Z_{\mathrm{scale}}, Z_{\mathrm{shape}})$ has a nondegenerate distribution, then

$$
\sqrt{n(1 - \tau_n)} \frac{\overline{\mathrm{ES}}_n^*(\tau_n') - \mathrm{ES}(\tau_n')}{a((1 - \tau_n)^{-1}) \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s} \\
\xrightarrow{d} \gamma_-^2 Z_{\mathrm{loc}} - \frac{\gamma_-}{1 - \gamma_-} Z_{\mathrm{scale}} + \left( \frac{1}{1 - \gamma} - \frac{\gamma_-}{(1 - \gamma_-)^2} \right) Z_{\mathrm{shape}} - \lambda \frac{\gamma_- (1 - 2\gamma_- - \rho)}{(1 - \gamma_-)(1 - \gamma_- - \rho)(\gamma_- + \rho)}.
$$

The following corollary will be convenient to deal with the specific case $\gamma \ge 0$.

**Corollary 2** (Weak convergence of the extrapolated estimator $\overline{\mathrm{ES}}_n^*(\tau_n')$, light or heavy tails). With the notation and under the conditions of Theorem 3 with $\gamma \ge 0$, if actually

$$
\sqrt{n(1 - \tau_n)} \frac{\overline{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1 - \tau_n)^{-1})} = \mathbb{O}_P(1), \quad \sqrt{n(1 - \tau_n)} \left( \frac{\hat{a}_n((1 - \tau_n)^{-1})}{a((1 - \tau_n)^{-1})} - 1 \right) = \mathbb{O}_P(1)
$$

and $\sqrt{n(1 - \tau_n)}(\hat{\gamma}_n - \gamma) \xrightarrow{d} Z$ where $Z$ has a nondegenerate distribution, then

$$
\sqrt{n(1 - \tau_n)} \frac{\overline{\mathrm{ES}}_n^*(\tau_n') - \mathrm{ES}(\tau_n')}{a((1 - \tau_n)^{-1}) \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s} \xrightarrow{d} \frac{1}{1 - \gamma} Z.
$$
Theorem 3 reveals that the interplay between the estimation of ES($\tau_n$), at the intermediate level, and the estimation of the scale and shape parameters of the approximating Generalized Pareto distribution, is potentially more complex than in the estimation of extreme quantiles because, unlike $X_{[n\tau_n]:n}$, $\widehat{\mathrm{ES}}_n(\tau_n)$ will converge to ES($\tau_n$) at a rate slower than $1/\sqrt{n(1-\tau_n)}$ when $\gamma \in [1/2, 1)$. We make this explicit when $\hat{a}_n((1-\tau_n)^{-1}) = \hat{a}_n^{\mathrm{Mom}}((1-\tau_n)^{-1})$ and $\hat{\gamma}_n = \hat{\gamma}_n^{\mathrm{Mom}}$. The following second-order inequality controlling the asymptotic behavior of $\log U$ under condition $C_2(\gamma, a, \rho, A)$ is crucial for that purpose.

**Proposition 2** (On regular variation properties of $\log U$). Assume that $0 < U(\infty) = q(1) \le \infty$, and that condition $C_2(\gamma, a, \rho, A)$ holds. Then we have, for any $\delta, \varepsilon > 0$ sufficiently small, that for $t$ large enough and all $x > 1$,

$$ \left| \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} - \int_1^x s^{\gamma-1} \mathrm{d}s + \left( \frac{a(t)}{U(t)} - \gamma_+ \right) \int_1^x s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} \mathrm{d}u \right) \mathrm{d}s - A(t) x^{-\gamma_+} \int_1^x s^{\gamma-1} \left( \int_1^s u^{\rho-1} \mathrm{d}u \right) \mathrm{d}s \right| \le \varepsilon \left( \left| \frac{a(t)}{U(t)} - \gamma_+ \right| + |A(t)| \right) x^\delta. $$

Proposition 2 is valid regardless of the values of $\gamma$ and $\rho$ and features the functions $a$ and $A$ themselves rather than modifications of these functions, unlike the inequality stated at the top of p.104 in de Haan and Ferreira (2006). It is a weaker inequality since it is restricted to the range $x > 1$ and its right-hand side does not feature the multiplicative term $x^{\gamma-+\rho}$, but it shall be sufficient for our purpose of obtaining the asymptotic distribution of extrapolated Expected Shortfall estimators based on the moment estimators $\hat{a}_n^{\mathrm{Mom}}((1-\tau_n)^{-1})$ and $\hat{\gamma}_n^{\mathrm{Mom}}$. It is shown in Appendix B that Proposition 2 can indeed be seen as an extension of the second-order condition typically used in the literature.

We may now provide the asymptotic distribution of the estimator $\widehat{\mathrm{ES}}_n^*(\tau_n')$. Define

$$ B_1(\gamma, \rho) = \frac{1 + \gamma + \gamma^2 - \rho\gamma^2}{(1-\gamma)(1+\gamma)^2(1-\rho)^2} \mathbb{1}_{\{0 \le \gamma < 1\}} + \frac{\rho(1-\gamma)}{(1-\gamma-\rho)(1-2\gamma-\rho)(\gamma+\rho)} \mathbb{1}_{\{\gamma < 0\}}, $$

$$ B_2(\gamma) = \frac{\gamma}{(1-\gamma)(1+\gamma)^2} \mathbb{1}_{\{0 \le \gamma < 1\}} - \frac{\gamma(1-2\gamma-2\gamma^2+5\gamma^3)}{(1-\gamma)^3(1-2\gamma)(1-3\gamma)} \mathbb{1}_{\{\gamma < 0\}} $$

$$ \text{and } V(\gamma) = \frac{\gamma^2 + 1}{(1-\gamma)^2} \mathbb{1}_{\{0 \le \gamma < 1\}} + \frac{(1-\gamma)^2(1-3\gamma+4\gamma^2)}{(1-2\gamma)(1-3\gamma)(1-4\gamma)} \mathbb{1}_{\{\gamma < 0\}}. $$

**Theorem 4** (Weak convergence of the moment-based estimator $\widehat{\mathrm{ES}}_n^*(\tau_n')$). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $0 < U(\infty) = q(1) \le \infty$, $\gamma < 1$, and $\rho < 0$ if $\gamma \ge 0$. Let $\tau_n, \tau'_n \uparrow 1$ be such that $n(1-\tau_n) \to \infty$ and $n(1-\tau'_n) = O(1)$. Assume finally that

$$ \bullet \sqrt{n(1-\tau_n)}A((1-\tau_n)^{-1}) \to \lambda \in \mathbb{R}, $$

$$ \bullet \sqrt{n(1-\tau_n)}(a((1-\tau_n)^{-1})/q(\tau_n) - \gamma_+) \to \mu \in \mathbb{R}, $$

$$ \bullet \log((1-\tau_n)/(1-\tau'_n))/\sqrt{n(1-\tau_n)} \to 0. $$
Take $\hat{a}_n((1 - \tau_n)^{-1}) = \hat{a}_n^{\text{Mom}}((1 - \tau_n)^{-1})$ and $\hat{\gamma}_n = \hat{\gamma}_n^{\text{Mom}}$. Then

$$ \sqrt{n(1 - \tau_n)} \frac{\widehat{\text{ES}}_n^*(\tau_n') - \text{ES}(\tau_n')}{a((1 - \tau_n)^{-1}) \int_1^{(1-\tau_n)/(1-\tau_n')} s^{\gamma-1} \log(s) ds} \xrightarrow{d} \mathcal{N}(\lambda B_1(\gamma, \rho) + \mu B_2(\gamma), V(\gamma)). $$

Condition $\sqrt{n(1 - \tau_n)}(a((1 - \tau_n)^{-1})/q(\tau_n) - \gamma_+)$ $\to \mu \in \mathbb{R}$ is, compared to Theorem 3, an extra bias condition which appears due to the use of the moment estimator $\hat{\gamma}_n^{\text{Mom}}$, whose asymptotic normality requires an extrapolation formula on $\log U$ similar in spirit to (2). In view to Proposition 2, the use of this extrapolation formula on $\log U$ makes a bias term proportional to $(a((1 - \tau_n)^{-1})/q(\tau_n) - \gamma_+)$ appear. Note moreover that when $\gamma > 0$ and $n(1 - \tau_n') = O(1)$, which covers for instance the standard case when $\tau_n' = 1 - 1/n$ represents a quantile level in the neighborhood of the maximal observation in the sample, the asymptotic behavior of $\widehat{\text{ES}}_n^*(\tau_n')$ is dominated by that of the extreme value index estimator $\hat{\gamma}_n$. This may not be the case if this condition on $\tau_n'$ is violated when $\gamma \in [1/2, 1)$, because then $\widehat{\text{ES}}_n(\tau_n)$ converges to $\text{ES}(\tau_n)$ at a rate slower than $1/\sqrt{n(1 - \tau_n)}$.

Our final main result examines the asymptotic distribution of $\widehat{\text{ES}}_n^*(\tau_n')$. Define

$$ B_3(\gamma) = \frac{\gamma}{(1 - \gamma)(1 + \gamma)^2} \mathbb{1}_{\{0 \le \gamma < 1\}} - \frac{\gamma(1 - 3\gamma^2)}{(1 - \gamma)(1 - 2\gamma)(1 - 3\gamma)} \mathbb{1}_{\{\gamma < 0\}}. $$

**Theorem 5** (Weak convergence of the class of estimators $\widehat{\text{ES}}_n^*(\tau_n')$). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $0 < U(\infty) = q(1) \le \infty$, $\gamma < 1$, and $\rho < 0$ if $\gamma \ge 0$. Let $\tau_n, \tau_n' \uparrow 1$ be such that $n(1 - \tau_n) \to \infty$ and $(1 - \tau_n')/(1 - \tau_n) \to 0$. Assume finally that

* $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1}) \to \lambda \in \mathbb{R}$,

* $\sqrt{n(1 - \tau_n)}(a((1 - \tau_n)^{-1})/q(\tau_n) - \gamma_+)$ $\to \mu \in \mathbb{R}$,

* $\log((1 - \tau_n)/(1 - \tau_n'))/\sqrt{n(1 - \tau_n)} \to 0$.

Take $\hat{a}_n((1 - \tau_n)^{-1}) = \hat{a}_n^{\text{Mom}}((1 - \tau_n)^{-1})$ and $\hat{\gamma}_n = \hat{\gamma}_n^{\text{Mom}}$. Then

$$ \sqrt{n(1 - \tau_n)} \frac{\widehat{\text{ES}}_n^*(\tau_n') - \text{ES}(\tau_n')}{a((1 - \tau_n)^{-1}) \int_1^{(1-\tau_n)/(1-\tau_n')} s^{\gamma-1} \log(s) ds} \xrightarrow{d} \mathcal{N}(\lambda B_1(\gamma, \rho) + \mu B_3(\gamma), V(\gamma)). $$

Theorem 3, Corollary 1 and Corollary 2 are, like Theorem 2, high-level results valid under serial dependence. As a consequence, an analogue of Theorem 5, which is a consequence of Theorems 2 and 3 together with the joint convergence of the extreme value moment estimators, will hold under mixing assumptions. Theorem 4 (in the case $\gamma < 1/2$) also holds under mixing settings and with an enlarged asymptotic variance. These asymptotic results and their proofs are the basis for the construction of accurate and computationally straightforward confidence intervals about $\text{ES}(\tau_n')$, as discussed below.
## 4 Corrected asymptotic inference

We first highlight various defects of the standard asymptotic confidence intervals for the extreme Expected Shortfall derived from our results, and we build upon our insight to design corrected versions of these intervals whose coverage is close to the nominal level in the general case $\gamma < 1$. Our methods are implemented in the freely available R package Expectrem, see Section C.1 for a description of the relevant commands. We rely on the moment estimators $\hat{a}_n((1 - \tau_n)^{-1}) = \hat{a}_n^{\text{Mom}}((1 - \tau_n)^{-1})$ of the scale and $\hat{\gamma}_n = \hat{\gamma}_n^{\text{Mom}}$ of the shape extreme value parameters, respectively; we shall use the fact that if $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1}) \to 0$ and $\sqrt{n(1 - \tau_n)}(a((1 - \tau_n)^{-1})/q(\tau_n) - \gamma_+) \to 0$, then

$$ \sqrt{n(1 - \tau_n)} \left( \frac{X_{[n\tau_n]:n} - q(\tau_n)}{a((1 - \tau_n)^{-1})}, \frac{\hat{a}_n((1 - \tau_n)^{-1})}{a((1 - \tau_n)^{-1})} - 1, \hat{\gamma}_n - \gamma \right) \xrightarrow{d} (N_{\text{loc}}, N_{\text{scale}}, N_{\text{shape}}), $$

where $(N_{\text{loc}}, N_{\text{scale}}, N_{\text{shape}})$ is trivariate Gaussian centered and has covariance matrix

$$ \Sigma = \Sigma(\gamma) = \begin{pmatrix} 1 & \gamma & 0 \\ \gamma & v_1(\gamma) & c(\gamma) \\ 0 & c(\gamma) & v_2(\gamma) \end{pmatrix}, $$

with

$$ v_1(\gamma) = (\gamma^2 + 2) \mathbb{1}_{\{\gamma \ge 0\}} + \frac{2 - 16\gamma + 51\gamma^2 - 69\gamma^3 + 50\gamma^4 - 24\gamma^5}{(1 - 2\gamma)(1 - 3\gamma)(1 - 4\gamma)} \mathbb{1}_{\{\gamma < 0\}}, $$

$$ v_2(\gamma) = (\gamma^2 + 1) \mathbb{1}_{\{\gamma \ge 0\}} + \frac{(1 - \gamma)^2(1 - 2\gamma)(1 - \gamma + 6\gamma^2)}{(1 - 3\gamma)(1 - 4\gamma)} \mathbb{1}_{\{\gamma < 0\}}, $$

$$ \text{and } c(\gamma) = -(1 - \gamma) \mathbb{1}_{\{\gamma \ge 0\}} - \frac{(1 - \gamma)^2(1 - 4\gamma + 12\gamma^2)}{(1 - 3\gamma)(1 - 4\gamma)} \mathbb{1}_{\{\gamma < 0\}}. $$

See the proof of Theorem 5 in the case $\gamma < 0$, and Corollary 4.2.2 on p.133 in de Haan and Ferreira (2006) when $\gamma \ge 0$. Throughout this section we let $Y_1, \dots, Y_n$ be independent and identically unit Pareto distributed (i.e. with distribution function $y \mapsto 1 - 1/y$ for $y \ge 1$) and we recall that $(X_1, \dots, X_n) \stackrel{d}{=} (U(Y_1), \dots, U(Y_n))$.

We consider the case where the probability level $\tau'_n \uparrow 1$ is such that $n(1 - \tau'_n)$ is bounded, which is the typical scenario in extreme value practice. The case of an intermediate level $\tau_n = 1 - k_n/n$, where $k_n \to \infty$ and $k_n/n \to 0$, is discussed in Section C.2 of the Appendix. We first construct confidence intervals based on $\hat{\text{ES}}_n^*(\tau'_n)$, the extrapolated version of the empirical Expected Shortfall estimator. For a heavy-tailed distribution with finite variance, an extensively studied competitor is the Weissman-type (after Weissman, 1978) estimator

$$ \hat{\text{ES}}_n^W(\tau'_n) = \left(\frac{1 - \tau'_n}{1 - \tau_n}\right)^{-\hat{\gamma}_n^H} \hat{\text{ES}}_n(\tau_n) = \left(\frac{k_n}{n(1 - \tau'_n)}\right)^{\hat{\gamma}_n^H} \hat{\text{ES}}_n(1 - k_n/n). $$
Here $\hat{\gamma}_n^H = M_n^{(1)}$ is the Hill estimator of $\gamma$ Hill (1975). For $\gamma \in (0, 1/2)$, the asymptotic behavior of this estimator is dominated by that of $\hat{\gamma}_n^H$ (see El Methni and Stupfler, 2017), so that

$$ \frac{\sqrt{k_n}}{\log(k_n/(n(1-\tau'_n)))} \log \frac{\widehat{\mathrm{ES}}_n^W(\tau'_n)}{\mathrm{ES}(\tau'_n)} \xrightarrow{d} \mathcal{N}(0, \gamma^2). $$

The associated 100(1 - $\alpha$)% asymptotic confidence interval is

$$ \hat{I}_1^*(\alpha) = \left[ \widehat{\mathrm{ES}}_n^W(\tau'_n) \exp \left( \pm \hat{\gamma}_n^H \frac{\log(k_n/(n(1-\tau'_n)))}{\sqrt{k_n}} z_{1-\alpha/2} \right) \right] $$

where $z_{1-\alpha/2}$ is the quantile of level $1 - \alpha/2$ of the standard Gaussian distribution. Lifting the restriction $\gamma \in (0, 1/2)$ can be done using the semiparametric estimator $\widehat{\mathrm{ES}}_n^*(\tau'_n)$ instead of $\widehat{\mathrm{ES}}_n^W(\tau'_n)$, with $\hat{a}_n(n/k_n) = \hat{a}_n^{\mathrm{Mom}}(n/k_n)$ and $\hat{\gamma}_n = \hat{\gamma}_n^{\mathrm{Mom}}$. Applying Theorem 4 (with the notation therein) suggests, when $\gamma < 1$, another confidence interval for $\mathrm{ES}(\tau'_n)$:

$$ \hat{I}_2^*(\alpha) = \left[ \widehat{\mathrm{ES}}_n^*(\tau'_n) \pm \frac{\hat{a}_n(n/k_n) \int_1^{k_n/(n(1-\tau'_n))} s^{\hat{\gamma}_n-1} \log(s) ds}{\sqrt{k_n} \sqrt{V(\hat{\gamma}_n) z_{1-\alpha/2}}} \right]. $$

This confidence interval is theoretically valid for $\gamma < 1$. In the proof of Theorem 4, the limiting distribution of $\widehat{\mathrm{ES}}_n^*(\tau'_n)$ is obtained, when $\gamma \in [0, 1)$, by neglecting the finite-sample uncertainty in $\widehat{\mathrm{ES}}_n(1 - k_n/n)$ and $\hat{a}_n(n/k_n)$. To construct an accurate approximation to the distribution of $\widehat{\mathrm{ES}}_n^*(\tau'_n)$, we propose a sampling approach. Let $d_n = k_n/(n(1-\tau'_n))$; up to the additive bias term $\mathrm{ES}(\tau'_n) - \mathrm{ES}(1 - k_n/n) - \frac{a(n/k_n)}{1-\gamma} \int_1^{d_n} s^{\gamma-1} ds$ that we neglect,

$$ \frac{\widehat{\mathrm{ES}}_n^*(\tau'_n) - \mathrm{ES}(\tau'_n)}{\hat{a}_n(n/k_n)} \approx \frac{\widehat{\mathrm{ES}}_n(1 - k_n/n) - \mathrm{ES}(1 - k_n/n)}{\hat{a}_n(n/k_n)} - \frac{\int_1^{d_n} s^{\gamma-1} ds}{1 - \gamma} \left( \frac{a(n/k_n)}{\hat{a}_n(n/k_n)} - 1 \right) + \left( \frac{\int_1^{d_n} s^{\gamma-1} ds}{(1-\gamma)^2} + \frac{\int_1^{d_n} \log(s) s^{\gamma-1} ds}{1-\gamma} \right) (\hat{\gamma}_n - \gamma). $$

We obtain an asymptotic representation of each term of the right-hand side in the above approximation. Up to a negligible bias term,

$$ \begin{aligned} \frac{\widehat{\mathrm{ES}}_n(1 - k_n/n) - \mathrm{ES}(1 - k_n/n)}{a(n/k_n)} &\approx \frac{\widehat{\mathrm{ES}}_n(1 - k_n/n) - X_{n-k_n:n}}{a(n/k_n)} - \frac{1}{1-\gamma} + \frac{X_{n-k_n:n} - U(n/k_n)}{a(n/k_n)} \\ &\approx \frac{a(Y_{n-k_n:n})}{a(n/k_n)} \times \frac{1}{k_n} \sum_{i=1}^{k_n} \left( \frac{U(Y_{n-i+1:n}) - U(Y_{n-k_n:n})}{a(Y_{n-k_n:n})} - \frac{1}{1-\gamma} \right) \\ &\quad + \frac{1}{1-\gamma} \left( \frac{a(Y_{n-k_n:n})}{a(n/k_n)} - 1 \right) + \frac{U(Y_{n-k_n:n}) - U(n/k_n)}{a(n/k_n)}. \end{aligned} $$

Using condition $C_2(\gamma, a, \rho, A)$, the Rényi representation (see (A.15) and (A.16) in the Ap-
pendix) and the regular variation property of $a(\cdot)$ we get, with $D_r(x) = \int_1^x s^{r-1}ds$,

$$ \frac{\widehat{\mathrm{ES}}_n(1-k_n/n) - \mathrm{ES}(1-k_n/n)}{a(n/k_n)} \stackrel{\mathrm{d}}{\approx} \left(\frac{k_n}{n} Y_{n-k_n:n}\right)^\gamma \times \frac{1}{k_n} \sum_{i=1}^{k_n} \left(D_\gamma\left(Y_{n-i+1:n}/Y_{n-k_n:n}\right) - \frac{1}{1-\gamma}\right) \\ + \frac{1}{1-\gamma} \left(\left(\frac{k_n}{n} Y_{n-k_n:n}\right)^\gamma - 1\right) + D_\gamma\left(\frac{k_n}{n} Y_{n-k_n:n}\right) \\ = \left(\frac{k_n}{n} \exp\left(\sum_{i=k_n+1}^n \frac{\log(Y_i)}{i}\right)\right)^\gamma \left(\frac{1}{k_n} \sum_{i=1}^{k_n} D_\gamma(Y_i) - \frac{1}{1-\gamma}\right) \\ + \frac{1}{1-\gamma} D_\gamma\left(\frac{k_n}{n} \exp\left(\sum_{i=k_n+1}^n \frac{\log(Y_i)}{i}\right)\right) =: -G_n^{(1)}(Y_1, \dots, Y_n, \gamma). $$

Besides

$$ \frac{\hat{a}_n(n/k_n)}{a(n/k_n)} = \frac{X_{n-k_n:n} - U(n/k_n)}{a(n/k_n)} M_n^{(1)}(1 - \hat{\gamma}_{n,-}) + \left\{ \frac{U(n/k_n)}{a(n/k_n)} M_n^{(1)} \right\} (1 - \hat{\gamma}_{n,-}) $$

and as such, it follows from (A.58), (A.60), (A.61) in the proof of Lemma A.8 (see the Appendix) and the convergence $M_n^{(1)}(1 - \hat{\gamma}_{n,-}) \xrightarrow{\mathbb{P}} \gamma_+$ that

$$ \frac{\hat{a}_n(n/k_n)}{a(n/k_n)} \stackrel{\mathrm{d}}{\approx} \gamma_+ D_\gamma \left( \frac{k_n}{n} \exp \left( \sum_{i=k_n+1}^n \frac{\log(Y_i)}{i} \right) \right) \\ + \frac{1}{2} \left( \frac{1}{k_n} \sum_{i=1}^{k_n} D_{\gamma-}(Y_i) \right) \left( 1 - \frac{\left( \frac{1}{k_n} \sum_{i=1}^{k_n} D_{\gamma-}(Y_i) \right)^2}{\frac{1}{k_n} \sum_{i=1}^{k_n} (D_{\gamma-}(Y_i))^2} \right)^{-1} =: G_n^{(2)}(Y_1, \dots, Y_n, \gamma). $$

Finally,

$$ \hat{\gamma}_n - \gamma = \frac{a(n/k_n)}{U(n/k_n)} \left\{ \frac{U(n/k_n)}{a(n/k_n)} M_n^{(1)} \right\} - \gamma_+ + \left\{ 1 - \frac{1}{2} \left( 1 - \frac{(M_n^{(1)})^2}{M_n^{(2)}} \right)^{-1} - \gamma_- \right\} \\ \approx \gamma_+ \left( \left\{ \frac{U(n/k_n)}{a(n/k_n)} M_n^{(1)} \right\} - 1 \right) + \left\{ 1 - \frac{1}{2} \left( 1 - \frac{(M_n^{(1)})^2}{M_n^{(2)}} \right)^{-1} - \gamma_- \right\} $$

so using again the Rényi representation and Equations (A.60) and (A.61) in the proof of Lemma A.8 (see the Appendix) leads to

$$ \left( \frac{\widehat{\mathrm{ES}}_n(1-k_n/n) - \mathrm{ES}(1-k_n/n)}{\hat{a}_n(n/k_n)}, \frac{a(n/k_n)}{\hat{a}_n(n/k_n)} - 1, \hat{\gamma}_n - \gamma \right) \\ \stackrel{\mathrm{d}}{\approx} \left( -G_n(Y_1, \dots, Y_n, \gamma), \frac{1}{G_n^{(2)}(Y_1, \dots, Y_n, \gamma)} - 1, H_n(Y_1, \dots, Y_n, \gamma) \right) $$
with $G_n(Y_1, \dots, Y_n, \gamma) := G_n^{(1)}(Y_1, \dots, Y_n, \gamma)/G_n^{(2)}(Y_1, \dots, Y_n, \gamma)$ and

$$H_n(Y_1, \dots, Y_n, \gamma) := \gamma_+ \left( \frac{1}{k_n} \sum_{i=1}^{k_n} D_{\gamma_-}(Y_i) - 1 \right) + 1 - \frac{1}{2} \left( 1 - \frac{\left( \frac{1}{k_n} \sum_{i=1}^{k_n} D_{\gamma_-}(Y_i) \right)^2}{\frac{1}{k_n} \sum_{i=1}^{k_n} D_{\gamma_-}(Y_i)^2} \right)^{-1} - \gamma_-.$$

Conclude that

$$\frac{\widehat{\mathrm{ES}}_n^*(\tau'_n) - \mathrm{ES}(\tau'_n)}{\hat{a}_n(n/k_n)} \stackrel{\mathrm{d}}{=} -G_n(Y_1, \dots, Y_n, \gamma) - \frac{\int_1^{d_n} s^{\gamma-1} ds}{1-\gamma} \left( \frac{1}{G_n^{(2)}(Y_1, \dots, Y_n, \gamma)} - 1 \right) + \left( \frac{\int_1^{d_n} s^{\gamma-1} ds}{(1-\gamma)^2} + \frac{\int_1^{d_n} \log(s) s^{\gamma-1} ds}{1-\gamma} \right) H_n(Y_1, \dots, Y_n, \gamma) =: -E_n(Y_1, \dots, Y_n, \gamma).$$

The random variable $E_n(Y_1, \dots, Y_n, \gamma)$ is very easy to simulate and its distribution can thus be tabulated; this gives rise to the confidence interval

$$\hat{\mathrm{I}}_3^*(\alpha) = \left[ \widehat{\mathrm{ES}}_n^*(\tau'_n) + \hat{a}_n(n/k_n) e_{n,\alpha/2}(\hat{\gamma}_n), \widehat{\mathrm{ES}}_n^*(\tau'_n) + \hat{a}_n(n/k_n) e_{n,1-\alpha/2}(\hat{\gamma}_n) \right]$$

where $e_{n,\tau}(\gamma)$ is the $\tau$th quantile of $E_n(Y_1, \dots, Y_n, \gamma)$. Our experience is that the performance of this interval is adversely affected by ignoring the finite-sample uncertainty in the plug-in step of replacing $\gamma$ by $\hat{\gamma}_n$ when the tail of $X$ is very heavy. Since $\sqrt{k_n}(\hat{\gamma}_n - \gamma) \stackrel{\mathrm{d}}{=} \mathcal{N}(0, v_2(\gamma))$, we propose to deal with this issue of uncertainty quantification by computing directly the quantiles of $E_n(Y_1, \dots, Y_n, \tilde{\gamma}_n)$ for $\tilde{\gamma}_n = \hat{\gamma}_n + Z\sqrt{v_2(\hat{\gamma}_n)/k_n}$, where $Z \sim \mathcal{N}(0, 1)$ is independent from the data; in addition, we retain only those values of $\tilde{\gamma}_n$ that are smaller than 1, i.e. we resample given $\tilde{\gamma}_n < 1$. This gives rise to an alternative interval $\hat{\mathrm{I}}_4^*(\alpha)$. Algorithm 1 gives all the details concerning the actual computation of both $\hat{\mathrm{I}}_3^*(\alpha)$ and $\hat{\mathrm{I}}_4^*(\alpha)$ (here and throughout the function $\Phi$ denotes the standard normal distribution function).

We turn to extrapolated versions of the semiparametric Expected Shortfall estimator. Again, in the heavy-tailed setting $\gamma \in (0, 1)$, one may consider the Weissman-type estimator

$$\widehat{\mathrm{ES}}_n^W(\tau'_n) = \left( \frac{1 - \tau'_n}{1 - \tau_n} \right)^{-\hat{\gamma}_n^H} \widehat{\mathrm{ES}}_n^H(\tau_n) = \left( \frac{k_n}{n(1 - \tau'_n)} \right)^{\hat{\gamma}_n^H} \frac{X_{n-k_n:n}}{1 - \hat{\gamma}_n^H}$$

built upon the intermediate estimator $\widehat{\mathrm{ES}}_n^H(\tau_n) = X_{n-k_n:n}/(1 - \hat{\gamma}_n^H)$ motivated by the approximation $\mathrm{ES}(\tau) \approx q(\tau)/(1 - \tau)$ as $\tau \uparrow 1$, see Proposition 1(iii). It is also readily seen that the asymptotic behavior of $\hat{\gamma}_n^H$ dominates, leading to a 100(1 - $\alpha$)% confidence interval analogous to $\hat{\mathrm{I}}_1^*(\alpha)$:

$$\tilde{\mathrm{I}}_1^*(\alpha) = \left[ \widehat{\mathrm{ES}}_n^W(\tau'_n) \exp \left( \pm \hat{\gamma}_n^H \frac{\log(k_n/(n(1-\tau'_n)))}{\sqrt{k_n}} z_{1-\alpha/2} \right) \right].$$

Just as in the construction of $\hat{\mathrm{I}}_2^*(\alpha)$ as opposed to $\hat{\mathrm{I}}_1^*(\alpha)$, one may construct a more widely applicable confidence interval $\tilde{\mathrm{I}}_2^*(\alpha)$ by using the estimator $\widehat{\mathrm{ES}}_n^*(\tau'_n)$ instead of $\widehat{\mathrm{ES}}_n^W(\tau'_n)$. The-
**Algorithm 1** Confidence intervals for ES($\tau'_n$) - Extrapolated empirical estimator

**Require:** $N \ge 1$, $\alpha \in (0, 1)$, $\widehat{\text{ES}}_n(1 - k_n/n)$, $\hat{a}_n(n/k_n) = \hat{a}_n^{\text{Mom}}(n/k_n)$, $\hat{\gamma}_n = \hat{\gamma}_n^{\text{Mom}}(k_n)$

**Ensure:** $\hat{\gamma}_n < 1$ and $1 - \tau'_n < k_n/n$

Compute $\widehat{\text{ES}}_n^*(\tau'_n) = \widehat{\text{ES}}_n(1 - k_n/n) + \hat{a}_n(n/k_n) \times \frac{1}{1 - \hat{\gamma}_n} \int_1^{k_n/(n(1-\tau'_n))} s^{\hat{\gamma}_n-1} ds$

Simulate $N$ replications $U_1, \dots, U_N$ from a uniform distribution on $[0, 1]$

for $i \in \{1, \dots, N\}$ do

Calculate $\tilde{\gamma}_{n,i} = \hat{\gamma}_n + \sqrt{\frac{v_2(\hat{\gamma}_n)}{k_n}} \Phi^{-1} \left( U_i \Phi \left( (1 - \hat{\gamma}_n) \sqrt{\frac{k_n}{v_2(\hat{\gamma}_n)}} \right) \right)$

Simulate $n$ replications $Y_1, \dots, Y_n$ from a unit Pareto distribution

Compute $E_i = E_n(Y_1, \dots, Y_n, \hat{\gamma}_n)$ and $\tilde{E}_i = E_n(Y_1, \dots, Y_n, \tilde{\gamma}_{n,i})$

end for

Compute $\hat{I}_2^*(\alpha) = \hat{I}_2^*(\alpha) = E_{\lfloor N(1-\alpha/2) \rfloor:N}$ and $\hat{I}_3^*(\alpha) = \hat{I}_3^*(\alpha) = E_{\lfloor N(1-\alpha/2) \rfloor:N}$

return $\hat{I}_4^*(\alpha) = \hat{I}_4^*(\alpha) = \left[ \widehat{\text{ES}}_n^*(\tau'_n) + \hat{a}_n(n/k_n) \tilde{E}_{\text{down}}, \widehat{\text{ES}}_n^*(\tau'_n) + \hat{a}_n(n/k_n) \tilde{E}_{\text{up}} \right]$

orem 5 suggests the interval

$$ \tilde{I}_2^*(\alpha) = \left[ \widehat{\text{ES}}_n^*(\tau'_n) \pm \frac{\hat{a}_n(n/k_n) \int_1^{k_n/(n(1-\tau'_n))} s^{\hat{\gamma}_n-1} \log(s) ds}{\sqrt{k_n} \sqrt{V(\hat{\gamma}_n) z_{1-\alpha/2}}} \right] $$

which closely resembles $\hat{I}_2^*(\alpha)$, the only difference being that $\widehat{\text{ES}}_n^*(\tau'_n)$ is replaced by $\widehat{\text{ES}}_n^*(\tau'_n)$. Like $\hat{I}_2^*(\alpha)$, this confidence interval is theoretically valid for $\gamma < 1$, but its finite-sample performance suffers when the underlying distribution is heavy-tailed because it neglects the statistical uncertainty in $\widehat{\text{ES}}_n(1 - k_n/n)$ and $\hat{a}_n(n/k_n)$. We devise a workaround based on viewing $\widehat{\text{ES}}_n^*(\tau'_n)$ as a quadratic form of a Gaussian random vector. Set $J_{n,1}(\gamma) = \int_1^{k_n/(n(1-\tau'_n))} s^{\gamma-1} ds$ and $J_{n,2}(\gamma) = \int_1^{k_n/(n(1-\tau'_n))} s^{\gamma-1} \log(s) ds$, and set

$$ \tilde{Z}_n^* = \sqrt{k_n} \frac{\widehat{\text{ES}}_n^*(\tau'_n) - \text{ES}(\tau'_n)}{\hat{a}_n(n/k_n)} = \sqrt{k_n} \frac{\widehat{\text{ES}}_n^*(\tau'_n) - \text{ES}(\tau'_n)}{a(n/k_n)} \frac{a(n/k_n)}{\hat{a}_n(n/k_n)}. $$

Since

$$ \frac{\widehat{\text{ES}}_n^*(\tau'_n) - \text{ES}(\tau'_n)}{a(n/k_n)} \approx \frac{\widehat{\text{ES}}_n(1 - k_n/n) - \text{ES}(1 - k_n/n)}{a(n/k_n)} + \frac{\int_1^{d_n} s^{\gamma-1} ds}{1 - \gamma} \left( \frac{\hat{a}_n(n/k_n)}{a(n/k_n)} - 1 \right) + \left( \frac{\int_1^{d_n} s^{\gamma-1} ds}{(1-\gamma)^2} + \frac{\int_1^{d_n} \log(s) s^{\gamma-1} ds}{1-\gamma} \right) (\hat{\gamma}_n - \gamma), $$
recalling Theorem 2 suggests the approximation

$$ \tilde{Z}_n^* \approx \left( N_{\text{loc}} + \frac{1 + J_{n,1}(\gamma)}{1 - \gamma} N_{\text{scale}} + \frac{1 + J_{n,1}(\gamma) + (1 - \gamma)J_{n,2}(\gamma)}{(1 - \gamma)^2} N_{\text{shape}} \right) \left( 1 - \frac{N_{\text{scale}}}{\sqrt{k_n}} \right) \\ = \mathbf{u}^*(\gamma)^{\top} \mathbf{N} + \frac{\mathbf{N}^{\top} \mathbf{S}^*(\gamma) \mathbf{N}}{\sqrt{k_n}}, $$

where $\mathbf{N} = (N_{\text{loc}}, N_{\text{scale}}, N_{\text{shape}})^{\top}$ follows a trivariate centered normal distribution with co-variance matrix $\mathbf{\Sigma}$,

$$ \mathbf{u}^*(\gamma) = \begin{pmatrix} 1 \\ \frac{1+J_{n,1}(\gamma)}{1-\gamma} \\ \frac{1+J_{n,1}(\gamma)+(1-\gamma)J_{n,2}(\gamma)}{(1-\gamma)^2} \end{pmatrix} $$

$$ \text{and } \mathbf{S}^*(\gamma) = \begin{pmatrix} 0 & -\frac{1}{2} & 0 \\ -\frac{1}{2} & -\frac{1+J_{n,1}(\gamma)}{1-\gamma} & -\frac{1+J_{n,1}(\gamma)+(1-\gamma)J_{n,2}(\gamma)}{2(1-\gamma)^2} \\ 0 & -\frac{1+J_{n,1}(\gamma)+(1-\gamma)J_{n,2}(\gamma)}{2(1-\gamma)^2} & 0 \end{pmatrix}. $$

Straightforward calculations show that, if $\mathbf{Z} = (Z_1, \dots, Z_p)$ is a $p$-dimensional random vector of independent centered unit Gaussian random variables and $\mathbf{M}$ is a $p \times p$ symmetric matrix, one has $\mathbb{E}(\mathbf{Z}^{\top}\mathbf{M}\mathbf{Z}) = \text{tr}(\mathbf{M})$ and $\mathbb{E}((\mathbf{Z}^{\top}\mathbf{M}\mathbf{Z})^2) = 2\text{tr}(\mathbf{M}^2) + (\text{tr}(\mathbf{M}))^2$. The mean $m^*(\gamma)$ and standard deviation $s^*(\gamma)$ of $\mathbf{u}^*(\gamma)^{\top}\mathbf{N} + \mathbf{N}^{\top}\mathbf{S}^*(\gamma)\mathbf{N}/\sqrt{k_n}$ are then

$$ m^*(\gamma) = \frac{\text{tr}(\mathbf{S}^*(\gamma)\mathbf{\Sigma}(\gamma))}{\sqrt{k_n}} \quad \text{and} \quad s^*(\gamma) = \sqrt{\mathbf{u}^*(\gamma)^{\top}\mathbf{\Sigma}(\gamma)\mathbf{u}^*(\gamma) + 2\frac{\text{tr}(\mathbf{S}^*(\gamma)\mathbf{\Sigma}(\gamma)\mathbf{S}^*(\gamma)\mathbf{\Sigma}(\gamma))}{k_n}}. $$

Approximating $\tilde{Z}_n^*$ by a Gaussian random variable with mean $m^*(\hat{\gamma}_n)$ and standard deviation $s^*(\hat{\gamma}_n)$ suggests the confidence interval

$$ \tilde{\mathbf{I}}_3^*(\alpha) = \left[ \tilde{\mathbb{E}}\mathbf{S}_n^*(\tau_n') - \frac{\hat{a}_n(n/k_n)}{\sqrt{k_n}} m^*(\hat{\gamma}_n) \pm \frac{\hat{a}_n(n/k_n)}{\sqrt{k_n}} s^*(\hat{\gamma}_n) z_{1-\alpha/2} \right]. $$

Similarly to $\tilde{\mathbf{I}}_3^*(\alpha)$, the finite-sample performance of $\tilde{\mathbf{I}}_3^*(\alpha)$ is compromised because the statistical uncertainty of the estimator $\hat{\gamma}_n$ plugged into $m^*$ and $s^*$ is not accounted for. To take this uncertainty into account, we analytically derive a correction term that should be added due to this plug-in step. Let $\mathbf{\Sigma} = \mathbf{\Lambda}\mathbf{\Lambda}^{\top}$ be the Cholesky decomposition of $\mathbf{\Sigma}$, where

$$ \mathbf{\Lambda} = \mathbf{\Lambda}(\gamma) = \begin{pmatrix} 1 & 0 & 0 \\ \gamma & \sqrt{v_1(\gamma) - \gamma^2} & 0 \\ 0 & \frac{c(\gamma)}{\sqrt{v_1(\gamma) - \gamma^2}} & \sqrt{v_2(\gamma) - \frac{c^2(\gamma)}{v_1(\gamma) - \gamma^2}} \end{pmatrix} $$

and note that $\mathbf{N} = \mathbf{\Lambda}\mathbf{Z}$ where $\mathbf{Z}$ is made of three independent centered unit Gaussian random
variables. Recalling that $\sqrt{k_n}(\hat{\gamma}_n - \gamma) \approx N_{\text{shape}}$, a Taylor expansion yields

$$ \tilde{Z}_n^* \stackrel{d}{=} \mathbf{u}^*(\gamma)^\top \mathbf{\Lambda}(\gamma) \mathbf{Z} + \frac{\mathbf{Z}^\top \mathbf{\Lambda}(\gamma)^\top \mathbf{S}^*(\gamma) \mathbf{\Lambda}(\gamma) \mathbf{Z}}{\sqrt{k_n}} + o_P\left(\frac{1}{\sqrt{k_n}}\right) \\ = \mathbf{u}^*(\hat{\gamma}_n)^\top \mathbf{\Lambda}(\hat{\gamma}_n) \mathbf{Z} + \frac{\mathbf{Z}^\top \mathbf{\Lambda}(\hat{\gamma}_n)^\top \mathbf{S}^*(\hat{\gamma}_n) \mathbf{\Lambda}(\hat{\gamma}_n) \mathbf{Z}}{\sqrt{k_n}} - \mathbf{\theta}^*(\hat{\gamma}_n)^\top \frac{N_{\text{shape}}}{\sqrt{k_n}} \mathbf{Z} + o_P\left(\frac{1}{\sqrt{k_n}}\right) $$

where $\mathbf{\theta}^*(\gamma) = \frac{d\mathbf{u}^*}{d\gamma}(\gamma)^\top \mathbf{\Lambda}(\gamma) + \mathbf{u}^*(\gamma)^\top \frac{d\mathbf{\Lambda}}{d\gamma}(\gamma)$.

Since $\mathbf{N} = \mathbf{\Lambda}\mathbf{Z}$, one has $N_{\text{shape}} = \Lambda_{23}(\gamma)Z_2 + \Lambda_{33}(\gamma)Z_3 \stackrel{d}{=} \Lambda_{23}(\hat{\gamma}_n)Z_2 + \Lambda_{33}(\hat{\gamma}_n)Z_3 + o_P(1)$. Then

$$ \tilde{Z}_n^* \stackrel{d}{=} \mathbf{w}^*(\hat{\gamma}_n)^\top \mathbf{Z} + \frac{\mathbf{Z}^\top \mathbf{W}^*(\hat{\gamma}_n) \mathbf{Z}}{\sqrt{k_n}} + o_P\left(\frac{1}{\sqrt{k_n}}\right), $$

where $\mathbf{w}^*(\gamma) = \mathbf{\Lambda}^\top(\gamma)\mathbf{u}^*(\gamma)$, and

$$ \mathbf{W}^*(\gamma) = \mathbf{\Lambda}(\gamma)^\top \mathbf{S}^*(\gamma) \mathbf{\Lambda}(\gamma) - \frac{1}{2} \begin{pmatrix} 0 & \theta_1^*(\gamma) \Lambda_{23}(\gamma) & \theta_1^*(\gamma) \Lambda_{33}(\gamma) \\ \theta_1^*(\gamma) \Lambda_{23}(\gamma) & 2\theta_2^*(\gamma) \Lambda_{23}(\gamma) & \theta_2^*(\gamma) \Lambda_{33}(\gamma) + \theta_3^*(\gamma) \Lambda_{23}(\gamma) \\ \theta_1^*(\gamma) \Lambda_{33}(\gamma) & \theta_2^*(\gamma) \Lambda_{33}(\gamma) + \theta_3^*(\gamma) \Lambda_{23}(\gamma) & 2\theta_3^*(\gamma) \Lambda_{33}(\gamma) \end{pmatrix} $$

We omit the explicit expressions of $\mathbf{w}^*(\gamma)$ and $\mathbf{W}^*(\gamma)$ for the sake of brevity. As in the construction of $\tilde{\mathbf{I}}_3^*(\alpha)$, we then approximate the distribution of the random variable $\mathbf{w}^*(\gamma_0)^\top \mathbf{Z} + \mathbf{Z}^\top \mathbf{W}^*(\gamma_0)\mathbf{Z}/\sqrt{k_n}$ (for any fixed $\gamma_0 < 1$) by a Gaussian distribution with mean $\text{tr}(\mathbf{W}^*(\gamma_0))/\sqrt{k_n}$ and variance $\|\mathbf{w}^*(\gamma_0)\|_2^2 + 2 \text{tr}((\mathbf{W}^*(\gamma_0))^2)/k_n$. This suggests our final confidence interval

$$ \tilde{\mathbf{I}}_4^*(\alpha) = \left[ \tilde{\text{ES}}_n^*(\tau_n') + \frac{\hat{a}_n(n/k_n)}{\sqrt{k_n}} \left( -\frac{\text{tr}(\mathbf{W}^*(\hat{\gamma}_n))}{\sqrt{k_n}} \pm \sqrt{\|\mathbf{w}^*(\hat{\gamma}_n)\|_2^2 + 2 \frac{\text{tr}((\mathbf{W}^*(\hat{\gamma}_n))^2)}{k_n}} \right) z_{1-\alpha/2} \right]. $$

We next examine and compare the finite-sample performance of these eight intervals.

# 5 Numerical experiments

## 5.1 Finite-sample simulation study

We consider the Kumaraswamy, Reverse-Burr, Gumbel, Exponential, Pareto and Fréchet distributions, see Table 1. In each setting, we simulate $N = 10,000$ replications of an i.i.d. sample of size $n = 1,000$ from the chosen distribution. We estimate and infer the quantity $\text{ES}(\tau_n') = \text{ES}(1 - 1/n) = \text{ES}(0.999)$ using the estimators $\tilde{\text{ES}}_n^*(\tau_n')$ and $\tilde{\text{ES}}_n^*(\tau_n')$, and the confidence intervals $\tilde{\mathbf{I}}_1^*(\alpha)$, $\tilde{\mathbf{I}}_2^*(\alpha)$, $\tilde{\mathbf{I}}_3^*(\alpha)$, $\tilde{\mathbf{I}}_4^*(\alpha)$, $\tilde{\mathbf{I}}_1^*(\alpha)$, $\tilde{\mathbf{I}}_2^*(\alpha)$, $\tilde{\mathbf{I}}_3^*(\alpha)$ and $\tilde{\mathbf{I}}_4^*(\alpha)$ with $\alpha = 0.05$, i.e. at the nominal confidence level $100(1 - 0.05)\% = 95\%$, and with base intermediate level $\tau_n = 1 - k_n/n$ with $k_n = 200$. The true values of $\text{ES}(0.999)$, computed using Table E.1 (except
for the Reverse-Burr distribution where the true value was obtained numerically using the cubintegrate routine from the R package cubature), and the empirical coverage probabilities of the competing intervals are provided in Table 2. We carried out a similar inferential exercise for the intermediate value ES($\tau_n$) = ES(1 - $k_n/n$) = ES(0.8); since this is arguably less relevant for statistical practice than inference at extreme levels near or beyond the limits of the sample, we defer the results of the analysis at the intermediate level to Section D of the Appendix.

The interval $\hat{I}_4^*(\alpha)$ appears to have coverage probability closest to the nominal level overall when the underlying distribution has a finite variance; note that $\tilde{I}_4^*(\alpha)$ performs worse when $\gamma < 0$, due to the fact that its construction neglects various sources of bias that are nonetheless substantial in that setting. By contrast, $\tilde{I}_4^*(\alpha)$ seems to offer the best trade-off between coverage and interval length for infinite-variance distributions, whereas $\hat{I}_4^*(\alpha)$ has slightly better coverage at the expense of being much wider. The naive Gaussian intervals $\hat{I}_1^*(\alpha)$, $\hat{I}_2^*(\alpha)$, $\tilde{I}_1^*(\alpha)$ and $\tilde{I}_2^*(\alpha)$ do not perform well even when the conditions for their theoretical validity are satisfied. The interpretation here is that the Gaussian asymptotic behavior of $\hat{ES}_n^*$ and $\tilde{ES}_n^*$, giving rise to the naive Gaussian confidence intervals, is obtained by neglecting the uncertainty in $\hat{ES}_n$, $\tilde{ES}_n$ and $\hat{a}_n$, even though it is substantial in finite-sample settings.

## 5.2 Real data application 1: The OpenFEMA dataset

The OpenFEMA dataset, provided by the US government<fnref n="1" />, contains a historical database of records of residential flood insurance claims in the USA updated at a monthly frequency. Our goal is to infer the average value of an extremely high claim, defined here as happening with probability 0.5%. To alleviate concerns linked to possible non-stationarity of the data, we consider the claims consecutive to floods caused by a stream, river, or lake overflow that occurred in 2012 (sample 1), on the one hand, and 2017 (sample 2), on the other hand. We further stratify along floods rated A in the database (each corresponding to a so-called Special Flood with no Base Flood Elevation on the insurance rate map) and those rated B (each corresponding to a so-called Moderate Flood from primary water source). A scatterplot of the locations of the claims, superimposed on a map of the USA, is given in Figure 1.

After data cleaning, this yields four samples of flood insurance claims, expressed in thousands of USD, denoted by A-2012 ($n = 555$), B-2012 ($n = 402$), A-2017 ($n = 1,131$) and B-2017 ($n = 925$). The large cluster of claims in New Jersey and New York in 2012 is linked to Hurricane Sandy in late October 2012. The two large clusters of 2017 claims are related to Hurricane Harvey in southeast Texas and Louisiana in August 2017, and to Hurricane Irma in Florida in September 2017. We give in Figure 2 moment extreme value index estimates

<footnote n="1">¹Freely available at https://www.fema.gov/openfema-data-page/fima-nfip-redacted-claims-v2</footnote>
and extrapolated Expected Shortfall estimates at level 0.995, along with the asymptotic con-
fidence intervals $\hat{I}_4^*(\alpha)$ and $\tilde{I}_4^*(\alpha)$ constructed in Section 4 with $\alpha = 0.05$, i.e. at the nominal
confidence level 95%. The analysis of sample A-2012 reveals a heavy right tail with an infinite
variance, with the estimates $\hat{\mathrm{ES}}_n^*$ and $\tilde{\mathrm{ES}}_n^*$ agreeing with the Weissman-Hill estimates $\hat{\mathrm{ES}}_n^W$
and $\tilde{\mathrm{ES}}_n^W$. The analysis of sample A-2017 is less conclusive, with the extreme value index
estimate not significantly different from 0 at the 95% asymptotic confidence level, although
the point estimate is always positive. In this situation, it is better to avoid assuming that the
right tail is heavy, and therefore to use the estimates $\hat{\mathrm{ES}}_n^*$ and $\tilde{\mathrm{ES}}_n^*$ instead of the Weissman-
Hill estimates. Samples B-2012 and B-2017 are pretty clearly light-tailed ($\gamma = 0$), and we
observe that for such samples the Weissman-Hill estimates are more sensitive to the choice of
$k$ and very quickly drift away from the estimates $\hat{\mathrm{ES}}_n^*$ and $\tilde{\mathrm{ES}}_n^*$ because the Hill estimate of $\gamma$
takes substantially higher values than its moment estimate on most of the range of values of
$k$ considered.

From an applied standpoint, it is reasonable to conclude that while there are clear dif-
ferences between the tails along zone types, it is not obvious that the extreme Expected
Shortfall changes across years within a zone even if the uncertainty about estimates is af-
fected by changing sample sizes: a rough point estimate of ES(0.995) is 500,000 USD for zone
A, and 400,000 USD for zone B. We carried out a similar analysis with different subsamples
of the data and in different years, and found that likewise, the validity of the assumption of
a finite variance in the claim size distribution or, on the contrary, of its heavy-tailedness is
dubious; see Figure 6 in Section E of the Appendix.

5.3 Real data application 2: Daily financial loss returns

We examine the extreme value behavior of two series of daily loss returns (i.e. negative log-
returns) for the CAC 40 and FTSE 100 indices, from 9 December 2008 to 12 July 2016
for the former and from 6 July 2005 to 10 June 2013 for the latter, covering the subprime
financial crisis. These time series may feature substantial serial dependence. To handle this
dependence in a dynamic setting, it is usual practice to apply a filter such as an ARMA-
GARCH model and then, subject to the residuals from the fitted model passing goodness-
of-fit checks, to treat the residuals from this model as i.i.d. observations to which extreme
value theory may be applied. This kind of two-stage procedure, first fitting an ARMA-
GARCH model to retrieve standardized residuals and then employing extreme value theory
to estimate tail-related risk under the assumption that they can statistically be considered
i.i.d., has been used since at least McNeil and Frey (2000). To the best of our knowledge,
most of the recent methodological contributions in this framework assume that the residuals
can be considered identically distributed and even heavy-tailed: see, among others, He and
Einmahl (2016), Girard et al. (2021) and Kaibuchi et al. (2022). Despite the theoretical impossibility to guarantee the homogeneity of residuals, this assumption remains a reasonable approximation for the purposes of extreme value analysis: indeed, since the focus is solely on extreme residuals, minor deviations from this assumption in the central part of the distribution will not materially affect the results as long as the extremes of the residuals can be considered homogeneous. However, our assessments hereafter provide empirical evidence of the fact that the residuals may have light-tailed or short-tailed distributions.

We first provide more details about our stationarity and goodness-of-fit checks in order to back the use of the ARMA-GARCH filter. For each observed time series, we consider successive rolling windows of length $n = 1,500$, which results in 438 (resp. 503) samples of $n$ daily loss returns for the CAC 40 (resp. FTSE 100) data. In Figure 3, panels (A) and (E) show the results of the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for each rolling window, illustrating the stationarity of the CAC 40 and FTSE 100 time series samples across all rolling windows, which is a prerequisite for ARMA-GARCH modeling. Following the methodology of Girard et al. (2022), we then estimate an ARMA(1,1)-GARCH(1,1) model on the successive rolling windows for each dataset. The corresponding residuals form 438 (resp. 503) samples of size $n$ for the CAC 40 (resp. FTSE 100) data. Panels (B) and (F) of Figure 3 examine the independence of the residuals within each sample using the Ljung-Box test on both the residuals and their squares. As a complement to the Ljung-Box test, panels (C) and (G) showcase the results of the Lagrange Multiplier test, as proposed by Engle (1982), to confirm the absence of remaining ARCH effects in the residuals and evaluate the adequacy of the fitted ARMA-GARCH model. Finally, the results of the sign bias test of Engle and Ng (1993), presented in panels (D) and (H), indicate the absence of leverage effects in the data, at the significance level 0.05, and thus that our conditional variance structure is not misspecified.

We then estimate the extreme value index $\gamma$ of each sample of residuals using the estimator $\hat{\gamma}_n^{\text{Mom}}$. As illustrated in the first two series of panels in Figure 4, the heavy right tail model $\gamma > 0$ can reasonably be excluded for these samples, and depending on the rolling window chosen, the appropriate model seems to be either light-tailed or short-tailed, although it is often difficult to firmly rule out a light-tailed model. This is especially clear when considering the final rolling window in each dataset, the moment estimator plot of which is indicated in cyan in the leftmost panels (A) and (E) in Figure 4. Assuming again that the residuals within this rolling window can be statistically considered i.i.d. and the second-order condition $C_2(\gamma, a, \rho, A)$ holds, we know by Lemma A.10 in the Appendix that, if $\sqrt{k}A(n/k) \to 0$ and $\sqrt{k}(a(n/k)/q(1-k/n) - \gamma_+) \to 0$, then the moment estimator $\hat{\gamma}_n^{\text{Mom}}$ of the extreme value
index $\gamma$ of the sample of residuals satisfies

$$ \sqrt{k}(\hat{\gamma}_n^{\text{Mom}} - \gamma) \xrightarrow{d} \begin{cases} \mathcal{N}(0, \gamma^2 + 1) & \text{if } \gamma \ge 0, \\ \mathcal{N}\left(0, \frac{(1-\gamma)^2(1-2\gamma)(1-\gamma+6\gamma^2)}{(1-3\gamma)(1-4\gamma)}\right) & \text{if } \gamma < 0. \end{cases} $$

This asymptotic normality result straightforwardly opens the door to the construction of asymptotic Gaussian 95% confidence intervals for $\gamma$. The moment estimator plot relative to the final rolling window is represented in panels (B) and (F), along with the aforementioned asymptotic 95% confidence intervals: it is reasonable to conclude from these panels that the residuals are, respectively, light-tailed for the CAC 40 and short-tailed for the FTSE 100 over this window. We arrived at the same conclusion using a more parsimonious GARCH(1,1) model and/or a different sample size $n$ for the rolling window. Although this variety of tail behaviors does not seem to have been appreciated before in the statistical literature, it is of utmost interest to practitioners concerned with the accuracy of daily forecasts; as we showed in our simulation study, even when the data is heavy-tailed, there is no substantial drawback in using the general estimators $\widehat{\text{ES}}_n^*$ and $\widehat{\text{ES}}_n^*$ we provide, whereas the Weissman-Hill estimators $\widehat{\text{ES}}_n^W$ and $\widehat{\text{ES}}_n^W$ perform poorly outside of the heavy right tail model.

Of course, as highlighted above, this analysis requires that the residuals can be statistically considered to satisfy the second-order condition $C_2(\gamma, a, \rho, A)$, along with the bias conditions $\sqrt{k}A(n/k) \to \lambda = 0$ and $\sqrt{k}(a(n/k)/q(1-k/n) - \gamma_+) \to \mu = 0$. To check that this body of assumptions (homogeneity of the tail residuals and bias components $\lambda = 0$ and $\mu = 0$) is indeed reasonable, we note that according to Propositions 1(i) and 2, if a random variable $X$, with tail quantile function $U$ and right endpoint $q(1) \in (0, \infty]$, satisfies condition $C_2(\gamma, a, \rho, A)$, then for $t, tx$ large enough,

$$ \frac{U(tx) - U(t)}{a(t)} \approx \int_{1}^{x} s^{\gamma-1}ds + A(t) \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1}du \right) ds \\ \text{and } \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} \approx \int_{1}^{x} s^{\gamma-1}ds - \left( \frac{a(t)}{U(t)} - \gamma_{+} \right) \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{-|\gamma|-1}du \right) ds \\ + A(t)x^{-\gamma_{+}} \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1}du \right) ds. $$

It follows from these two approximations that:

* On the one hand, the distribution of the exceedance $X - z$ given that $X > z$ is, as $z$ tends to $q(1)$, approximately Generalized Pareto with scale parameter $a(1/(1-F(z)))$ and shape parameter $\gamma$, with the quality of the approximation controlled by the function $A$;

* On the other hand, the distribution of the log-exceedance $\log X - \log z$ given that $X > z$ is, as $z$ tends to $q(1)$, approximately Generalized Pareto with scale parameter $a(1/(1-\gamma))$.
$F(z))/U(1/(1-F(z))) \approx a(1/(1-F(z)))/z$ and shape parameter $\gamma_-$, with the quality of the approximation controlled by the function $A$ and the function $a/U - \gamma_+$.

In other words, the assumptions of homogeneity and negligible bias components can be checked by constructing Generalized Pareto QQ-plots of the $k$ exceedances (resp. log-exceedances) over the threshold $X_{n-k:n}$, with scale and shape parameter values taken as the estimates $\hat{a}_n^{\text{Mom}}(n/k)$ and $\hat{\gamma}_n^{\text{Mom}} = \hat{\gamma}_n^{\text{Mom}}(k)$ (resp. $\hat{a}_n^{\text{Mom}}(n/k)/X_{n-k:n}$ and $\hat{\gamma}_{n,-}^{\text{Mom}} = \hat{\gamma}_{n,-}^{\text{Mom}}(k)$): if these QQ-plots align closely with the $y=x$ line, showing no substantial deviation in the right end, then both the assumption of homogeneity of extremes and the assumption that $\lambda = \mu = 0$ can be considered reasonable. We constructed an overlay of these Generalized Pareto QQ-plots of the residuals (Figure 5, top panels) and log-residuals (Figure 5, bottom panels) corresponding to each rolling window in the CAC 40 and FTSE 100 datasets. It can be seen from these graphs that the two assumptions indeed appear to be satisfied generally over all rolling windows, with the particular example of the last rolling window highlighted using red circles.

The resulting residual-based estimates $\widehat{\text{ES}}_n^*(1-1/n)$ and $\widehat{\text{ES}}_n^*(1-1/n)$ obtained over the last rolling window are represented in panels (C) for CAC 40 and (G) for FTSE 100, along with their 95% asymptotic confidence intervals $\hat{I}_4^*(\alpha)$ and $\tilde{I}_4^*(\alpha)$ with $\alpha = 0.05$, respectively. They point towards very similar estimates and confidence intervals, with clearly identified stable regions around $k = 200$ resulting in pointwise ES(1-1/n) estimates around 5.3 for CAC 40 residuals and 3.9 for FTSE 100 residuals. Dynamic predictions of the extreme ES for the next day can then easily be obtained for the raw data using the ARMA(1,1)-GARCH(1,1) structure. We do so by adapting the procedure described in Daouia et al. (2024b) and we provide, in the rightmost panels of Figure 4, the point forecast $\widehat{\text{ES}}_n^*(1-1/n)$ in (D) and $\widehat{\text{ES}}_n^*(1-1/n)$ in (H) of ES(1-1/n) for the daily loss returns over the observation period, along with their 95% asymptotic confidence intervals and the realization of the future observation.

## Acknowledgments

This research was supported by the French National Research Agency under the grants ANR-19-CE40-0013 (ExtremReg project), ANR-23-CE40-0009 (EXSTA project), ANR-17-EURE-0010 (EUR CHESS) and ANR-11-LABX-0020-01 (Centre Henri Lebesgue). A. Daouia and G. Stupfler acknowledge financial support from the TSE-HEC ACPR Chair “Regulation and systemic risks”. G. Stupfler acknowledges further support from the Chair Stress Test, RISK Management and Financial Steering of the Foundation Ecole Polytechnique.
References

Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. *Journal of Banking & Finance*, 26(7):1505-1518.

Acerbi, C. and Tasche, D. (2002a). Expected shortfall: A natural coherent alternative to value at risk. *Economic Notes*, 31(2):379-388.

Acerbi, C. and Tasche, D. (2002b). On the coherence of expected shortfall. *Journal of Banking & Finance*, 26(7):1487-1503.

Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999). Coherent measures of risk. *Mathematical Finance*, 9(3):203-228.

Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J. (2004). *Statistics of Extremes: Theory and Applications*. Wiley.

Brazauskas, V., Jones, B. L., Puri, M. L., and Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view. *Journal of Statistical Planning and Inference*, 138(11):3590-3604.

Bücher, A. and Zhou, C. (2021). A horse race between the Block Maxima method and the Peak-over-Threshold approach. *Statistical Science*, 36(3):360-378.

Caeiro, F., Gomes, M. I., and Pestana, D. (2005). Direct reduction of bias of the classical Hill estimator. *REVSTAT – Statistical Journal*, 3(2):113-136.

Cai, J.-J., de Haan, L., and Zhou, C. (2013). Bias correction in extreme value statistics with index around zero. *Extremes*, 16(2):173-201.

Daouia, A., Padoan, S. A., and Stupfler, G. (2024a). Extreme expectile estimation for short-tailed data. *Journal of Econometrics*, 241(2):105770.

Daouia, A., Stupfler, G., and Usseglio-Carleve, A. (2024b). Bias-reduced and variance-corrected asymptotic Gaussian inference about extreme expectiles. *Statistics & Computing*, 34(4):130.

de Haan, L. and Ferreira, A. (2006). *Extreme Value Theory: An Introduction*. Springer, New York.

de Haan, L. and Resnick, S. (1996). Second-order regular variation and rates of convergence in extreme value theory. *The Annals of Probability*, 24(1):97-124.
Dekkers, A. L. M., Einmahl, J. H. J., and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. *Annals of Statistics*, 17(4):1833–1855.

Diebolt, J., Guillou, A., and Rached, I. (2007). Approximation of the distribution of ex-
cesses through a generalized probability-weighted moments method. *Journal of Statistical
Planning and Inference*, 137(3):841–857.

Dombry, C. and Ferreira, A. (2019). Maximum likelihood estimators based on the block maxima method. *Bernoulli*, 25(3):1690–1723.

Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. *Bernoulli*, 9(4):617–657.

Drees, H., Ferreira, A., and de Haan, L. (2004). On maximum likelihood estimation of the extreme value index. *Annals of Applied Probability*, 14(3):1179–1201.

El Methni, J., Gardes, L., and Girard, S. (2014). Nonparametric estimation of extreme risks from conditional heavy-tailed distributions. *Scandinavian Journal of Statistics*, 41(4):988–1012.

El Methni, J., Gardes, L., and Girard, S. (2018). Kernel estimation of extreme regression risk measures. *Electronic Journal of Statistics*, 12(1):359–398.

El Methni, J. and Stupfler, G. (2017). Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. *Statistica Sinica*, 27(2):907–930.

El Methni, J. and Stupfler, G. (2018). Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions. *Econometrics and Statistics*, 6:129–148.

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. *Econometrica*, 50(4):987–1007.

Engle, R. F. and Ng, V. K. (1993). Measuring and testing the impact of news on volatility.
*The Journal of Finance*, 48(5):1749–1778.

Falk, M., Hüsler, J., and Reiss, R.-D. (2011). *Laws of Small Numbers: Extremes and Rare Events (third, revised and extended edition)*. Birkhäuser.

Feller, W. (1971). *An Introduction to Probability Theory and its Applications – Volume II* (second edition). Wiley, New York.

Föllmer, H. and Schied, A. (2004). *Stochastic Finance – An Introduction in Discrete Time* (second edition). de Gruyter, Berlin.
Fraga Alves, M. I., Gomes, M. I., and de Haan, L. (2003). A new class of semi-parametric estimators of the second order parameter. *Portugaliae Mathematica*, 60(2):193-213.

Fraga Alves, M. I., Gomes, M. I., de Haan, L., and Neves, C. (2007). A note on second order conditions in extreme value theory: Linking general and heavy tail conditions. *REVSTAT*, 5(3):285-304.

Fung, T. and Seneta, E. (2018). Quantile function expansion using regularly varying functions. *Methodology and Computing in Applied Probability*, 20(4):1091-1103.

Geluk, J. L. and de Haan, L. (2000). Stable probability distributions and their domains of attraction: A direct approach. *Probability and Mathematical Statistics*, 20(1):169-188.

Girard, S., Stupfler, G., and Usseglio-Carleve, A. (2021). Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models. *Annals of Statistics*, 49(6):3358-3382.

Girard, S., Stupfler, G., and Usseglio-Carleve, A. (2022). On automatic bias reduction for extreme expectile estimation. *Statistics & Computing*, 32(4):64.

Goegebeur, Y., Guillou, A., Pedersen, T., and Qin, J. (2022). Extreme-value based estimation of the conditional tail moment with application to reinsurance rating. *Insurance: Mathematics and Economics*, 107:102-122.

Gomes, M. I. and Martins, M. J. (2002). “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. *Extremes*, 5(1):5-31.

He, Y. and Einmahl, J. H. J. (2016). Estimation of extreme depth-based quantile regions. *Journal of the Royal Statistical Society Series B*, 79(2):449-461.

Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. *Annals of Statistics*, 3(5):1163-1174.

Hosking, J. R. M., Wallis, J. R., and Wood, E. F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. *Technometrics*, 27(3):251-261.

Hsing, T. (1991). On tail index estimation using dependent data. *Annals of Statistics*, 19(3):1547-1569.

Kaibuchi, H., Kawasaki, Y., and Stupfler, G. (2022). GARCH-UGH: a bias-reduced approach for dynamic extreme Value-at-Risk estimation in financial time series. *Quantitative Finance*, 22(7):1277-1294.
McNeil, A. J. and Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. *Journal of Empirical Finance*, 7(3–4):271–300.

Nagaraja, H. N. (2006). Order statistics from independent exponential random variables and the sum of the top order statistics. In Balakrishnan, N., Sarabia, J. M., and Castillo, E., editors, *Advances in Distribution Theory, Order Statistics, and Inference*, Statistics for Industry and Technology, chapter 11, pages 173–185. Birkhäuser Boston.

Rockafellar, R. T. and Uryasev, S. (2002). Conditional Value-at-Risk for general loss distributions. *Journal of Banking & Finance*, 26(7):1443–1471.

Smith, R. L. (1987). Estimating tails of probability distributions. *Annals of Statistics*, 15(3):1174–1207.

Stupfler, G. and Usseglio-Carleve, A. (2025). Simple sufficient criteria for second-order extended regular variation in the Gumbel domain of attraction: The case of Weibull-tailed distributions. *Extremes*, 28(2):393–435.

Weissman, I. (1978). Estimation of parameters and large quantiles based on the $k$ largest observations. *Journal of the American Statistical Association*, 73(364):812–815.
<table><caption>Table 1: A list of standard, unit-scale continuous distributions satisfying $C_2(\gamma, \alpha, \rho, A)$, with the associated values of $\gamma$, $\rho$ and the functions $a$ and $A$. The parameters $\alpha$ and $\beta$ are positive, $\Gamma: x \in (0, \infty) \mapsto \Gamma(x) = \int_0^\infty x^{\gamma-1}e^{-t}dt$ denotes Euler's Gamma function, and $B: (x, y) \in (0, \infty)^2 \mapsto B(x, y) = \int_0^1 x^{\gamma-1}(1-t)^{y-1}dt$ denotes Euler's Beta function. When, for all $x > 0$, $(U(tx) - U(t))/a(t) = (x^\gamma - 1)/\gamma$ for $t$ large enough (i.e. when the Generalized Pareto model for the right tail is actually exact), we set $\rho = -\infty$ and $A(t) \equiv 0$. For the Student distribution with $\alpha$ degrees of freedom, we set $C_\alpha = (\Gamma((\alpha+1)/2)\alpha(\alpha-1)/2)^{1/\alpha}$. [The Burr distribution with equal parameters $\alpha = \beta$ and the Dagum distribution with parameters $\alpha = \beta = 1$ are equal to the Generalized Pareto distribution with unit extreme value index. The Cauchy distribution is the Student distribution with 1 degree of freedom, the exponential distribution is the Gamma distribution with unit shape parameter, and the Arcsine distribution is the Beta(1/2, 1/2) distribution.]</caption><thead><tr><th>Distribution</th><th>F(t) = 1 - F(t) = P(X > t)</th><th>γ</th><th>a(t)<br>1/α t<sup>1/α</sup><br>t<sup>1/α</sup></th><th>ρ</th><th>A(t)</th></tr></thead><tbody><tr><td>Pareto</td><td>t<sup>α</sup>, t > 1</td><td>1/α</td><td></td><td>-∞</td><td>0</td></tr><tr><td>Generalized Pareto</td><td>(1 + t/α)<sup>-α</sup>, t > 0</td><td>1/α</td><td></td><td>-∞</td><td>0</td></tr><tr><td>Burr (α ≠ β)</td><td>(1 + t<sup>β/γ</sup>)<sup>-1/β</sup>, t > 0</td><td>α</td><td>αt<sup>α</sup> - α/β (α - β)t<sup>α-β</sup></td><td>-β</td><td>(α - β)t<sup>-β</sup></td></tr><tr><td>Dagum (α ≠ 1)</td><td>1 - (1 + t<sup>-α</sup>)<sup>-β</sup>, t > 0</td><td>1/α</td><td>β<sup>1/α</sup> t<sup>1/α</sup> - (1/α - 1) (1/β + 1) β<sup>1/α</sup> t<sup>1/α-1</sup></td><td>-1</td><td>1/2 (1/α - 1) (1/β + 1) t<sup>-1</sup></td></tr><tr><td>Dagum (α = 1, β ≠ 1)</td><td>1 - (1 + t<sup>-1</sup>)<sup>-β</sup>, t > 0</td><td>1</td><td>βt - β/12 (1/β + 1) (1/β - 1) t<sup>-1</sup></td><td>-2</td><td>1/6 (1/β + 1) (1/β - 1) t<sup>-2</sup></td></tr><tr><td>Fréchet (α ≠ 1)</td><td>1 - exp(-t<sup>-α</sup>), t > 0</td><td>1/α</td><td>1/α t<sup>1/α</sup> - 1/2α (1/α - 1) t<sup>1/α-1</sup></td><td>-1</td><td>1/2 (1/α - 1) t<sup>-1</sup></td></tr><tr><td>Fréchet (α = 1)</td><td>1 - exp(-t<sup>-1</sup>), t > 0</td><td>1</td><td>t + 1/12 t<sup>-1</sup></td><td>-2</td><td>-1/6 t<sup>-2</sup></td></tr><tr><td>Inverse-Gamma</td><td>∫<sup>1/t</sup><sub>0</sub> 1/(α<sup>α-1</sup> e<sup>-α</sup> dx, t > 0</td><td>1/α</td><td>1/(α(α+1)) t<sup>1/α</sup> + (α(α+1))<sup>1/α</sup> t<sup>-1/α</sup></td><td>-2/α</td><td>- (Γ(α+1))<sup>2/α</sup> t<sup>-2/α</sup></td></tr><tr><td>Cauchy</td><td>1/2 - arctan(t)/π</td><td>1</td><td>1/π t + π/3 t<sup>-1</sup></td><td>-2</td><td>- 2π<sup>2</sup> t<sup>-2</sup></td></tr><tr><td>Student</td><td>∫<sup>∞</sup><sub>∫<sub>απ</sub>(α/2)</sub> Γ((α+1)/2) (1 + x<sup>2</sup>)<sup>-((α+1)/2)</sup> dx</td><td>1/α</td><td>C<sub>α</sub> t<sup>1/α</sup> + α<sup>1/2</sup> C<sub>α</sub> t<sup>-1/α</sup></td><td>-2/α</td><td>- α<sup>1/2</sup> C<sub>α</sub> t<sup>-2/α</sup></td></tr><tr><td>Exponential</td><td>exp(-t), t > 0</td><td>0</td><td>1</td><td>-∞</td><td>0</td></tr><tr><td>Gamma (α ≠ 1)</td><td>∫<sup>∞</sup><sub>t</sub> 1/(α<sup>α-1</sup> e<sup>-α</sup> dx, t > 0</td><td>0</td><td>1 + α<sup>-1</sup> log(t) - (α - 1)<sup>2</sup> log log(t) + (α - 1) log Γ(α) + α<sup>-2</sup> log<sup>2</sup>(t)</td><td>0</td><td>- α<sup>-1</sup> log<sup>2</sup>(t)</td></tr><tr><td>Weibull (β ≠ 1)</td><td>exp(-t<sup>β</sup>), t > 0</td><td>0</td><td>1/β log(t) t<sup>1/β-1</sup></td><td>0</td><td>(1/β - 1) log(t)</td></tr><tr><td>Gumbel</td><td>1 - exp(-exp(-t))</td><td>0</td><td>1 + 1/2 t<sup>-1</sup></td><td>-1</td><td>-1/2 t<sup>-1</sup></td></tr><tr><td>Logistic</td><td>1/(1 + exp(t))</td><td>0</td><td>1 + t<sup>-1</sup></td><td>-1</td><td>-t<sup>-1</sup></td></tr><tr><td>Normal</td><td>1 - Φ(t) = ∫<sup>∞</sup><sub>t</sub> 1/√2π e<sup>-x<sup>2</sup>/2</sup> dx</td><td>0</td><td>1/√2 log(t) (1 + log log(t) + (1/2 log(4π) - 1) 1/2 log(t))</td><td>0</td><td>- 1/2 log(t)</td></tr><tr><td>Arcsine</td><td>1 - 2 arcsin(√t)/π, t ∈ [0, 1]</td><td>-2</td><td>π<sup>2</sup> t<sup>-2</sup> - π<sup>4</sup> t<sup>-4</sup> / 12</td><td>-2</td><td>π<sup>2</sup> t<sup>-2</sup></td></tr><tr><td>Beta (α ≠ 1)</td><td>∫<sup>1</sup><sub>t</sub> 1/(α<sub>β</sub>) x<sup>α-1</sup> (1 - x)<sup>β-1</sup> dx, t ∈ [0, 1]</td><td>-1/β</td><td>(βB(α,β))<sup>1/β</sup> t<sup>-1/β</sup> - 2(1-α)(βB(α,β))<sup>2/β</sup> t<sup>-2/β</sup> / β(β+1)</td><td>-1/β</td><td>2(1-α)(βB(α,β))<sup>1/β</sup> t<sup>-1/β</sup> / β(β+1)</td></tr><tr><td>Beta (α = 1)</td><td>(1 - t)<sup>β</sup>, t ∈ [0, 1]</td><td>-1/β</td><td>1/β t<sup>-1/β</sup></td><td>-∞</td><td>0</td></tr><tr><td>Kumaraswamy (α ≠ 1)</td><td>(1 - t<sup>α</sup>)<sup>β</sup>, t ∈ [0, 1]</td><td>-1/β</td><td>1/αβ t<sup>-1/β</sup> - 1/αβ (1 - 1) t<sup>-2/β</sup></td><td>-1/β</td><td>-1/β</td></tr><tr><td>Reverse-Burr</td><td>(1 + (1 - t)<sup>-β/α</sup>)<sup>-1/β</sup>, t < 1</td><td>-α</td><td>αt<sup>-α</sup> + α/β (α + β)t<sup>-β-α</sup></td><td>-β</td><td>- (α + β)t<sup>-β</sup></td></tr><tr><td>Reverse-Fréchet</td><td>1 - exp(- (1 - t)<sup>α</sup>), t < 1</td><td>-1/α</td><td>1/α t<sup>-1/α</sup> + 1/2α (1/α + 1) t<sup>-1/α-1</sup></td><td>-1</td><td>-1/2 (1/α + 1) t<sup>-1</sup></td></tr><tr><td>Triangular</td><td>1 - t<sup>2</sup>/2, t ∈ [0, 1] and (2-t)<sup>2</sup>/2, t ∈ [1, 2]</td><td>-1/2</td><td>(2t)<sup>-1/2</sup></td><td>-∞</td><td>0</td></tr></tbody></table>
Stream, river, or lake overflows in 2012

<figure><img src="image_3.png" /></figure>

Stream, river, or lake overflows in 2017

<figure><img src="image_4.png" /></figure>

Figure 1: OpenFEMA Flood Insurance data – Locations of flood claims consecutive to a stream, river, or lake overflow in 2012 (top) and 2017 (bottom) in the United States. A red cross denotes a claim in an area rated A (Special Flood with no Base Flood Elevation) and a green cross denotes a claim in an area rated B (Moderate Flood from primary water source).
<figure><img src="image_5.png" /><figcaption>Figure 2: OpenFEMA Flood Insurance data – Left panels: Histograms of the claims. Middle panels: Extreme value index estimates $\hat{\gamma}_n^{\text{Mom}}$ (solid black) with 95% asymptotic Gaussian confidence intervals (black ribbon), and Hill estimates (solid red) as functions of $k$. Right panels: Expected Shortfall estimates $\hat{\text{ES}}_n^*(0.995)$ and $\tilde{\text{ES}}_n^*(0.995)$ (solid blue and green, respectively) with 95% asymptotic confidence intervals $\hat{I}_4^*(\alpha)$ (blue ribbon) and $\tilde{I}_4^*(\alpha)$ (green ribbon) for $\alpha = 0.05$, and Weissman-Hill Expected Shortfall estimates $\hat{\text{ES}}_n^W(0.995)$ (solid red) and $\tilde{\text{ES}}_n^W(0.995)$ (dotted red) as functions of $k$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 3: Daily loss returns data – Top panels: CAC 40 data, bottom panels: FTSE 100 data. Panels (A) and (E): Kwiatkowski-Phillips-Schmidt-Shin test of stationarity. Panels (B) and (F): Ljung-Box test of independence on the residuals and their squares. Panels (C) and (G): Lagrange Multiplier test for ARCH effects. Panels (D) and (H): Sign bias test for leverage effects.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 4: Daily loss returns data – Top panels: CAC 40 data, bottom panels: FTSE 100 data. Panels (A) and (E): Estimates $\hat{\gamma}_{n}^{\text{Mom}}$ for the ARMA-GARCH residuals over successive rolling windows of length $n = 1,500$. Panels (B) and (F): Over the highlighted rolling window (26 August 2010-12 July 2016 for CAC 40, 2 July 2007-10 June 2013 for FTSE 100), estimate $\hat{\gamma}_{n}^{\text{Mom}}$ (red curve) with 95% asymptotic Gaussian confidence intervals (black curves). Panels (C) and (G): Over this same window, estimate $\hat{\text{ES}}_{n}^{*}(\tau_{n}')$ (solid orange) of ES($\tau_{n}'$) at level $\tau_{n}' = 1 - 1/1,500$ with 95% asymptotic confidence intervals $\hat{I}_{4}^{*}(\alpha)$ (dashed orange), along with the estimate $\hat{\text{ES}}_{n}^{*}(\tau_{n}')$ and $\hat{I}_{4}^{*}(\alpha)$ (blue curves) with $\alpha = 0.05$. Panels (D) and (H): Next-day forecasts (black curve) $\hat{\text{ES}}_{n}^{*}(\tau_{n}')$ in panel (D) and $\hat{\text{ES}}_{n}^{*}(\tau_{n}')$ in panel (H) of ES($\tau_{n}'$) for the original loss returns computed over each window, with 95% lower (blue curve) and upper (magenta curve) asymptotic confidence bounds, along with the realization of the future observation (red curve).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5: (Top panels) Overlay of QQ-plots from all rolling windows (grey points) for the GPD fitted to exceedances of the residuals over their 90th percentile threshold, along with the QQ-plot from the final rolling window (red circles), for the CAC 40 dataset (left panel) and FTSE 100 dataset (right panel). (Bottom panels) Overlay of QQ-plots from all rolling windows (grey points) for the GPD fitted to exceedances of the log-residuals over their 90th percentile threshold, along with the QQ-plot from the final rolling window (red circles), for the CAC 40 dataset (left panel) and FTSE 100 dataset (right panel).</figcaption></figure>
# Appendix to the paper “Corrected inference about the extreme Expected Shortfall in the general Max-Domain of Attraction”

Abdelaati Daouia, Gilles Stupfler & Antoine Usseglio-Carleve

This appendix contains all necessary auxiliary results, their proofs and the proofs of our main results, and provides extra finite-sample results about our simulation study. Unless specified otherwise, we denote throughout by $F$ and $\bar{F} = 1 - F$ the distribution function and survival function of the random variable of interest $X$. The associated quantile function is $q: \tau \mapsto \inf\{x \in \mathbb{R} | F(x) \ge \tau\}$, and the tail quantile function is $U: t \mapsto q(1 - 1/t)$, for $t > 1$.

## A Auxiliary results and proofs

Our first result gives simple sufficient conditions under which condition $C_2(\gamma, a, \rho, A)$ holds. They are formulated in terms of asymptotic expansions of the tail quantile function.

**Lemma A.1** (Sufficient conditions for $C_2(\gamma, a, \rho, A)$). Let $U$ be the tail quantile function of the random variable $X$.

* (Heavy tails) Assume that

$$U(t) = t^{\alpha}(C + Dt^{-\beta} + o(t^{-\beta})) \text{ as } t \to \infty$$

with $\alpha, \beta, C > 0$ and $D \neq 0$.

1. If $\alpha \neq \beta$, then condition $C_2(\gamma, a, \rho, A)$ holds with $\gamma = \alpha, \rho = -\beta$,

$$A(t) = -\frac{\beta(\alpha - \beta)}{\alpha} \frac{D}{C} t^{-\beta} \quad \text{and} \quad a(t) = \alpha C t^{\alpha} \left(1 - \frac{1}{\beta} A(t)\right) = \alpha C t^{\alpha} \left(1 + \frac{\alpha - \beta}{\alpha} \frac{D}{C} t^{-\beta}\right).$$

2. If $\alpha = \beta$ and moreover

$$U(t) = t^{\alpha}(C + Dt^{-\alpha} + D't^{-\alpha-\alpha'} + o(t^{-\alpha-\alpha'})) \text{ as } t \to \infty$$

with $\alpha' > 0$ and $D' \neq 0$, then condition $C_2(\gamma, a, \rho, A)$ holds with $\gamma = \alpha, \rho = -\alpha - \alpha'$,

$$A(t) = \frac{\alpha'(\alpha + \alpha')}{\alpha} \frac{D'}{C} t^{-\alpha-\alpha'} \quad \text{and} \quad a(t) = \alpha C t^{\alpha} \left(1 - \frac{\alpha' D'}{\alpha C} t^{-\alpha-\alpha'}\right).$$

* (Light tails) Assume that

$$U(t) = \log^{\alpha}(t)(C + Dt^{-\beta} \log^{\delta}(t) + o(t^{-\beta} \log^{\delta}(t))) \text{ as } t \to \infty$$

with $\alpha, \beta, C > 0$ and $D, \delta \in \mathbb{R}$.
1. If $\alpha = 1$ and $D \neq 0$, then condition $C_2(\gamma, a, \rho, A)$ holds with $\gamma = 0, \rho = -\beta$,

$$A(t) = \beta^2 \frac{D}{C} t^{-\beta} \log^{1+\delta}(t) \quad \text{and} \quad a(t) = C \left(1 - \frac{1}{\beta}A(t)\right) = C \left(1 - \beta \frac{D}{C} t^{-\beta} \log^{1+\delta}(t)\right).$$

2. If $\alpha \neq 1$, then condition $C_2(\gamma, a, \rho, A)$ holds with $\gamma = \rho = 0$,

$$A(t) = \frac{\alpha - 1}{\log(t)} \quad \text{and} \quad a(t) = \alpha C \log^{\alpha-1}(t).$$

• (Short tails) Assume that $U(\infty) < +\infty$ and

$$U(\infty) - U(t) = t^{-\alpha}(C + Dt^{-\beta} + o(t^{-\beta})) \quad \text{as } t \to \infty$$

with $\alpha, \beta, C > 0$ and $D \neq 0$. Then condition $C_2(\gamma, a, \rho, A)$ holds with $\gamma = -\alpha, \rho = -\beta$,

$$A(t) = -\frac{\beta(\alpha + \beta)}{\alpha} \frac{D}{C} t^{-\beta} \quad \text{and} \quad a(t) = \alpha C t^{-\alpha} \left(1 - \frac{1}{\beta}A(t)\right) = \alpha C t^{-\alpha} \left(1 + \frac{\alpha + \beta}{\alpha} \frac{D}{C} t^{-\beta}\right).$$

*Proof of Lemma A.1.* We prove each statement separately.

(Heavy tails) Assume that $U(t) = t^{\alpha}(C + Dt^{-\beta} + o(t^{-\beta}))$ as $t \to \infty$, with $\alpha, \beta, C > 0$ and $D \neq 0$. If $\alpha \neq \beta$, set

$$A(t) = -\frac{\beta(\alpha - \beta)}{\alpha} \frac{D}{C} t^{-\beta} \quad \text{and} \quad a(t) = \alpha C t^{\alpha} \left(1 + \frac{\alpha - \beta}{\alpha} \frac{D}{C} t^{-\beta}\right).$$

Then

$$\frac{U(tx) - U(t)}{a(t)} - \frac{x^{\alpha} - 1}{\alpha} = -\frac{\alpha - \beta}{\alpha} \frac{D}{C} t^{-\beta} \frac{x^{\alpha} - 1}{\alpha} + \frac{\alpha - \beta}{\alpha} \frac{D}{C} t^{-\beta} \frac{x^{\alpha-\beta} - 1}{\alpha - \beta} + o(t^{-\beta}) \\ = A(t) \int_{1}^{x} s^{\alpha-1} \left( \int_{1}^{s} u^{-\beta-1} du \right) ds + o(|A(t)|)$$

as announced. If in fact $\alpha = \beta$ and

$$U(t) = t^{\alpha}(C + Dt^{-\alpha} + D't^{-\alpha-\alpha'} + o(t^{-\alpha-\alpha'})) = D + t^{\alpha}(C + D't^{-\alpha-\alpha'} + o(t^{-\alpha-\alpha'}))$$

as $t \to \infty$, then $U(t) - D = t^{\alpha}(C + D't^{-\alpha-\alpha'} + o(t^{-\alpha-\alpha'}))$ as $t \to \infty$ and the above calculation applies with $D$ replaced by $D'$ and $\beta$ replaced by $\alpha + \alpha'$.

(Light tails) Assume that $U(t) = \log^{\alpha}(t)(C + Dt^{-\beta}\log^{\delta}(t) + o(t^{-\beta}\log^{\delta}(t)))$ as $t \to \infty$ with $\alpha, \beta, C > 0$ and $D, \delta \in \mathbb{R}$. If $\alpha = 1$, then for any $x > 0$,

$$\begin{aligned} U(tx) - U(t) &= C \log(x) + D(x^{-\beta} - 1) t^{-\beta} \log^{1+\delta}(t) + o(t^{-\beta} \log^{1+\delta}(t)) \\ &= C \left(1 - \beta \frac{D}{C} t^{-\beta} \log^{1+\delta}(t)\right) \log(x) \\ &\quad + C \times \beta^2 \frac{D}{C} t^{-\beta} \log^{1+\delta}(t) \int_{1}^{x} s^{-1} \left( \int_{1}^{s} u^{-\beta-1} du \right) ds + o(t^{-\beta} \log^{1+\delta}(t)). \end{aligned}$$
This proves the result in this case. If in fact $\alpha \neq 1$, then for any $x > 0$,

$$U(tx) = C \log^{\alpha}(t) \left( 1 + \alpha \frac{\log(x)}{\log(t)} + \frac{\alpha(\alpha - 1) \log^2(x)}{2 \log^2(t)} + o\left(\frac{1}{\log^2(t)}\right) \right)$$

and therefore

$$\begin{aligned} \frac{U(tx) - U(t)}{\alpha C \log^{\alpha-1}(t)} &= \log(x) + \frac{\alpha-1}{\log(t)} \times \frac{1}{2} \log^2(x) + o\left(\frac{1}{\log(t)}\right) \\ &= \log(x) + \frac{\alpha-1}{\log(t)} \int_1^x s^{-1} \left( \int_1^s u^{-1} du \right) ds + o\left(\frac{1}{\log(t)}\right) \end{aligned}$$

as required.

(Short tails) Assume that $U(\infty) < +\infty$ and $U(\infty) - U(t) = t^{-\alpha}(C + Dt^{-\beta} + o(t^{-\beta}))$ as $t \to \infty$ with $\alpha, \beta, C > 0$ and $D \neq 0$. Set

$$A(t) = -\frac{\beta(\alpha + \beta)}{\alpha} \frac{D}{C} t^{-\beta} \quad \text{and} \quad a(t) = \alpha C t^{-\alpha} \left(1 - \frac{1}{\beta} A(t)\right) = \alpha C t^{-\alpha} \left(1 + \frac{\alpha + \beta D}{\alpha C} t^{-\beta}\right).$$

Then

$$\begin{aligned} \frac{U(tx) - U(t)}{a(t)} - \frac{x^{-\alpha} - 1}{-\alpha} &= -\frac{\alpha + \beta D}{\alpha C} t^{-\beta} \frac{x^{-\alpha} - 1}{-\alpha} + \frac{\alpha + \beta D}{\alpha C} t^{-\beta} \frac{x^{-\alpha-\beta} - 1}{-\alpha - \beta} + o(t^{-\beta}) \\ &= A(t) \int_{1}^{x} s^{-\alpha-1} \left( \int_{1}^{s} u^{-\beta-1} du \right) ds + o(|A(t)|) \end{aligned}$$

as announced.

<figure><img src="image_9.png" /></figure>

The next two lemmas are dedicated to checking condition $C_2(\gamma, a, \rho, A)$ for the Gamma distribution with unit scale and for the standard Gaussian distribution. This is done by computing asymptotic expansions of the tail quantile functions of these distributions that are of independent interest; in particular, we obtain approximations that are more precise than those proven in Sections 4.1 and 4.3 in Fung and Seneta (2018).

**Lemma A.2** (Asymptotic expansion of the Gamma quantile function). Let U be the tail quantile function of the Gamma distribution with shape parameter $\alpha \neq 1$, whose distribution function is

$$F(x) = \int_{0}^{x} \frac{1}{\Gamma(\alpha)} t^{\alpha-1} e^{-t} dt \quad \text{for } x > 0.$$

Then

$$U(t) = \log(t) \left( 1 + (\alpha - 1) \frac{\log \log(t)}{\log(t)} - \frac{\log \Gamma(\alpha)}{\log(t)} + (\alpha - 1)^2 \frac{\log \log(t)}{\log^2(t)} - \frac{(\alpha - 1)(\log \Gamma(\alpha) - 1)}{\log^2(t)} + o\left(\frac{1}{\log^2(t)}\right) \right) \quad \text{as } t \to \infty.$$

In particular, the Gamma distribution satisfies condition $C_2(\gamma, a, \rho, A)$, with $\gamma = \rho = 0$,

$$a(t) = 1 + \frac{\alpha - 1}{\log(t)} - (\alpha - 1)^2 \frac{\log \log(t)}{\log^2(t)} + (\alpha - 1) \frac{\log \Gamma(\alpha) + \alpha - 2}{\log^2(t)} \quad \text{and} \quad A(t) = - \frac{\alpha - 1}{\log^2(t)}.$$
Proof of Lemma A.2. The key is to note that for any $\beta > 0$ and any $x > 0$,

$$ \int_x^\infty t^{\beta-1} e^{-t} dt = x^{\beta-1} e^{-x} + (\beta-1) \int_x^\infty t^{\beta-2} e^{-t} dt $$

by an integration by parts, and since

$$ 0 \le \int_x^\infty t^{\beta-2} e^{-t} dt \le \frac{1}{x} \int_x^\infty t^{\beta-1} e^{-t} dt, $$

one has

$$ \forall \beta > 0, \quad \int_x^\infty t^{\beta-1} e^{-t} dt = x^{\beta-1} e^{-x} \left( 1 + O\left(\frac{1}{x}\right) \right) \quad \text{as } x \to \infty. $$

Consequently

$$ \begin{aligned} 1 - F(x) &= \int_x^\infty \frac{1}{\Gamma(\alpha)} t^{\alpha-1} e^{-t} dt \\ &= \frac{x^{\alpha-1} e^{-x}}{\Gamma(\alpha)} + \frac{(\alpha-1)x^{\alpha-2} e^{-x}}{\Gamma(\alpha)} + \frac{(\alpha-1)(\alpha-2)}{\Gamma(\alpha)} \int_x^\infty t^{\alpha-3} e^{-t} dt \\ &= \frac{x^{\alpha-1} e^{-x}}{\Gamma(\alpha)} \left( 1 + \frac{\alpha-1}{x} + \frac{(\alpha-1)(\alpha-2)}{x^2} + O\left(\frac{1}{x^3}\right) \right) \quad \text{as } x \to \infty. \end{aligned} $$

The Gamma distribution is unbounded to the right, so $U(t) \to \infty$ as $t \to \infty$, and has a continuous distribution function, so $1 - F(U(t)) = 1/t$ for any $t > 1$. Plugging $x = U(t)$ in the above asymptotic expansion leads in particular to

$$ \log(t) = U(t) - (\alpha - 1) \log(U(t)) + \log \Gamma(\alpha) - \frac{\alpha - 1}{U(t)} + O\left(\frac{1}{(U(t))^2}\right) \quad \text{as } t \to \infty. \quad (A.1) $$

We repeatedly use (A.1) in order to get the successive terms appearing in the desired asymptotic expansion of $U(t)$.

**Asymptotic equivalent of $U(t)$:** The leading term in the right-hand side of (A.1) is obviously $U(t)$, so $U(t) \sim \log(t)$.

**First term in the asymptotic expansion of $U(t)$:** Write $U(t) = \log(t)(1 + \varepsilon_1(t))$, with $\varepsilon_1(t) \to 0$ as $t \to \infty$. Then (A.1) yields

$$ 0 = \log(t)\varepsilon_1(t) - (\alpha - 1) \log \log(t) (1 + o(1)) $$

and therefore $\varepsilon_1(t) \sim (\alpha - 1) \frac{\log \log(t)}{\log(t)}$.

**Second term in the asymptotic expansion of $U(t)$:** Write

$$ U(t) = \log(t) \left( 1 + (\alpha - 1) \frac{\log \log(t)}{\log(t)} (1 + \varepsilon_2(t)) \right), $$

with $\varepsilon_2(t) \to 0$ as $t \to \infty$. We find, using (A.1), that

$$ 0 = (\alpha - 1) \log \log(t) \varepsilon_2(t) + \log \Gamma(\alpha) + o(1) $$
and therefore $\varepsilon_2(t) \sim -\frac{\log \Gamma(\alpha)}{\alpha-1} \frac{1}{\log \log(t)}$.

**Third term in the asymptotic expansion of $U(t)$:** Write

$$U(t) = \log(t) \left( 1 + (\alpha - 1) \frac{\log \log(t)}{\log(t)} - \frac{\log \Gamma(\alpha)}{\log(t)} (1 + \varepsilon_3(t)) \right),$$

with $\varepsilon_3(t) \to 0$ as $t \to \infty$. Then $1/U(t) \sim 1/\log(t) = o(\log \log(t)/\log(t))$ and

$$\log(U(t)) = \log \log(t) + (\alpha - 1) \frac{\log \log(t)}{\log(t)} + o\left(\frac{\log \log(t)}{\log(t)}\right).$$

Plugging these two relationships into (A.1) results in

$$0 = \log \Gamma(\alpha) \varepsilon_3(t) + (\alpha - 1)^2 \frac{\log \log(t)}{\log(t)} (1 + o(1))$$

and therefore $\varepsilon_3(t) \sim -\frac{(\alpha-1)^2 \log \log(t)}{\log \Gamma(\alpha) \log(t)}$.

**Fourth term in the asymptotic expansion of $U(t)$:** Write finally

$$U(t) = \log(t) \left( 1 + (\alpha - 1) \frac{\log \log(t)}{\log(t)} - \frac{\log \Gamma(\alpha)}{\log(t)} + (\alpha - 1)^2 \frac{\log \log(t)}{\log^2(t)} (1 + \varepsilon_4(t)) \right),$$

with $\varepsilon_4(t) \to 0$ as $t \to \infty$. One has $1/U(t) \sim 1/\log(t)$ and

$$\log(U(t)) = \log \log(t) + (\alpha - 1) \frac{\log \log(t)}{\log(t)} - \frac{\log \Gamma(\alpha)}{\log(t)} + o\left(\frac{1}{\log(t)}\right).$$

Plugging these two relationships into (A.1) results in

$$0 = (\alpha - 1) \log \log(t) \varepsilon_4(t) + \log \Gamma(\alpha) - 1 + o(1)$$

and therefore $\varepsilon_4(t) \sim -\frac{\log \Gamma(\alpha) - 1}{\alpha - 1} \frac{1}{\log \log(t)}$. Hence the desired asymptotic expansion of $U$.

To prove that $U$ satisfies condition $C_2(\gamma, a, \rho, A)$, note that since

$$1 - F(x) = \frac{x^{\alpha-1}e^{-x}}{\Gamma(\alpha)} \left( 1 + \frac{\alpha-1}{x} + \frac{(\alpha-1)(\alpha-2)}{x^2} + O\left(\frac{1}{x^3}\right) \right) \quad \text{as } x \to \infty,$$

one may write the following stronger version of (A.1):

$$\log(t) = U(t) - (\alpha - 1) \log(U(t)) + \log \Gamma(\alpha) \\ - \log \left( 1 + \frac{\alpha - 1}{U(t)} + \frac{(\alpha - 1)(\alpha - 2)}{(U(t))^2} + o\left(\frac{1}{(U(t))^2}\right) \right) \quad \text{as } t \to \infty.$$

In particular, for any $x > 0$,

$$U(tx) - U(t) = \log(x) + (\alpha - 1) \log \left( \frac{U(tx)}{U(t)} \right) \\ + \log \left( \frac{1 + (\alpha - 1)/U(tx) + (\alpha - 1)(\alpha - 2)/(U(tx))^2 + o(1/\log^2(t))}{1 + (\alpha - 1)/U(t) + (\alpha - 1)(\alpha - 2)/(U(t))^2 + o(1/\log^2(t))} \right). \quad (A.2)$$
Since $U(tx)/U(t) \to 1$, this yields, first of all, the convergence $U(tx) - U(t) \to \log(x)$ as $t \to \infty$. Then straightforward calculations based on the asymptotic expansion of $U(t)$ we have previously obtained entail

$$ 
\begin{aligned}
\frac{U(tx)}{\log(tx)} - \frac{U(t)}{\log(t)} &= (\alpha - 1) \left( \frac{\log \log(tx)}{\log(tx)} - \frac{\log \log(t)}{\log(t)} \right) - \left( \frac{\log \Gamma(\alpha)}{\log(tx)} - \frac{\log \Gamma(\alpha)}{\log(t)} \right) + o \left( \frac{1}{\log^2(t)} \right) \\
&= \left( -(\alpha - 1) \frac{\log \log(t)}{\log^2(t)} + \frac{\log \Gamma(\alpha) + \alpha - 1}{\log^2(t)} \right) \log(x) + o \left( \frac{1}{\log^2(t)} \right) \quad (\text{A.3})
\end{aligned}
$$

and, using the fact that $U(t) \sim \log(t)$,

$$ 
\begin{aligned}
& \frac{1 + (\alpha - 1)/U(tx) + (\alpha - 1)(\alpha - 2)/(U(tx))^2 + o(1/\log^2(t))}{1 + (\alpha - 1)/U(t) + (\alpha - 1)(\alpha - 2)/(U(t))^2 + o(1/\log^2(t))} - 1 \\
&= (\alpha - 1) \left( \frac{1}{U(tx)} - \frac{1}{U(t)} \right) + o \left( \frac{1}{\log^2(t)} \right) \\
&= -(\alpha - 1) \frac{\log(x)}{\log^2(t)} + o \left( \frac{1}{\log^2(t)} \right).
\end{aligned}
\quad (\text{A.4})
$$

Combine (A.2), (A.3) and (A.4) and the fact that $U(t)/\log(t) = 1 + O(\log \log(t)/\log(t))$ to get

$$ 
\begin{aligned}
U(tx) - U(t) &= \log(x) + (\alpha - 1) \log \left( 1 + \frac{\log(x)}{\log(t)} \right) \\
&\quad + (\alpha - 1) \log \left( 1 + \frac{\log(t)}{U(t)} \left( \frac{U(tx)}{\log(tx)} - \frac{U(t)}{\log(t)} \right) \right) \\
&\quad + \log \left( \frac{1 + (\alpha - 1)/U(tx) + (\alpha - 1)(\alpha - 2)/(U(tx))^2 + o(1/\log^2(t))}{1 + (\alpha - 1)/U(t) + (\alpha - 1)(\alpha - 2)/(U(t))^2 + o(1/\log^2(t))} \right) \\
&= \log(x) \left( 1 + \frac{\alpha - 1}{\log(t)} - (\alpha - 1)^2 \frac{\log \log(t)}{\log^2(t)} + (\alpha - 1) \frac{\log \Gamma(\alpha) + \alpha - 2}{\log^2(t)} \right) \\
&\quad - \frac{\alpha - 1}{\log^2(t)} \times \frac{1}{2} \log^2(x) + o \left( \frac{1}{\log^2(t)} \right).
\end{aligned}
$$

The conclusion directly follows. $\square$

**Lemma A.3 (Asymptotic expansion of the Gaussian quantile function).** Let $U$ be the tail quantile function of the standard Gaussian distribution. Then

$$ U(t) = \sqrt{2 \log(t)} \left( 1 - \frac{\log \log(t)}{4 \log(t)} - \frac{\log(4\pi)}{4 \log(t)} - \frac{\log^2(\log(t))}{32 \log^2(t)} - \frac{1}{8} \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{\log \log(t)}{\log^2(t)} \right. \\ \left. - \frac{1}{4} \left( \frac{1}{8} \log^2(4\pi) - \frac{1}{2} \log(4\pi) + 1 \right) \frac{1}{\log^2(t)} + o \left( \frac{1}{\log^2(t)} \right) \right) \quad \text{as } t \to \infty. $$

In particular, the standard Gaussian distribution satisfies condition $C_2(\gamma, a, \rho, A)$, with $\gamma = \rho = 0$,

$$ a(t) = \frac{1}{\sqrt{2 \log(t)}} \left( 1 + \frac{\log \log(t)}{4 \log(t)} + \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{1}{2 \log(t)} \right) \quad \text{and} \quad A(t) = -\frac{1}{2 \log(t)}. $$
Proof of Lemma A.3. Let $\Phi$ be the distribution function of the standard Gaussian distribution. The starting point is to integrate by parts twice in order to get

$$1 - \Phi(x) = \int_{x}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} du = \frac{1}{\sqrt{2\pi}} \frac{e^{-x^2/2}}{x} \left(1 - \frac{1}{x^2}\right) + \int_{x}^{\infty} \frac{3}{\sqrt{2\pi}} e^{-u^2/2} \frac{du}{u^4} \\ = \frac{1}{\sqrt{2\pi}} \frac{e^{-x^2/2}}{x} \left(1 - \frac{1}{x^2} + O\left(\frac{1}{x^4}\right)\right) \text{ as } x \to \infty.$$

Consequently

$$-\log(1 - \Phi(x)) = \frac{x^2}{2} + \log(x) + \frac{1}{2}\log(2\pi) + \frac{1}{x^2} + O\left(\frac{1}{x^4}\right) \text{ as } x \to \infty.$$

The standard Gaussian distribution is unbounded to the right, so $U(t) \to \infty$ as $t \to \infty$, and $\Phi$ is continuous, so $1 - \Phi(U(t)) = 1/t$ for any $t > 1$. Plugging $x = U(t)$ in the above asymptotic expansion leads to

$$\log(t) = \frac{(U(t))^2}{2} + \log(U(t)) + \frac{1}{2}\log(2\pi) + \frac{1}{(U(t))^2} + O\left(\frac{1}{(U(t))^4}\right) \text{ as } t \to \infty. \quad (A.5)$$

As in the proof of Lemma A.2, we repeatedly use (A.5) in order to get the successive terms appearing in the desired asymptotic expansion of $U(t)$.

**Asymptotic equivalent of $U(t)$:** The leading term in the right-hand side of (A.5) is obviously $(U(t))^2/2$, which yields $U(t) \sim \sqrt{2\log(t)}$.

**First term in the asymptotic expansion of $U(t)$:** Writing $U(t) = \sqrt{2\log(t)}(1 + \varepsilon_1(t))$, with $\varepsilon_1(t) \to 0$ as $t \to \infty$, one obtains from (A.5) that

$$0 = 2\log(t)\varepsilon_1(t)(1 + o(1)) + \frac{1}{2}\log\log(t) + O(1)$$

and therefore $\varepsilon_1(t) \sim -\frac{\log\log(t)}{4\log(t)}$.

**Second term in the asymptotic expansion of $U(t)$:** Writing

$$U(t) = \sqrt{2\log(t)} \left( 1 - \frac{\log\log(t)}{4\log(t)} (1 + \varepsilon_2(t)) \right),$$

with $\varepsilon_2(t) \to 0$ as $t \to \infty$, we get using (A.5)

$$0 = -\frac{1}{2}\log\log(t)\varepsilon_2(t) + \frac{1}{2}\log(4\pi) + o(1)$$

and therefore $\varepsilon_2(t) \sim \frac{\log(4\pi)}{\log\log(t)}$.

**Third term in the asymptotic expansion of $U(t)$:** Write

$$U(t) = \sqrt{2\log(t)} \left( 1 - \frac{\log\log(t)}{4\log(t)} - \frac{\log(4\pi)}{4\log(t)}(1 + \varepsilon_3(t)) \right),$$
with $\varepsilon_3(t) \to 0$ as $t \to \infty$. Then

$$ \frac{(U(t))^2}{2} - \log(t) = -\frac{1}{2} \log \log(t) - \frac{1}{2} \log(4\pi)(1 + \varepsilon_3(t)) + \frac{\log^2(\log(t))}{16 \log(t)}(1 + o(1)) $$

and

$$ \log(U(t)) = \frac{1}{2} \log \log(t) + \frac{1}{2} \log(2) + o\left(\frac{\log^2(\log(t))}{\log(t)}\right). $$

Plugging these two asymptotic expansions into (A.5) results in

$$ 0 = -\frac{1}{2} \log(4\pi)\varepsilon_3(t) + \frac{\log^2(\log(t))}{16 \log(t)}(1 + o(1)) $$

and therefore $\varepsilon_3(t) \sim \frac{\log^2(\log(t))}{8 \log(4\pi) \log(t)}$.

**Fourth term in the asymptotic expansion of $U(t)$:** Write

$$ U(t) = \sqrt{2 \log(t)} \left( 1 - \frac{\log \log(t)}{4 \log(t)} - \frac{\log(4\pi)}{4 \log(t)} - \frac{\log^2(\log(t))}{32 \log^2(t)} (1 + \varepsilon_4(t)) \right), $$

with $\varepsilon_4(t) \to 0$ as $t \to \infty$. One has

$$ \frac{(U(t))^2}{2} - \log(t) = -\frac{1}{2} \log \log(t) - \frac{1}{2} \log(4\pi) - \frac{\log^2(\log(t))}{16 \log(t)}\varepsilon_4(t) + \frac{\log(4\pi) \log \log(t)}{8 \log(t)}(1 + o(1)) $$

and

$$ \log(U(t)) = \frac{1}{2} \log \log(t) + \frac{1}{2} \log(2) - \frac{\log \log(t)}{4 \log(t)}(1 + o(1)). $$

Plugging these two asymptotic expansions into (A.5) results in

$$ 0 = -\frac{\log^2(\log(t))}{16 \log(t)}\varepsilon_4(t) + \frac{1}{4}\left(\frac{1}{2}\log(4\pi) - 1\right)\frac{\log \log(t)}{\log(t)} $$

and therefore $\varepsilon_4(t) \sim 4(\frac{1}{2}\log(4\pi) - 1)\frac{1}{\log \log(t)}$.

**Fifth term in the asymptotic expansion of $U(t)$:** Write finally

$$ U(t) = \sqrt{2 \log(t)} \left( 1 - \frac{\log \log(t)}{4 \log(t)} - \frac{\log(4\pi)}{4 \log(t)} - \frac{\log^2(\log(t))}{32 \log^2(t)} \right. \\ \left. - \frac{1}{8} \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{\log \log(t)}{\log^2(t)} (1 + \varepsilon_5(t)) \right) $$

with $\varepsilon_5(t) \to 0$ as $t \to \infty$. One has

$$ \frac{(U(t))^2}{2} - \log(t) = -\frac{1}{2} \log \log(t) - \frac{1}{2} \log(4\pi) - \frac{1}{4} \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{\log \log(t)}{\log^2(t)} \varepsilon_5(t) \\ + \frac{\log \log(t)}{4 \log(t)} + \frac{\log^2(4\pi)}{16 \log(t)}(1 + o(1)) $$
and

$$ \log(U(t)) = \frac{1}{2} \log \log(t) + \frac{1}{2} \log(2) - \frac{\log \log(t)}{4 \log(t)} - \frac{\log(4\pi)}{4 \log(t)} (1 + o(1)). $$

Plugging the two above asymptotic expansions along with the asymptotic equivalent $1/(U(t))^2 \sim 1/(2 \log(t))$ into (A.5) yields

$$ 0 = -\frac{1}{4} \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{\log \log(t)}{\log(t)} \varepsilon_5(t) + \frac{1}{2} \left( \frac{1}{8} \log^2(4\pi) - \frac{1}{2} \log(4\pi) + 1 \right) \frac{1}{\log(t)} (1 + o(1)) $$

and therefore

$$ \frac{1}{8} \left( \frac{1}{2} \log(4\pi) - 1 \right) \varepsilon_5(t) \sim \frac{1}{4} \left( \frac{1}{8} \log^2(4\pi) - \frac{1}{2} \log(4\pi) + 1 \right) \frac{1}{\log \log(t)}. $$

This results in the desired asymptotic expansion of $U$.

To prove that $U$ satisfies condition $C_2(\gamma, a, \rho, A)$, note that the asymptotic expansion we have just shown gives, for any $x > 0$,

$$ \begin{aligned} U(tx) &= \sqrt{2 \log(t)} \left( 1 + \frac{\log(x)}{2 \log(t)} - \frac{\log^2(x)}{8 \log^2(t)} + o\left(\frac{1}{\log^2(t)}\right) \right) \\ &\quad \times \left( 1 - \frac{\log \log(t)}{4 \log(t)} \left\{ 1 - \frac{\log(x)}{\log(t)} \right\} - \frac{\log(x)}{4 \log^2(t)} - \frac{\log(4\pi)}{4 \log(t)} \left\{ 1 - \frac{\log(x)}{\log(t)} \right\} - \frac{\log^2(\log(t))}{32 \log^2(t)} \right. \\ &\quad \left. - \frac{1}{8} \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{\log \log(t)}{\log^2(t)} - \frac{1}{4} \left( \frac{1}{8} \log^2(4\pi) - \frac{1}{2} \log(4\pi) + 1 \right) \frac{1}{\log^2(t)} (1 + o(1)) \right) \\ &= U(t) + \sqrt{2 \log(t)} \left( \frac{\log(x)}{2 \log(t)} + \frac{\log \log(t)}{8 \log^2(t)} \log(x) + \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{\log(x)}{4 \log^2(t)} \right. \\ &\quad \left. - \frac{\log^2(x)}{8 \log^2(t)} + o\left(\frac{1}{\log^2(t)}\right) \right) \end{aligned} $$

as $t \to \infty$. Consequently

$$ U(tx) - U(t) = \frac{1}{\sqrt{2 \log(t)}} \left( \left\{ 1 + \frac{\log \log(t)}{4 \log(t)} + \left( \frac{1}{2} \log(4\pi) - 1 \right) \frac{1}{2 \log(t)} \right\} \log(x) \right. \\ \left. - \frac{1}{2 \log(t)} \times \frac{1}{2} \log^2(x) + o\left(\frac{1}{\log(t)}\right) \right) $$

from which the result follows immediately. $\square$

We next introduce notation linked to the second-order condition $C_2(\gamma, a, \rho, A)$ that shall be used several times in subsequent proofs. If this condition holds, define

$$ a_{\star}(t) = \begin{cases} a(t) \left(1 - \frac{1}{\rho}A(t)\right), & \rho < 0, \\ a(t) \left(1 - \frac{1}{\gamma}A(t)\right), & \gamma \neq 0, \rho = 0, \\ a(t), & \gamma = 0, \rho = 0, \end{cases} \quad \text{and} \quad A_{\star}(t) = \begin{cases} \frac{1}{\rho}A(t), & \rho < 0, \\ A(t), & \rho = 0 \end{cases} \quad (A.6) $$
as well as

$$ \Psi_{\gamma, \rho}(x) = \begin{cases} \frac{x^{\gamma+\rho}-1}{\gamma+\rho}, & \gamma+\rho \neq 0, \rho < 0, \\ \log(x), & \gamma+\rho = 0, \rho < 0, \\ \frac{1}{\gamma}x^{\gamma}\log(x), & \gamma \neq 0, \rho = 0, \\ \frac{1}{2}\log^2(x), & \gamma = 0, \rho = 0. \end{cases} \qquad (A.7) $$

Then, up to replacing $a_*$ and $A_*$ by suitable functions $a_{\star,0}$ and $A_{\star,0}$ such that $a_{\star,0}(t) = a_*(t)(1+o(|A(t)|))$ and $A_{\star,0}(t) = A_*(t)(1+o(1))$ as $t \to \infty$, Theorem 2.3.6 p.46 in de Haan and Ferreira (2006) guarantees, for any $\delta, \varepsilon > 0$, that $t, tx$ can be chosen so large that

$$ \left| \frac{U(tx) - U(t)}{a_{\star}(t)} - \int_{1}^{x} s^{\gamma-1}ds - A_{\star}(t)\Psi_{\gamma, \rho}(x) \right| \leq \varepsilon |A(t)|x^{\gamma+\rho} \max(x^{\delta}, x^{-\delta}). \quad (A.8) $$

Note also that $a_*$ is asymptotically equivalent to $a$ in a neighborhood of infinity and $a_*(t)/U(t)-\gamma_+ = O(|a(t)/U(t)-\gamma_+|) + O(|A(t)|)$ as $t \to \infty$. We now have the material necessary to prove Proposition 1.

*Proof of Proposition 1.* (i) One has

$$ \frac{U(tx) - U(t)}{a(t)} - \int_{1}^{x} s^{\gamma-1}ds = \frac{U(tx) - U(t)}{a_{\star}(t)} - \int_{1}^{x} s^{\gamma-1}ds + \left(\frac{a_{\star}(t)}{a(t)} - 1\right) \int_{1}^{x} s^{\gamma-1}ds \\ + \left(\frac{a_{\star}(t)}{a(t)} - 1\right) \left(\frac{U(tx) - U(t)}{a_{\star}(t)} - \int_{1}^{x} s^{\gamma-1}ds\right). \quad (A.9) $$

Fix $\delta, \varepsilon > 0$ sufficiently small. Inequality (A.8) yields, for $t$ large enough and all $x > 1$,

$$ \left| \left( \frac{a_{\star}(t)}{a(t)} - 1 \right) \left( \frac{U(tx) - U(t)}{a_{\star}(t)} - \int_{1}^{x} s^{\gamma-1}ds \right) \right| \leq \frac{\varepsilon}{3} |A(t)| x^{\gamma_{+} + \delta}. \quad (A.10) $$

Note also the relationship $a_*(t)/a(t) - 1 = c(\gamma, \rho)A(t)(1+o(|A(t)|))$ as $t \to \infty$, where $c(\gamma, \rho) = -1/\rho$ if $\rho < 0$, $-1/\gamma$ if $\rho = 0$ and $\gamma \neq 0$, and $0$ if $\gamma = 0$ and $\rho = 0$. Then, for $t$ large enough and all $x > 1$,

$$ \left| \left( \frac{a_{\star}(t)}{a(t)} - 1 \right) \int_{1}^{x} s^{\gamma-1}ds - c(\gamma, \rho)A(t) \int_{1}^{x} s^{\gamma-1}ds \right| \leq \frac{\varepsilon}{3} |A(t)| x^{\gamma_{+} + \delta}. \quad (A.11) $$

Set finally $d(\rho) = 1/\rho$ if $\rho < 0$ and $1$ if $\rho = 0$, and combine inequality (A.8) again with (A.9), (A.10) and (A.11) to obtain that for $t$ large enough and all $x > 1$,

$$ \left| \frac{U(tx) - U(t)}{a(t)} - \int_{1}^{x} s^{\gamma-1}ds - c(\gamma, \rho)A(t) \int_{1}^{x} s^{\gamma-1}ds - d(\rho)A(t)\Psi_{\gamma, \rho}(x) \right| \leq \varepsilon |A(t)| x^{\gamma_{+} + \delta}. $$

It is then sufficient to prove the identity

$$ c(\gamma, \rho) \int_{1}^{x} s^{\gamma-1}ds + d(\rho)\Psi_{\gamma, \rho}(x) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1}du \right) ds. \quad (A.12) $$
We prove this identity by exhaustion. For this we recall that

$$ \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds = \begin{cases} \frac{1}{\rho} \left( \frac{x^{\gamma+\rho}-1}{\gamma+\rho} - \frac{x^{\gamma}-1}{\gamma} \right) & \text{if } \gamma \neq 0, \rho < 0, \gamma+\rho \neq 0, \\ \frac{1}{\rho} \left( \log(x) - \frac{x^{\gamma}-1}{\gamma} \right) & \text{if } \gamma \neq 0, \rho < 0, \gamma+\rho = 0, \\ \frac{1}{\rho} \left( \frac{x^{\rho}-1}{\rho} - \log(x) \right) & \text{if } \gamma = 0, \rho < 0, \\ \frac{1}{\gamma} \left( x^{\gamma} \log(x) - \frac{x^{\gamma}-1}{\gamma} \right) & \text{if } \gamma \neq 0, \rho = 0, \\ \frac{1}{2} \log^2(x) & \text{if } \gamma = 0, \rho = 0. \end{cases} $$

If $\gamma \neq 0$, $\rho < 0$ and $\gamma + \rho \neq 0$, the left-hand side of (A.12) is

$$ \frac{1}{\rho} \left( \frac{x^{\gamma+\rho}-1}{\gamma+\rho} - \frac{x^{\gamma}-1}{\gamma} \right) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds, $$

and if instead $\gamma + \rho = 0$, the left-hand side of (A.12) is

$$ \frac{1}{\rho} \left( \log(x) - \frac{x^{\gamma}-1}{\gamma} \right) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds. $$

When $\gamma = 0$ and $\rho < 0$, the left-hand side of (A.12) is

$$ \frac{1}{\rho} \left( \frac{x^{\rho}-1}{\rho} - \log(x) \right) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds. $$

If $\gamma \neq 0$ and $\rho = 0$, the left-hand side of (A.12) is

$$ \frac{1}{\gamma} \left( x^{\gamma} \log(x) - \frac{x^{\gamma}-1}{\gamma} \right) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds, $$

and when $\gamma = 0$ and $\rho = 0$, it is

$$ \frac{1}{2} \log^2(x) = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} du \right) ds. $$

Equation (A.12) follows and the proof of (i) is complete.

(ii) Since $X$ has the same distribution as $U(Y)$, where $Y$ has a unit Pareto distribution (with distribution function $y \mapsto 1 - 1/y$, for $y > 1$), one may write

$$ \text{CTE}(\tau) - q(\tau) = \mathbb{E}(U(Y) - U(t) | U(Y) > U(t)) $$

where throughout $t = t(\tau) = (1 - \tau)^{-1} \to \infty$ as $\tau \uparrow 1$. Note also that condition $C_2(\gamma, a, \rho, A)$ is equivalent to the following second-order condition on $\bar{F}$:

$$ \lim_{s \uparrow q(1)} \frac{1}{A(1/\bar{F}(s))} \left( \frac{\bar{F}(s + x a(1/\bar{F}(s)))}{\bar{F}(s)} - Q_{\gamma}(x) \right) = G_{\gamma, \rho}(x), \quad \text{when } 1 + \gamma x > 0, $$
where $Q_γ$ and $G_{γ,ρ}$ are functions that are continuous at 0 with $Q_γ(0) = 1$ and $G_{γ,ρ}(0) = 0$, see Theorem 2.3.8 p.48 in de Haan and Ferreira (2006). This equivalent second-order condition is known to be true locally uniformly in $x$, see Theorem B.3.19 p.401 in de Haan and Ferreira (2006). Taking $s = U(t)$ and $x = εA(t)$ for $|ε|$ arbitrarily small, mimicking the first seven lines of the proof of this Theorem B.3.19 p.401 in de Haan and Ferreira (2006) leads to $t\bar{F}(U(t)) = 1 + o(|A(t)|)$ and then to

$$ 
\begin{aligned}
\text{CTE}(\tau) - q(\tau) &= \frac{\mathbb{E}((U(Y) - U(t))1_{\{Y>1/\bar{F}(U(t))\}})}{\bar{F}(U(t))} \\
&= t \mathbb{E}((U(Y) - U(t))1_{\{Y>1/\bar{F}(U(t))\}})(1 + o(|A(t)|)).
\end{aligned}
$$

Moreover

$$ 
\begin{aligned}
&|\mathbb{E}((U(Y) - U(t))1_{\{Y>1/\bar{F}(U(t))\}}) - \mathbb{E}((U(Y) - U(t))1_{\{Y>t\}})| \\
&\le |U(1/\bar{F}(U(t))) - U(t)| \times \frac{1}{t}|t\bar{F}(U(t)) - 1| = o\left(\frac{a(t)}{t}|A(t)|\right),
\end{aligned}
$$

using the fact that the extended regular variation property satisfied by $U$ holds in fact locally uniformly in $x > 0$, see Theorem B.2.18 p.383 in de Haan and Ferreira (2006), and taking $x = x(t) = 1/(t\bar{F}(U(t))) \to 1$ as $t \to \infty$. This entails

$$ 
\begin{aligned}
\text{CTE}(\tau) - q(\tau) &= t \mathbb{E}((U(Y) - U(t))1_{\{Y>t\}}) + o(a(t)|A(t)|) \\
&= t \int_t^\infty (U(y) - U(t)) \frac{dy}{y^2} + o(a(t)|A(t)|).
\end{aligned}
$$

Since

$$ t \int_t^\infty (U(y) - U(t)) \frac{dy}{y^2} = \frac{1}{1-\tau} \int_\tau^1 (q(s) - q(\tau))ds = \text{ES}(\tau) - q(\tau) $$

this shows that

$$ \frac{\text{CTE}(\tau) - q(\tau)}{a((1-\tau)^{-1})} = \frac{\text{ES}(\tau) - q(\tau)}{a((1-\tau)^{-1})} + o(|A((1-\tau)^{-1})|). $$

The change of variables $y = tx$ then yields

$$ \frac{\text{ES}(\tau) - q(\tau)}{a((1-\tau)^{-1})} = \int_{1}^{\infty} \frac{U(tx) - U(t)}{a(t)} \frac{dx}{x^2} + o(|A(t)|). $$

Combining (i) with condition $γ < 1$, we obtain

$$ 
\begin{aligned}
\frac{\text{ES}(\tau) - q(\tau)}{a((1-\tau)^{-1})} &= \int_{1}^{\infty} \left( \int_{1}^{x} s^{\gamma-1} \mathrm{d}s \right) \frac{\mathrm{d}x}{x^2} \\
&\quad + A(t) \int_{1}^{\infty} \left\{ \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} \mathrm{d}u \right) \mathrm{d}s \right\} \frac{\mathrm{d}x}{x^2} + o(|A(t)|).
\end{aligned}
$$

Use finally the identity

$$ \int_{1}^{\infty} \left\{ \int_{1}^{x} s^{b-1} \left( \int_{1}^{s} u^{c-1} \mathrm{d}u \right) \mathrm{d}s \right\} \frac{\mathrm{d}x}{x^2} = \frac{1}{(1-b)(1-b-c)} $$
valid for $b < 1$ and $b + c < 1$ to conclude the proof of (ii).

(iii) This is a consequence of (ii) and the convergence $a(t)/U(t) \to \gamma_+ = \gamma$ as $t \to \infty$. $\square$

Throughout this appendix, for $1 < \alpha < 2$, we denote by $Z_\alpha$ a unit, right-skewed stable random variable having Fourier transform

$$ \mathbb{E}(e^{itZ_{\alpha}}) = \exp\left(-|t|^{\alpha}\left\{1 - i \tan\left(\frac{\pi\alpha}{2}\right) \operatorname{sign}(t)\right\}\right). $$

We also recall the notation $\Gamma : x \in (0, \infty) \mapsto \Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt$ for Euler's Gamma function.

The following lemma is a central limit theorem for sums of Pareto random variables. It is the fundamental tool for the proof of Proposition A.1 below.

**Lemma A.4 (Limiting behavior of sample averages of Pareto random variables).** Let $(Y_i)$ be a sequence of independent unit Pareto random variables.

* [Case $\gamma < 1/2$, finite second moment] When $\gamma < 1/2$, we have

$$ \sqrt{k} \left( \frac{1}{k} \sum_{i=1}^{k} Y_i^{\gamma} - \frac{1}{1-\gamma} \right) \xrightarrow{d} N \left( 0, \frac{\gamma^2}{(1-\gamma)^2(1-2\gamma)} \right). $$

* [Case $\gamma = 1/2$, infinite second moment, phase transition] We have

$$ \frac{\sqrt{k}}{\sqrt{\log k}} \left( \frac{1}{k} \sum_{i=1}^{k} \sqrt{Y_i} - 2 \right) \xrightarrow{d} N(0, 1). $$

* [Case $1/2 < \gamma < 1$, infinite second moment] When $1/2 < \gamma < 1$, we have

$$ k^{1-\gamma} \left( \frac{1}{k} \sum_{i=1}^{k} Y_i^{\gamma} - \frac{1}{1-\gamma} \right) \xrightarrow{d} \left\{ -\frac{\Gamma(2-1/\gamma)}{1/\gamma - 1} \cos\left(\frac{\pi}{2\gamma}\right) \right\}^{\gamma} Z_{1/\gamma}. $$

*Proof of Lemma A.4.* First and foremost, for any $\alpha < 1$, $\mathbb{E}(Y^\alpha) = 1/(1-\alpha)$. The first convergence is then an obvious consequence of the standard central limit theorem.

To show the second and third convergences, we follow Section XVII.5 of Feller (1971), which provides a unified treatment of the two cases. Equivalent results can be obtained by applying Theorems 1 and 2 in Geluk and de Haan (2000). Let, for any fixed $\gamma \in [1/2, 1)$ and $m \in \mathbb{R}$, $\mu_{\gamma,m}$ be the truncated second moment function of $Y^\gamma - m$, defined by

$$ \forall x > 0, \mu_{\gamma,m}(x) := \int_{-x}^{x} z^2 \mathbb{P}(Y^\gamma - m \in dz). $$
Then, for $x > |1 - m|$, $\mu_{\gamma,m}(x) = \mathbb{E}((Y^\gamma - m)^2 1_{\{|Y^\gamma - m| \le x\}}) = \mathbb{E}((Y^\gamma - m)^2 1_{\{Y^\gamma \le x+m\}})$ and therefore

$$|\mu_{\gamma,m}(x) - \mathbb{E}((Y^\gamma - m)^2 1_{\{Y^\gamma \le x\}})| \le \mathbb{E}((Y^\gamma - m)^2 1_{\{Y^\gamma \in (x, x+m]\}}).$$

Clearly, for $x > |1 - m|$ and $x > 1$,

$$ \begin{aligned} \mathbb{E}((Y^\gamma - m)^2 1_{\{Y^\gamma \in (x, x+m]\}}) &\le \max(x^2, (x-m)^2) \mathbb{P}(Y^\gamma \in (x, x+m]) \\ &= \max(x^2, (x-m)^2) \left| \mathbb{P}(Y > x^{1/\gamma}) - \mathbb{P}(Y > (x+m)^{1/\gamma}) \right| \\ &= x^{-1/\gamma} \max(x^2, (x-m)^2) \left| 1 - \left(1 + \frac{m}{x}\right)^{-1/\gamma} \right| \\ &= o(x^{2-1/\gamma}) \quad \text{as } x \to \infty. \end{aligned} $$

Moreover, in view of the fact that $Y^\gamma$ has a finite first moment and an infinite second moment, the monotone convergence theorem yields

$$ \begin{aligned} \mathbb{E}((Y^\gamma - m)^2 1_{\{Y^\gamma \le x\}}) &= \mathbb{E}(Y^{2\gamma} 1_{\{Y^\gamma \le x\}}) - 2m\mathbb{E}(Y^\gamma 1_{\{Y^\gamma \le x\}}) + m^2\mathbb{P}(Y^\gamma \le x) \\ &= \mathbb{E}(Y^{2\gamma} 1_{\{|Y^\gamma| \le x\}}) + O(1) = \mu_{\gamma,0}(x)(1 + o(1)) \quad \text{as } x \to \infty. \end{aligned} $$

Let $\mu = \mu_{\gamma,0}$ be the truncated second moment function of $Y^\gamma$. Since

$$ \forall x > 1, \mu(x) = \frac{1}{\gamma} \int_{1}^{x} y^{1-1/\gamma} dy = \begin{cases} \frac{1}{2\gamma - 1}(x^{2-1/\gamma} - 1) & \text{if } \gamma \in (1/2, 1), \\ 2\log x & \text{if } \gamma = 1/2, \end{cases} $$

we find that for any $m \in \mathbb{R}$,

$$ \mu_{\gamma,m}(x) = \mu(x)(1 + o(1)) = \begin{cases} \frac{x^{2-1/\gamma}}{2\gamma - 1}(1 + o(1)) & \text{if } \gamma \in (1/2, 1), \\ 2\log(x)(1 + o(1)) & \text{if } \gamma = 1/2. \end{cases} $$

Then the sequence

$$ a_k := \begin{cases} (2\gamma - 1)^{-\gamma} k^{\gamma} & \text{if } \gamma \in (1/2, 1), \\ \sqrt{k \log k} & \text{if } \gamma = 1/2 \end{cases} $$

clearly satisfies $a_k \to \infty$ and $k\mu(a_k)/a_k^2 \to 1$. Furthermore, the random variable $Y^\gamma$ is such that

$$ \forall y > 1, \mathbb{P}(Y^\gamma \le -y) = 0 \quad \text{and} \quad \mathbb{P}(Y^\gamma > y) = y^{-1/\gamma}. $$

The conditions of Theorem 2 in Section XVII.5 of Feller (1971) are therefore satisfied; note that its balanced tails condition (5.18) implicitly allows that one tail is regularly varying and
dominates the other, as can be seen by comparing this condition with the setup of (3.17) and (3.18) in Section XVII.3 of Feller (1971). By Theorem 3 in Section XVII.5 therein,

$$ \frac{1}{a_k} \left( \sum_{i=1}^{k} \{Y_i^{\gamma} - \mathbb{E}(Y^{\gamma})\} \right) \xrightarrow{d} W_{1/\gamma}, \qquad (A.13) $$

where $W_{1/\gamma}$ has a stable distribution with Fourier transform

$$ \mathbb{E}(e^{itW_{1/\gamma}}) = \exp \left( |t|^{1/\gamma} \frac{\Gamma(3-1/\gamma)}{(1/\gamma)(1/\gamma-1)} \left\{ \cos\left(\frac{\pi}{2\gamma}\right) - i \sin\left(\frac{\pi}{2\gamma}\right) \operatorname{sign}(t) \right\} \right). $$

[Note the typo below (3.19) in Section XVII.3 of Feller (1971), which should read “here the lower sign applies when $\zeta > 0$, the upper for $\zeta < 0$”, as can be seen by comparing (3.19) with (3.11) therein.] The result is then immediate for $\gamma = 1/2$. When $\gamma \in (1/2, 1)$, the above Fourier transform is also

$$ \mathbb{E}(e^{itW_{1/\gamma}}) = \exp \left( -|t|^{1/\gamma} \left\{ -(2\gamma - 1) \frac{\Gamma(2 - 1/\gamma)}{1/\gamma - 1} \cos\left(\frac{\pi}{2\gamma}\right) \right\} \left\{ 1 - i \tan\left(\frac{\pi}{2\gamma}\right) \operatorname{sign}(t) \right\} \right). $$

Rephrase then convergence (A.13) as

$$ k^{1-\gamma} \left( \frac{1}{k} \sum_{i=1}^{k} Y_i^{\gamma} - \frac{1}{1-\gamma} \right) \xrightarrow{d} (2\gamma-1)^{-\gamma} W_{1/\gamma} \stackrel{d}{=} \left\{ -\frac{\Gamma(2-1/\gamma)}{1/\gamma-1} \cos\left(\frac{\pi}{2\gamma}\right) \right\}^{\gamma} Z_{1/\gamma} \quad (A.14) $$

to conclude the proof. □

Proposition A.1 below gives the joint asymptotic behavior of the empirical counterpart of the Expected Shortfall and the empirical quantile at intermediate levels. It is the key to the proof of Theorem 1. Before proceeding to the statement and proof of Proposition A.1 it will be useful to recall a couple of results linked to Rényi's representation of order statistics from an independent unit exponential sample (see p.37 in de Haan and Ferreira, 2006), as this will be an important tool in some of our proofs below. Let $(E_1, \dots, E_n)$ be independent unit exponential random variables: one has

$$ (E_{i:n})_{1 \le i \le n} \stackrel{d}{=} \left( \sum_{j=1}^{i} \frac{E_j}{n-j+1} \right)_{1 \le i \le n}. $$

Consequently

$$ ((E_{n-i+1:n} - E_{n-k:n})_{1 \le i \le k}, E_{n-k:n}) \stackrel{d}{=} \left( \left( \sum_{j=n-k+1}^{n-i+1} \frac{E_j}{n-j+1} \right)_{1 \le i \le k}, \sum_{j=1}^{n-k} \frac{E_j}{n-j+1} \right). \qquad (A.15) $$
It follows in particular that $(E_{n-i+1:n} - E_{n-k:n})_{1\le i\le k} \stackrel{d}{=} (E_{k-i+1:k})_{1\le i\le k}$ and thus that, for any measurable function $f$ on $(0, \infty)$,

$$ \frac{1}{k} \sum_{i=1}^{k} f(E_{n-i+1:n} - E_{n-k:n}) \stackrel{d}{=} \frac{1}{k} \sum_{i=1}^{k} f(E_i) \quad \text{and} \quad \frac{1}{k} \sum_{i=1}^{k} f(E_{n-i+1:n} - E_{n-k:n}) \perp E_{n-k:n}. $$

This can be equivalently rephrased in the following way: if $(Y_1, \dots, Y_n)$ are independent unit Pareto random variables, then for any measurable function $f$ on $(1, \infty)$,

$$ \frac{1}{k} \sum_{i=1}^{k} f(Y_{n-i+1:n}/Y_{n-k:n}) \stackrel{d}{=} \frac{1}{k} \sum_{i=1}^{k} f(Y_i) \quad \text{and} \quad \frac{1}{k} \sum_{i=1}^{k} f(Y_{n-i+1:n}/Y_{n-k:n}) \perp Y_{n-k:n}. \quad (A.16) $$

[This follows from the fact that if $Y$ is unit Pareto, then $\log Y$ is unit exponential.] A particular consequence of that is the law of large numbers

$$ \frac{1}{k} \sum_{i=1}^{k} f(Y_{n-i+1:n}/Y_{n-k:n}) \xrightarrow{\mathbb{P}} \mathbb{E}(f(Y)) = \int_{1}^{\infty} f(x) \frac{dx}{x^2} \quad (A.17) $$

valid as soon as $x \mapsto x^{-2}f(x)1_{\{x \ge 1\}}$ is integrable. This convergence will mostly be used with the following choices for $f$:

* For $f(x) = x^a \{\int_1^x s^{b-1} ds\}$ with $a < 1$ and $a + b < 1$, yielding

$$ \mathbb{E} \left( Y^a \left\{ \int_1^Y s^{b-1} ds \right\} \right) = \frac{1}{(1-a)(1-a-b)}. \quad (A.18) $$

* For $f(x) = x^a \{\int_1^x s^{b-1}(\int_1^s u^{c-1} du) ds\}$ with $a < 1$, $a + b < 1$ and $a + b + c < 1$, yielding

$$ \mathbb{E} \left( Y^a \left\{ \int_1^Y s^{b-1} \left( \int_1^s u^{c-1} du \right) ds \right\} \right) = \frac{1}{(1-a)(1-a-b)(1-a-b-c)}. \quad (A.19) $$

* For $f(x) = x^a \{\int_1^x s^{b-1}(\int_1^s u^{c-1} du) ds\}(\int_1^x s^{d-1} ds)$ with $a < 1$, $a + b < 1$, $a + b + c < 1$, $a + d < 1$, $a + b + d < 1$ and $a + b + c + d < 1$, yielding

$$ \begin{aligned} & \mathbb{E} \left( Y^a \left\{ \int_1^Y s^{b-1} \left( \int_1^s u^{c-1} du \right) ds \right\} \left( \int_1^Y s^{d-1} ds \right) \right) \\ & = \frac{(1-a-b-d)(1-a-b-c-d) + (1-a)(2-2a-2b-c-d)}{(1-a)(1-a-d)(1-a-b)(1-a-b-d)(1-a-b-c)(1-a-b-c-d)}. \end{aligned} \quad (A.20) $$

We also note the following two lemmas (possibly known elsewhere) used in the proof of Proposition A.1. We provide concise proofs of these auxiliary results for the sake of completeness.
**Lemma A.5.** Let $f$ be a measurable positive function on $(0, \infty)$. Assume that there are $\alpha \in \mathbb{R}$, $\beta \le 0$ and a measurable function $C$ having constant sign and converging to $0$ at infinity such that the following property holds: for any $\delta, \varepsilon > 0$, there is $t_0 > 0$ such that for $t, tx \ge t_0$,

$$ \left| \frac{f(tx)}{f(t)} - x^{\alpha} - C(t)x^{\alpha} \int_{1}^{x} s^{\beta-1} ds \right| \le \varepsilon |C(t)| x^{\alpha+\beta} \max(x^{\delta}, x^{-\delta}). $$

Then if $(Y_1, \dots, Y_n)$ are independent unit Pareto random variables, and if $k = k(n)$ is a sequence of integers with $k \to \infty$ and $k/n \to 0$, one has

$$ \frac{f(Y_{n-k:n})}{f(n/k)} - \left(\frac{k}{n}Y_{n-k:n}\right)^{\alpha} = o_{\mathbb{P}}(|C(n/k)|). $$

*Proof of Lemma A.5.* It follows from Corollary 2.2.2 p.41 in de Haan and Ferreira (2006) that $\frac{k}{n}Y_{n-k:n} = 1 + o_{\mathbb{P}}(1)$, and in particular that $Y_{n-k:n} \xrightarrow{\mathbb{P}} \infty$. Use the assumption on $f$ with $t = n/k \to \infty$ and $tx = Y_{n-k:n}$ to obtain, for any $\delta > 0$,

$$ \frac{f(Y_{n-k:n})}{f(n/k)} - \left(\frac{k}{n}Y_{n-k:n}\right)^{\alpha} = C(n/k) \left(\frac{k}{n}Y_{n-k:n}\right)^{\alpha} \int_{1}^{\frac{k}{n}Y_{n-k:n}} s^{\beta-1} ds \\ + o_{\mathbb{P}}\left(|C(n/k)| \left(\frac{k}{n}Y_{n-k:n}\right)^{\alpha+\beta+\delta}\right). $$

Use again the fact that $\frac{k}{n}Y_{n-k:n} = 1 + o_{\mathbb{P}}(1)$ to complete the proof. $\square$

**Lemma A.6 (Asymptotic (random) inversion lemma).** Let $X_1, \dots, X_n, \dots$ be independent and identically distributed random copies of a random variable $X$ satisfying condition $C_2(\gamma, a, \rho, A)$. Let $k = k(n) \to \infty$ be a sequence of integers such that $k/n \to 0$ and $\sqrt{k}A(n/k) = O(1)$. Define

$$ \hat{F}_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{1}_{\{X_i \le x\}} \quad \text{and} \quad \hat{\bar{F}}_n(x) = 1 - \hat{F}_n(x). $$

Then

$$ \sqrt{k} \left( \frac{n}{k} \hat{F}_n(X_{n-k:n}) - 1 \right) \xrightarrow{\mathbb{P}} 0. $$

[This sample analog of the identity $\bar{F}(q(\tau))/(1-\tau) = 1 + o(|A((1-\tau)^{-1})|)$ (see the proof of Proposition 1(ii)) is obvious if $F$ is continuous, since then $\hat{\bar{F}}_n(X_{n-k:n}) = k/n$.]

*Proof of Lemma A.6.* Fix $t > 0$ and write

$$ \sqrt{k} \left| \frac{n}{k} \hat{F}_n(X_{n-k:n}) - 1 \right| \ge t \Leftrightarrow \sqrt{k} \left( \frac{n}{k} \hat{F}_n(X_{n-k:n}) - 1 \right) \le -t $$

because $\hat{\bar{F}}_n(X_{n-k:n}) \le k/n$. Then clearly

$$ \sqrt{k} \left| \frac{n}{k} \hat{\bar{F}}_n(X_{n-k:n}) - 1 \right| \ge t \Leftrightarrow 1 - \frac{k}{n} \left( 1 - \frac{t}{\sqrt{k}} \right) \le \hat{\bar{F}}_n(X_{n-k:n}). $$
Since $\alpha \mapsto X_{[n\alpha]:n}$ is the left-continuous inverse of $\hat{F}_n$, one has $\alpha \le \hat{F}_n(x) \Leftrightarrow X_{[n\alpha]:n} \le x$ for any $\alpha \in (0,1)$ and $x \in \mathbb{R}$. We obtain

$$ \sqrt{k} \left| \frac{n}{k} \hat{F}_n(X_{n-k:n}) - 1 \right| \ge t \Leftrightarrow X_{n-\lfloor k(1-t/\sqrt{k}) \rfloor:n} \le X_{n-k:n} $$

or equivalently

$$ \mathbb{P} \left( \sqrt{k} \left| \frac{n}{k} \hat{F}_n(X_{n-k:n}) - 1 \right| \ge t \right) = \mathbb{P} \left( \sqrt{k} \frac{X_{n-k:n} - X_{n-\lfloor k(1-t/\sqrt{k}) \rfloor:n}}{a_0(n/k)} \ge 0 \right) $$

where $a_0$ is a function that is asymptotically equivalent to $a$ in a neighborhood of infinity and such that Corollary 2.4.6 p.52 in de Haan and Ferreira (2006) applies. Using this theorem, one finds

$$ \sqrt{k} \left( \frac{X_{n-k:n} - X_{n-\lfloor k(1-t/\sqrt{k}) \rfloor:n}}{a_0(n/k)} + \int_1^{(1-t/\sqrt{k})^{-1}} z^{\gamma-1} dz \right) \xrightarrow{\mathbb{P}} 0. $$

Consequently

$$ \sqrt{k} \frac{X_{n-k:n} - X_{n-\lfloor k(1-t/\sqrt{k}) \rfloor:n}}{a_0(n/k)} \xrightarrow{\mathbb{P}} -t < 0 $$

and then

$$ \mathbb{P} \left( \sqrt{k} \left| \frac{n}{k} \hat{F}_n(X_{n-k:n}) - 1 \right| \ge t \right) = \mathbb{P} \left( \sqrt{k} \frac{X_{n-k:n} - X_{n-\lfloor k(1-t/\sqrt{k}) \rfloor:n}}{a_0(n/k)} + t \ge t \right) \to 0 $$

as required.

□

Let here and throughout $X_1, \dots, X_n$ be independent and identically distributed random copies of a random variable $X$ satisfying condition $C_2(\gamma, a, \rho, A)$. We are now ready to state and prove Proposition A.1 and, in passing, that the estimator $\hat{\mathrm{ES}}_n(\tau_n)$ is asymptotically equivalent to the two, a priori different estimators

$$ \hat{\mathrm{ES}}_n^{(1)}(\tau_n) = \frac{\sum_{i=1}^n X_i \mathbb{1}_{\{X_i > X_{[n\tau_n]:n}\}}}{\sum_{i=1}^n \mathbb{1}_{\{X_i > X_{[n\tau_n]:n}\}}} \quad \text{and} \quad \hat{\mathrm{ES}}_n^{(2)}(\tau_n) = \frac{1}{1 - \tau_n} \int_{\tau_n}^1 \hat{q}_n(t) dt $$

when $\tau_n$ is intermediate.

**Proposition A.1** (Joint weak convergence of $\hat{\mathrm{ES}}_n(\tau_n)$ and the empirical quantile). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $\gamma < 1$. Let $\tau_n \uparrow 1$ with $n(1 - \tau_n) \to \infty$. Let also

$$ v_n(\gamma) = \begin{cases} \sqrt{n(1 - \tau_n)} & \text{if } \gamma < 1/2, \\ \sqrt{n(1 - \tau_n)/\sqrt{\log(n(1 - \tau_n))}} & \text{if } \gamma = 1/2, \\ (n(1 - \tau_n))^{1-\gamma} & \text{if } 1/2 < \gamma < 1. \end{cases} $$
Then, if $v_n(\gamma)A((1 - \tau_n)^{-1}) = O(1)$, one has

$$v_n(\gamma) \left( \frac{\widehat{\mathrm{ES}}_n(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} - \frac{1}{(1-\gamma)(1-\gamma-\rho)} A((1-\tau_n)^{-1}), \frac{X_{\lfloor n\tau_n \rfloor:n} - q(\tau_n)}{a((1-\tau_n)^{-1})} \right) \\ \xrightarrow{d} \left\{ \begin{array}{ll} \mathcal{N} \left( \mathbf{0}_2, \begin{pmatrix} \frac{1+\gamma+2\gamma^2}{(1-\gamma)(1-2\gamma)} & \frac{\gamma}{1-\gamma} \\ \frac{\gamma}{1-\gamma} & 1 \end{pmatrix} \right) & \text{if } \gamma < 1/2, \\ (\mathcal{N}(0,4), 0) & \text{if } \gamma = 1/2, \\ \left( \frac{1}{\gamma} \left\{ -\frac{\Gamma(2-1/\gamma)}{1/\gamma-1} \cos\left(\frac{\pi}{2\gamma}\right) \right\}^{\gamma} Z_{1/\gamma}, 0 \right) & \text{if } 1/2 < \gamma < 1. \end{array} \right. $$

If moreover $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1}) = O(1)$, then

$$v_n(\gamma) \left( \frac{\widehat{\mathrm{ES}}_n(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right) = v_n(\gamma) \left( \frac{\widehat{\mathrm{ES}}_n^{(1)}(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right) + o_{\mathbb{P}}(1) \\ = v_n(\gamma) \left( \frac{\widehat{\mathrm{ES}}_n^{(2)}(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right) + o_{\mathbb{P}}(1)$$

and the above bivariate convergence result holds with $\widehat{\mathrm{ES}}_n$ replaced by either $\widehat{\mathrm{ES}}_n^{(1)}$ or $\widehat{\mathrm{ES}}_n^{(2)}$.

**Proof of Proposition A.1.** Set $k = n - \lfloor n\tau_n \rfloor = \lfloor n(1 - \tau_n) \rfloor$, so that $k = k(n)$ is a sequence of integers asymptotically equivalent to $n(1 - \tau_n)$; in particular, $k \to \infty$ and $k/n \to 0$. Note that $X_{\lfloor n\tau_n \rfloor:n} = X_{n-k:n}$, and therefore $\widehat{\mathrm{ES}}_n(\tau_n) = \widehat{\mathrm{ES}}_n(1 - k/n)$. Besides, a combination of Theorem 2.3.3 p.44 in de Haan and Ferreira (2006) with Theorem B.2.18 p.383 in de Haan and Ferreira (2006) applied to the function $t \mapsto t^{-\gamma}a(t)$ yields, for any $\delta, \varepsilon > 0$ sufficiently small, that for $t, tx$ large enough,

$$\left| \frac{a(tx)}{a(t)} - x^{\gamma} - A_{0}(t)x^{\gamma} \int_{1}^{x} s^{\rho-1}ds \right| \leq \varepsilon |A_{0}(t)|x^{\gamma+\rho} \max(x^{\delta}, x^{-\delta}) \quad (A.21)$$

up to replacing the function $A$ by a suitable, asymptotically equivalent function $A_0$. Since clearly

$$\frac{n(1 - \tau_n)}{k} = 1 + o\left(\frac{1}{n(1 - \tau_n)}\right),$$

it follows from Proposition 1(i) that

$$\frac{q(\tau_n) - q(1 - k/n)}{a((1 - \tau_n)^{-1})} = O\left(\frac{1}{n(1 - \tau_n)}\right) + o(|A((1 - \tau_n)^{-1})|) = o\left(\frac{1}{v_n(\gamma)}\right), \quad (A.22)$$

and from (A.21) that

$$\frac{a((1 - \tau_n)^{-1})}{a(n/k)} - 1 = O\left(\frac{1}{n(1 - \tau_n)}\right) + o(|A((1 - \tau_n)^{-1})|) = o\left(\frac{1}{v_n(\gamma)}\right). \quad (A.23)$$
We split the proof into two parts.

Step 1: Convergence properties of $\widehat{\mathrm{ES}}_n(\tau_n)$

From Equations (A.22) and (A.23) and the convergence $A((1 - \tau_n)^{-1})/A(n/k) \to 1$ (coming as a consequence of the regular variation property of A) it follows that it suffices to work in the setting $\tau_n = 1 - k/n$ with $k = k(n)$ a sequence of integers such that $k \to \infty$ and $k/n \to 0$, in which case

$$ \widehat{\mathrm{ES}}_n(1 - k/n) - X_{n-k:n} = \frac{1}{k} \sum_{i=1}^{k} (X_{n-i+1:n} - X_{n-k:n}). $$

With this in mind, we consider two cases.

Case $\gamma < 1/2$: Recall the notation of (A.6) and (A.7), and use the versions of $a_\star$ and $A_\star$ such that (A.8) holds. According to Theorem 2.4.2 and Corollary 2.4.6 pp.51-52 in de Haan and Ferreira (2006), up to enlarging the underlying probability space, one can construct a sequence of standard Brownian motions ($W_n$) such that for any $\varepsilon > 0$,

$$ \sup_{0<s\le 1} \min(1, s^{\gamma+1/2+\varepsilon}) \left| \sqrt{k} \left( \frac{X_{n-\lfloor ks \rfloor:n} - X_{n-k:n}}{a_\star(n/k)} - \int_1^{1/s} z^{\gamma-1} dz \right) - (s^{-\gamma-1}W_n(s) - W_n(1)) - \sqrt{k} A_\star(n/k) \Psi_{\gamma,\rho}(s^{-1}) \right| \xrightarrow{\mathbb{P}} 0 \\ \text{and} \quad \left| \sqrt{k} \frac{X_{n-k:n} - q(1-k/n)}{a_\star(n/k)} - W_n(1) \right| \xrightarrow{\mathbb{P}} 0. $$

Then

$$ \sqrt{k} \frac{X_{n-k:n} - q(1 - k/n)}{a(n/k)} = W_n(1) + o_{\mathbb{P}}(1) $$

because $a_\star$ and $a$ are asymptotically equivalent in a neighborhood of infinity. Also

$$ \int_0^1 \left( \sqrt{k} \left( \frac{X_{n-\lfloor ks \rfloor:n} - X_{n-k:n}}{a_\star(n/k)} - \int_1^{1/s} z^{\gamma-1} dz \right) - \sqrt{k} A_\star(n/k) \Psi_{\gamma,\rho}(s^{-1}) \right) ds \\ = \int_0^1 (s^{-\gamma-1} W_n(s) - W_n(1)) ds + o_{\mathbb{P}}(1). $$

In particular

$$ \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a_{\star}(n/k)} = \int_{0}^{1} \frac{X_{n-\lfloor ks \rfloor:n} - X_{n-k:n}}{a_{\star}(n/k)} ds = \frac{1}{1-\gamma} + o_{\mathbb{P}}\left(\frac{1}{\sqrt{k}}\right). \quad (A.24) $$

Let, as in the proof of Proposition 1(i), $c(\gamma, \rho) = -1/\rho$ if $\rho < 0$, $-1/\gamma$ if $\rho = 0$ and $\gamma \ne 0$, and $0$ if $\gamma = 0$ and $\rho = 0$, as well as $d(\rho) = 1/\rho$ if $\rho < 0$ and $1$ if $\rho = 0$, so that $a_\star(t) = a(t)(1 + c(\gamma, \rho)A(t)(1 + o(1)))$ and $A_\star(t) = d(\rho)A(t)(1 + o(1))$. Combining (A.12), (A.18)
and (A.19), one obtains

$$
\begin{align*}
\frac{c(\gamma, \rho)}{1-\gamma} + d(\rho) \int_0^1 \Psi_{\gamma,\rho}(s^{-1})ds &= \int_1^\infty \left( c(\gamma, \rho) \int_1^x s^{\gamma-1}ds + d(\rho)\Psi_{\gamma,\rho}(x) \right) \frac{dx}{x^2} \\
&= \int_1^\infty \left\{ \int_1^x s^{\gamma-1} \left( \int_1^s u^{\rho-1} \right) ds \right\} \frac{dx}{x^2} \\
&= \frac{1}{(1-\gamma)(1-\gamma-\rho)}. \tag{A.25}
\end{align*}
$$

Combining (A.24) and (A.25) results in

$$
\begin{align*}
& \sqrt{k} \left( \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a(n/k)} - \frac{1}{1-\gamma} - \frac{1}{(1-\gamma)(1-\gamma-\rho)} A(n/k) \right) \\
&= \sqrt{k} \left( \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a_*(n/k)} - \frac{1}{1-\gamma} \right) (1 + o_{\mathbb{P}}(1)) \\
&\quad + \frac{c(\gamma, \rho)}{1-\gamma} \sqrt{k} A(n/k) - \frac{1}{(1-\gamma)(1-\gamma-\rho)} \sqrt{k} A(n/k) + o_{\mathbb{P}}(1) \\
&= \int_0^1 \left( \sqrt{k} \left( \frac{X_{n-[ks]:n} - X_{n-k:n}}{a_*(n/k)} - \int_1^{1/s} z^{\gamma-1} dz \right) - \sqrt{k} A_*(n/k) \Psi_{\gamma,\rho}(s^{-1}) \right) ds + o_{\mathbb{P}}(1) \\
&= \int_0^1 (s^{-\gamma-1} W_n(s) - W_n(1)) ds + o_{\mathbb{P}}(1).
\end{align*}
$$

Since Cov($W_n(s)$, $W_n(t)$) = min($s, t$) for any $s, t > 0$, it is readily checked that

$$
\left( \int_0^1 (s^{-\gamma-1} W_n(s) - W_n(1)) ds, W_n(1) \right) \stackrel{\mathrm{d}}{=} \mathcal{N} \left( \mathbf{0}_2, \begin{pmatrix} \frac{1+\gamma+2\gamma^2}{(1-\gamma)(1-2\gamma)} & \frac{\gamma}{1-\gamma} \\ \frac{\gamma}{1-\gamma} & 1 \end{pmatrix} \right).
$$

The conclusion follows in this case.

Case $1/2 \le \gamma < 1$: Let $(Y_i)$ be a sequence of independent unit Pareto random variables. Recall from Corollary 2.2.2 p.41 in de Haan and Ferreira (2006) that $\frac{k}{n}Y_{n-k:n} = 1 + o_{\mathbb{P}}(1/\sqrt{k})$, and in particular that $Y_{n-k:n} \xrightarrow{\mathbb{P}} \infty$. Apply Proposition 1(i) to get

$$
\frac{X_{n-k:n} - q(1 - k/n)}{a(n/k)} \stackrel{\text{d}}{=} \frac{U(Y_{n-k:n}) - U(n/k)}{a(n/k)} = \frac{\left(\frac{k}{n}Y_{n-k:n}\right)^{\gamma} - 1}{\gamma} + o_{\mathbb{P}}(|A(n/k)|). \quad (A.26)
$$

By Corollary 2.2.2 p.41 in de Haan and Ferreira (2006) again, we obtain

$$
v_n(\gamma) \frac{X_{n-k:n} - q(1 - k/n)}{a(n/k)} = o_{\mathbb{P}}\left(\frac{v_n(\gamma)}{\sqrt{k}}\right) + o_{\mathbb{P}}(v_n(\gamma)|A(n/k)|) = o_{\mathbb{P}}(1).
$$

Besides,

$$
\frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a(n/k)} \stackrel{\mathrm{d}}{=} \frac{a(Y_{n-k:n})}{a(n/k)} \times \frac{1}{k} \sum_{i=1}^{k} \frac{U(Y_{n-i+1:n}) - U(Y_{n-k:n})}{a(Y_{n-k:n})}. \quad (A.27)
$$
Pick $\delta > 0$ such that $\gamma + \delta < 1$. By Proposition 1(i) again,

$$ \frac{1}{k} \sum_{i=1}^{k} \frac{U(Y_{n-i+1:n}) - U(Y_{n-k:n})}{a(Y_{n-k:n})} = \frac{1}{k} \sum_{i=1}^{k} \frac{(Y_{n-i+1:n}/Y_{n-k:n})^{\gamma} - 1}{\gamma} \\ + A(Y_{n-k:n}) \times \frac{1}{k} \sum_{i=1}^{k} \int_{1}^{Y_{n-i+1:n}/Y_{n-k:n}} s^{\gamma-1} \left( \int_{1}^{s} u^{\rho-1} \right) ds \\ + o_{\mathbb{P}}(|A(Y_{n-k:n})|) \times \frac{1}{k} \sum_{i=1}^{k} \left( \frac{Y_{n-i+1:n}}{Y_{n-k:n}} \right)^{\gamma+\delta}. \quad (A.28) $$

Using Lemma A.5 applied to the function $a$ (which, by (A.21), satisfies the uniform inequality required in that Lemma), the fact that $\frac{k}{n}Y_{n-k:n} = 1 + O_{\mathbb{P}}(1/\sqrt{k})$, and the regular variation property of $A$, we obtain

$$ \frac{a(Y_{n-k:n})}{a(n/k)} = 1 + o_{\mathbb{P}}\left(\frac{1}{v_n(\gamma)}\right) \quad \text{and} \quad v_n(\gamma)A(Y_{n-k:n}) = v_n(\gamma)A(n/k)(1+o_{\mathbb{P}}(1)) = o_{\mathbb{P}}(1). \quad (A.29) $$

Combining (A.16), (A.18), (A.19), (A.27), (A.28) and (A.29), we find

$$ \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a(n/k)} \stackrel{\text{d}}{=} \left( \frac{1}{k} \sum_{i=1}^{k} \frac{Y_i^{\gamma} - 1}{\gamma} + \frac{1}{(1-\gamma)(1-\gamma-\rho)} A(n/k) + o_{\mathbb{P}}\left(\frac{1}{v_n(\gamma)}\right) \right) \times \left( 1 + o_{\mathbb{P}}\left(\frac{1}{v_n(\gamma)}\right) \right). $$

Apply Lemma A.4 to

$$ \frac{1}{k} \sum_{i=1}^{k} \frac{Y_i^{\gamma} - 1}{\gamma} - \frac{1}{1-\gamma} = \frac{1}{\gamma} \left( \frac{1}{k} \sum_{i=1}^{k} Y_i^{\gamma} - \frac{1}{1-\gamma} \right) $$

in order to complete the proof in this case.

**Step 2: Asymptotic equivalence between $\widehat{\text{ES}}_n(\tau_n)$, $\widehat{\text{ES}}_n^{(1)}(\tau_n)$ and $\widehat{\text{ES}}_n^{(2)}(\tau_n)$**

Recall that in this part of the result we assume moreover $\sqrt{n(1-\tau_n)}A((1-\tau_n)^{-1}) = O(1)$. Clearly $\widehat{\text{ES}}_n^{(1)}(\tau_n) = \widehat{\text{ES}}_n^{(1)}(1-k/n)$, and thus, by Equation (A.23),

$$ \frac{\widehat{\text{ES}}_n^{(1)}(\tau_n) - X_{\lceil n\tau_n \rceil:n}}{a((1-\tau_n)^{-1})} = \frac{\widehat{\text{ES}}_n^{(1)}(1-k/n) - X_{n-k:n}}{a(n/k)} \left( 1 + o\left( \frac{1}{\sqrt{n(1-\tau_n)}} \right) \right). $$

Furthermore,

$$ \widehat{\text{ES}}_n^{(1)}(1-k/n) - X_{n-k:n} = \frac{\sum_{i=1}^{n} (X_i - X_{n-k:n}) \mathbb{1}_{\{X_i > X_{n-k:n}\}}}{\sum_{i=1}^{n} \mathbb{1}_{\{X_i > X_{n-k:n}\}}} = \frac{\sum_{i=1}^{k} (X_{n-i+1:n} - X_{n-k:n})}{\sum_{i=1}^{n} \mathbb{1}_{\{X_i > X_{n-k:n}\}}} $$
so that, by Lemma A.6,

$$
\begin{aligned}
\widehat{\mathrm{ES}}_{n}^{(1)}(1-k/n) - X_{n-k:n} &= \frac{\frac{1}{k} \sum_{i=1}^{k} (X_{n-i+1:n} - X_{n-k:n})}{\frac{n}{k} \widehat{F}_{n}(X_{n-k:n})} \\
&= \frac{1}{k} \sum_{i=1}^{k} (X_{n-i+1:n} - X_{n-k:n}) \left( 1 + o_{\mathbb{P}}\left(\frac{1}{\sqrt{k}}\right) \right) \\
&= \left( \widehat{\mathrm{ES}}_{n}(\tau_{n}) - X_{[n\tau_{n}]:n} \right) \left( 1 + o_{\mathbb{P}}\left(\frac{1}{\sqrt{n(1-\tau_{n})}}\right) \right).
\end{aligned}
$$

Since obviously

$$
\begin{aligned}
\widehat{\mathrm{ES}}_n(\tau_n) - X_{\lfloor n\tau_n \rfloor:n} &= \frac{n}{k} \int_{1-k/n}^{1} (\hat{q}_n(t) - \hat{q}_n(1-k/n))dt \\
&= \frac{n(1-\tau_n)}{k} \times \frac{1}{1-\tau_n} \int_{\tau_n}^{1} (\hat{q}_n(t) - \hat{q}_n(\tau_n))dt \\
&= \left( \widehat{\mathrm{ES}}_n^{(2)}(\tau_n) - X_{\lfloor n\tau_n \rfloor:n} \right) \left( 1 + o\left( \frac{1}{\sqrt{n(1-\tau_n)}} \right) \right),
\end{aligned}
$$

one finds

$$
\begin{aligned}
\frac{\widehat{\mathrm{ES}}_n(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} &= \frac{\widehat{\mathrm{ES}}_n^{(1)}(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} \left( 1 + o_{\mathbb{P}}\left(\frac{1}{\sqrt{n(1-\tau_n)}}\right) \right) \\
&= \frac{\widehat{\mathrm{ES}}_n^{(2)}(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} \left( 1 + o_{\mathbb{P}}\left(\frac{1}{\sqrt{n(1-\tau_n)}}\right) \right).
\end{aligned}
$$

Applying the convergence result obtained as a first step completes the proof because in any of the cases considered $v_n(\gamma) = O(\sqrt{n(1-\tau_n)})$. $\square$

*Proof of Theorem 1.* Write

$$
\frac{\widehat{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} = \frac{\widehat{\mathrm{ES}}_n(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} + \frac{X_{\lfloor n\tau_n \rfloor:n} - q(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{\mathrm{ES}(\tau_n) - q(\tau_n)}{a((1-\tau_n)^{-1})}.
$$

Set, as in Proposition A.1,

$$
v_n(\gamma) = \begin{cases} \sqrt{n(1-\tau_n)} & \text{if } 0 < \gamma < 1/2, \\ \frac{\sqrt{n(1-\tau_n)}}{\sqrt{\log(n(1-\tau_n))}} & \text{if } \gamma = 1/2, \\ (n(1-\tau_n))^{1-\gamma} & \text{if } 1/2 < \gamma < 1. \end{cases}
$$
Combine Proposition 1(ii) and Proposition A.1 to get

$$
v_n(\gamma) \left( \frac{\widehat{\mathrm{ES}}_n(\tau_n) - X_{\lfloor n\tau_n \rfloor:n}}{a((1-\tau_n)^{-1})} - \frac{\mathrm{ES}(\tau_n) - q(\tau_n)}{a((1-\tau_n)^{-1})}, \frac{X_{\lfloor n\tau_n \rfloor:n} - q(\tau_n)}{a((1-\tau_n)^{-1})} \right) \\
\xrightarrow{d} \left\{
\begin{array}{ll}
\mathcal{N} \left( \mathbf{0}_2, \left( \begin{array}{cc} \frac{1+\gamma+2\gamma^2}{(1-\gamma)(1-2\gamma)} & \frac{\gamma}{1-\gamma} \\ \frac{\gamma}{1-\gamma} & 1 \end{array} \right) \right) & \text{if } \gamma < 1/2, \\
(\mathcal{N}(0,4), 0) & \text{if } \gamma = 1/2, \\
\left( \frac{1}{\gamma} \left\{ -\frac{\Gamma(2-1/\gamma)}{1/\gamma-1} \cos\left(\frac{\pi}{2\gamma}\right) \right\}^\gamma Z_{1/\gamma}, 0 \right) & \text{if } 1/2 < \gamma < 1.
\end{array}
\right.
$$

The result is now a simple consequence of this joint convergence property.

<figure><img src="image_10.png" /></figure>

*Proof of Theorem 2.* Note that

$$
\frac{\widehat{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} = \frac{X_{\lfloor n\tau_n \rfloor:n} - q(\tau_n)}{a((1-\tau_n)^{-1})} + \frac{1}{1-\hat{\gamma}_n} \left( \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) + \frac{\hat{\gamma}_n - \gamma}{(1-\hat{\gamma}_n)(1-\gamma)} \\
- \left\{ \frac{\mathrm{ES}(\tau_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right\}.
$$

The result is then a simple consequence of Proposition 1(ii).

<figure><img src="image_11.png" /></figure>

*Proof of Theorem 3.* Recall the notation $d_n = (1 - \tau_n)/(1 - \tau'_n)$ and start by writing

$$
\frac{\overline{\mathrm{ES}}_n^*(\tau'_n) - \mathrm{ES}(\tau'_n)}{a((1-\tau_n)^{-1})} = \frac{\overline{\mathrm{ES}}_n(\tau_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} + \left( \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) \frac{1}{1-\hat{\gamma}_n} \int_1^{d_n} s^{\hat{\gamma}_n-1} ds \\
+ \frac{\hat{\gamma}_n - \gamma}{(1-\hat{\gamma}_n)(1-\gamma)} \int_1^{d_n} s^{\hat{\gamma}_n-1} ds + \frac{1}{1-\gamma} \left( \int_1^{d_n} s^{\hat{\gamma}_n-1} ds - \int_1^{d_n} s^{\gamma-1} ds \right) \\
- \left( \frac{\mathrm{ES}(\tau'_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \int_1^{d_n} s^{\gamma-1} ds \right). \tag{A.30}
$$

Now, for any $t \in \mathbb{R}$,

$$
\left| \frac{\int_{1}^{d_n} s^{t-1} ds}{\int_{1}^{d_n} s^{\gamma-1} ds} - 1 \right| \le \sup_{1 \le s \le d_n} |s^{t-\gamma} - 1| \le |t-\gamma| \log(d_n) \exp(|t-\gamma| \log(d_n))
$$

by the mean value theorem. Since $\log(d_n)/\sqrt{n(1-\tau_n)} \to 0$ and $\hat{\gamma}_n$ is $\sqrt{n(1-\tau_n)}$-consistent,
this yields

$$
\frac{\int_{1}^{d_n} s^{\hat{\gamma}_n - 1} ds}{\int_{1}^{d_n} s^{\gamma - 1} ds} \xrightarrow{\mathbb{P}} 1.
$$

Besides, a straightforward calculation gives

$$
\int_1^{d_n} s^{\gamma-1} \log(s) ds = \begin{cases} \frac{1}{\gamma} \left( d_n^\gamma \log(d_n) - \frac{d_n^\gamma - 1}{\gamma} \right) & \text{if } \gamma \neq 0, \\ \frac{\log^2(d_n)}{2} & \text{if } \gamma = 0. \end{cases} \quad (A.31)
$$
As a consequence,

$$ \frac{1}{\int_{1}^{d_n} s^{\gamma-1} \log(s) ds} \to \gamma_-^2 \quad \text{and} \quad \frac{\int_{1}^{d_n} s^{\hat{\gamma}_n-1} ds}{\int_{1}^{d_n} s^{\gamma-1} \log(s) ds} \xrightarrow{\mathbb{P}} -\gamma-. \quad (A.32) $$

Furthermore, for any $s \in [1, d_n]$ and any $t \in \mathbb{R}$, a Taylor expansion yields

$$ \begin{aligned} |s^{t-1} - s^{\gamma-1} - (t-\gamma)s^{\gamma-1} \log(s)| &\le \frac{(t-\gamma)^2 \log^2(s)}{2} \max_{u \in [\gamma, t]} s^{u-1} \\ &\le \frac{(t-\gamma)^2}{2} (s^{\gamma-1} + s^{t-1}) \log^2(s). \end{aligned} $$

It follows that

$$ \left| \int_{1}^{d_n} s^{\hat{\gamma}_n-1} ds - \int_{1}^{d_n} s^{\gamma-1} ds - (\hat{\gamma}_n - \gamma) \int_{1}^{d_n} s^{\gamma-1} \log(s) ds \right| \\ \le \frac{(\hat{\gamma}_n - \gamma)^2 \log(d_n)}{2} \int_{1}^{d_n} (s^{\gamma-1} + s^{\hat{\gamma}_n-1}) \log(s) ds. $$

Since for any $t \in \mathbb{R}$,

$$ \left| \frac{\int_{1}^{d_n} s^{t-1} \log(s) ds}{\int_{1}^{d_n} s^{\gamma-1} \log(s) ds} - 1 \right| \le \sup_{1 \le s \le d_n} |s^{t-\gamma} - 1| \le |t-\gamma| \log(d_n) \exp(|t-\gamma| \log(d_n)) $$

by the mean value theorem again, one finds

$$ \frac{\int_{1}^{d_n} s^{\hat{\gamma}_n-1} \log(s) ds}{\int_{1}^{d_n} s^{\gamma-1} \log(s) ds} \xrightarrow{\mathbb{P}} 1 $$

and thus

$$ \frac{\sqrt{n(1-\tau_n)}}{\int_{1}^{d_n} s^{\gamma-1} \log(s) ds} \left| \int_{1}^{d_n} s^{\hat{\gamma}_n-1} ds - \int_{1}^{d_n} s^{\gamma-1} ds - (\hat{\gamma}_n - \gamma) \int_{1}^{d_n} s^{\gamma-1} \log(s) ds \right| \xrightarrow{\mathbb{P}} 0. \quad (A.33) $$

Conclude from (A.32) and (A.33) that

$$ \begin{aligned} & \frac{\overline{ES}_n(\tau_n) - ES(\tau_n)}{a((1-\tau_n)^{-1})} + \left( \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) \frac{1}{1-\hat{\gamma}_n} \int_1^{d_n} s^{\hat{\gamma}_n-1} ds \\ & \qquad + \frac{\hat{\gamma}_n - \gamma}{(1-\hat{\gamma}_n)(1-\gamma)} \int_1^{d_n} s^{\hat{\gamma}_n-1} ds + \frac{1}{1-\gamma} \left( \int_1^{d_n} s^{\hat{\gamma}_n-1} ds - \int_1^{d_n} s^{\gamma-1} ds \right) \\ & \stackrel{\mathrm{d}}{=} \frac{\overline{ES}_n(\tau_n) - ES(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{\gamma_-}{1-\gamma_-} \int_1^{d_n} s^{\gamma-1} \log(s) ds \left( \frac{\hat{a}_n((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) \\ & \qquad + \left( \frac{1}{1-\gamma} - \frac{\gamma_-}{(1-\gamma_-)^2} \right) \int_1^{d_n} s^{\gamma-1} \log(s) ds (\hat{\gamma}_n - \gamma) + o_{\mathbb{P}} \left( \frac{\int_1^{d_n} s^{\gamma-1} \log(s) ds}{\sqrt{n(1-\tau_n)}} \right). \end{aligned} \quad (A.34) $$

The final (bias) term in (A.30) is controlled by noting that, for any $\gamma \in \mathbb{R}$,

$$ \frac{\gamma}{1-\gamma} \int_{1}^{d_n} s^{\gamma-1} ds = \frac{1}{1-\gamma} \left( \left( \frac{1-\tau'_n}{1-\tau_n} \right)^{-\gamma} - 1 \right). $$
[Note that indeed both sides are 0 when $\gamma = 0$.] This makes it possible to write

$$ 
\begin{aligned}
& \frac{\mathrm{ES}(\tau'_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \int_1^{d_n} s^{\gamma-1} \mathrm{d}s \\
&= \frac{a((1-\tau'_n)^{-1})}{a((1-\tau_n)^{-1})} \left( \frac{\mathrm{ES}(\tau'_n) - q(\tau'_n)}{a((1-\tau'_n)^{-1})} - \frac{1}{1-\gamma} \right) + \frac{1}{1-\gamma} \left( \frac{a((1-\tau'_n)^{-1})}{a((1-\tau_n)^{-1})} - \left( \frac{1-\tau'_n}{1-\tau_n} \right)^{-\gamma} \right) \\
&\quad - \left( \frac{\mathrm{ES}(\tau_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right) + \left( \frac{q(\tau'_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} - \int_1^{d_n} s^{\gamma-1} \mathrm{d}s \right).
\end{aligned}
\quad (A.35) $$

Recalling (A.21) and using the assumption $\rho < 0$ when $\gamma \ge 0$ along with (A.31), we obtain

$$ 
\begin{aligned}
& \frac{a((1-\tau'_n)^{-1})}{a((1-\tau_n)^{-1})} - \left(\frac{1-\tau'_n}{1-\tau_n}\right)^{-\gamma} = \begin{cases} A((1-\tau_n)^{-1})d_n^{\gamma} \left(-\frac{1}{\rho} + o(1)\right) & \text{if } \gamma \ge 0, \rho < 0, \\ o(|A((1-\tau_n)^{-1})|) & \text{if } \gamma < 0 \end{cases} \\
& = o\left(\frac{1}{\sqrt{n(1-\tau_n)}} \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s\right).
\end{aligned}
\quad (A.36) $$

Combining Proposition 1(ii) and the assumption $\rho < 0$ when $\gamma \ge 0$ again with the above equality, Potter bounds (see de Haan and Ferreira, 2006, Proposition B.1.9.5 p.367) on the function $|A|$ and (A.31), we find

$$ 
\begin{aligned}
& \frac{a((1-\tau'_n)^{-1})}{a((1-\tau_n)^{-1})} \left( \frac{\mathrm{ES}(\tau'_n) - q(\tau'_n)}{a((1-\tau'_n)^{-1})} - \frac{1}{1-\gamma} \right) \\
&= O\left( \left( \frac{1-\tau'_n}{1-\tau_n} \right)^{-\gamma} |A((1-\tau'_n)^{-1})| \right) + o\left( \frac{1}{\sqrt{n(1-\tau_n)}} \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s \right) \\
&= o(|A((1-\tau_n)^{-1})|d_n^{\gamma}) + o\left( \frac{1}{\sqrt{n(1-\tau_n)}} \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s \right) \\
&= o\left( \frac{1}{\sqrt{n(1-\tau_n)}} \int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s \right).
\end{aligned}
\quad (A.37) $$

Besides, Proposition 1(ii) and (A.32) yield

$$ 
\begin{aligned}
& \frac{\sqrt{n(1-\tau_n)}}{\int_1^{d_n} s^{\gamma-1} \log(s) \mathrm{d}s} \left( \frac{\mathrm{ES}(\tau_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right) \\
&= \begin{cases} o(\sqrt{n(1-\tau_n)}|A((1-\tau_n)^{-1})|) = o(1) & \text{if } \gamma \ge 0, \rho < 0, \\ \lambda \frac{\gamma^2}{(1-\gamma)(1-\gamma-\rho)} + o(1) & \text{if } \gamma < 0 \end{cases} \\
&= \lambda \frac{\gamma^2}{(1-\gamma-\rho)(1-\gamma-\rho)} + o(1).
\end{aligned}
\quad (A.38) $$
It is, finally, shown on p.137 in de Haan and Ferreira (2006) that

$$ \frac{\sqrt{n(1-\tau_n)}}{\int_1^{d_n} s^{\gamma-1} \log(s) ds} \left\{ \frac{q(\tau'_n) - q(\tau_n)}{a((1-\tau_n)^{-1})} - \int_1^{d_n} s^{\gamma-1} ds \right\} = \lambda \frac{\gamma_-}{\gamma_- + \rho} + o(1). \quad (A.39) $$

Combine (A.35), (A.36), (A.37), (A.38) and (A.39) to get

$$ \frac{\sqrt{n(1-\tau_n)}}{\int_1^{d_n} s^{\gamma-1} \log(s) ds} \left( \frac{\mathrm{ES}(\tau'_n) - \mathrm{ES}(\tau_n)}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \int_1^{d_n} s^{\gamma-1} ds \right) \\ \rightarrow \lambda \frac{\gamma_-(1-2\gamma_- - \rho)}{(1-\gamma_-)(1-\gamma_- - \rho)(\gamma_- + \rho)}. $$

Report this together with (A.34) into (A.30) to complete the proof. □

Recall that the Dekkers et al. (1989) moment estimators of the scale parameter $a(t)$ at $t = (1 - \tau_n)^{-1}$ and of the shape parameter $\gamma$ are respectively

$$ \hat{a}_n^{\text{Mom}}((1 - \tau_n)^{-1}) = X_{\lfloor n\tau_n \rfloor:n} M_n^{(1)}(1 - \hat{\gamma}_{n,-}^{\text{Mom}}) $$

$$ \text{and } \hat{\gamma}_n^{\text{Mom}} = M_n^{(1)} + \hat{\gamma}_{n,-}^{\text{Mom}}, \text{ with } \hat{\gamma}_{n,-}^{\text{Mom}} = 1 - \frac{1}{2} \left( 1 - \frac{(M_n^{(1)})^2}{M_n^{(2)}} \right)^{-1}, $$

where

$$ M_n^{(1)} = \frac{1}{\lfloor n(1-\tau_n) \rfloor} \sum_{i=1}^{\lfloor n(1-\tau_n) \rfloor} \log \frac{X_{n-i+1:n}}{X_{\lfloor n\tau_n \rfloor:n}} \quad \text{and} \quad M_n^{(2)} = \frac{1}{\lfloor n(1-\tau_n) \rfloor} \sum_{i=1}^{\lfloor n(1-\tau_n) \rfloor} \log^2 \frac{X_{n-i+1:n}}{X_{\lfloor n\tau_n \rfloor:n}}. $$

The main technical ingredient for the asymptotic analysis of these moment estimators is Proposition 2, whose proof relies on the following lemma.

**Lemma A.7** (Uniform inequality on $\log U$). Assume that $0 < U(\infty) = q(1) \le \infty$, that condition $C_2(\gamma, a, \rho, A)$ holds, and recall the definitions of $a_\star$, $A_\star$ and $\Psi_{\gamma,\rho}$ as in Equations (A.6) and (A.7). Then, up to replacing the functions $a_\star$ and $A_\star$ by suitable functions $a_{\star,0}$ and $A_{\star,0}$ such that $a_{\star,0}(t) = a_\star(t)(1 + o(|A(t)|))$ and $A_{\star,0}(t) = A_\star(t)(1 + o(1))$ as $t \to \infty$, we have, for any $\delta, \varepsilon > 0$ sufficiently small, that for $t$ large enough and all $x > 1$,

$$ \left| \frac{\log U(tx) - \log U(t)}{a_\star(t)/U(t)} - \int_1^x s^{\gamma-1} ds + \left( \frac{a_\star(t)}{U(t)} - \gamma_+ \right) \int_1^x s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds - A_\star(t) x^{-\gamma_+} \Psi_{\gamma,\rho}(x) \right| \le \varepsilon \left( \left| \frac{a_\star(t)}{U(t)} - \gamma_+ \right| + |A(t)| \right) x^\delta. $$

*Proof of Lemma A.7.* Work throughout with the versions of the functions $a_\star = a_{\star,0}$ and $A_\star = A_{\star,0}$ that make (A.8) hold true. Take $t$ so large that $U(t) > 0$ and write, for any $x > 1$,

$$ \frac{\log U(tx) - \log U(t)}{a_\star(t)/U(t)} = \frac{U(t)}{a_\star(t)} \log \left( 1 + \frac{a_\star(t)}{U(t)} \frac{U(tx) - U(t)}{a_\star(t)} \right). $$
We treat the two cases $\gamma \le 0$ and $\gamma > 0$ separately, and when considering $\gamma > 0$ we shall deal with the subcases $\rho < 0$ and $\rho = 0$ separately. Fix $\delta, \varepsilon > 0$ small enough; in particular, we impose that $\delta < -\rho$ in the case $\gamma > 0$ and $\rho < 0$. Fix also $\varepsilon \in (0, 1)$.

**Case** $\gamma \le 0$: It follows from (A.8) that

$$ \forall \eta > 0, \sup_{x \ge 1} x^{-\eta} \left| \frac{U(tx) - U(t)}{a_{\star}(t)} - \int_{1}^{x} s^{\gamma-1} ds \right| = O(|A(t)|) \to 0 \text{ as } t \to \infty. \quad (A.40) $$

In particular,

$$ \forall \eta > 0, \sup_{x \ge 1} x^{-\eta} \frac{U(tx) - U(t)}{a_{\star}(t)} = O(1) \text{ as } t \to \infty. \quad (A.41) $$

These facts and the convergence $a_{\star}(t)/U(t) \to 0$ motivate writing the identity

$$ \begin{aligned} & \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \int_{1}^{x} s^{\gamma-1} ds \\ & = \frac{U(t)}{a_{\star}(t)} \left\{ \log \left( 1 + \frac{a_{\star}(t) U(tx) - U(t)}{a_{\star}(t)} \right) - \frac{a_{\star}(t) U(tx) - U(t)}{a_{\star}(t)} \right\} \\ & + \frac{U(tx) - U(t)}{a_{\star}(t)} - \int_{1}^{x} s^{\gamma-1} ds. \end{aligned} \quad (A.42) $$

We concentrate on the first term. By Taylor's theorem,

$$ \left| \log(1+h) - h + \frac{h^2}{2} \right| \le \frac{h^3}{3} \text{ for any } h > 0. $$

Combining this for $h = \frac{a_{\star}(t)}{U(t)} \frac{U(tx)-U(t)}{a_{\star}(t)} > 0$ with (A.41) results in

$$ \left| \frac{U(t)}{a_{\star}(t)} \left\{ \log \left( 1 + \frac{a_{\star}(t) U(tx) - U(t)}{a_{\star}(t)} \right) - \frac{a_{\star}(t) U(tx) - U(t)}{a_{\star}(t)} \right\} + \frac{1}{2} \frac{a_{\star}(t)}{U(t)} \left( \frac{U(tx) - U(t)}{a_{\star}(t)} \right)^2 \right| \le \frac{\varepsilon}{2} \frac{a(t)}{U(t)} x^{\delta} $$

for $t$ large enough, uniformly in $x > 1$. Then, using (A.40) and (A.41) again, we find

$$ \left| \frac{U(t)}{a_{\star}(t)} \left\{ \log \left( 1 + \frac{a_{\star}(t) U(tx) - U(t)}{a_{\star}(t)} \right) - \frac{a_{\star}(t) U(tx) - U(t)}{a_{\star}(t)} \right\} + \frac{1}{2} \frac{a_{\star}(t)}{U(t)} \left( \int_{1}^{x} s^{\gamma-1} ds \right)^2 \right| \le \varepsilon \frac{a(t)}{U(t)} x^{\delta} \quad (A.43) $$

for $t$ large enough, uniformly in $x > 1$. Combining (A.8), (A.42) and (A.43), we get, for $t$ large enough,

$$ \left| \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \int_{1}^{x} s^{\gamma-1} ds + \frac{1}{2} \frac{a_{\star}(t)}{U(t)} \left( \int_{1}^{x} s^{\gamma-1} ds \right)^2 - A_{\star}(t) \Psi_{\gamma, \rho}(x) \right| \le \varepsilon \left( \frac{a(t)}{U(t)} x^{\delta} + |A(t)| x^{\gamma+\rho+\delta} \right) \le \varepsilon \left( \frac{a(t)}{U(t)} + |A(t)| \right) x^{\delta} \text{ for all } x > 1. $$
Since, when $\gamma \le 0$,

$$ \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{-|\gamma|-1} du \right) ds = \int_{1}^{x} s^{\gamma-1} \left( \int_{1}^{s} u^{\gamma-1} du \right) ds = \frac{1}{2} \left( \int_{1}^{x} s^{\gamma-1} ds \right)^{2}, $$

the desired result follows in this case.

**Case $\gamma > 0$:** One has

$$ \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \log(x) = \left(\frac{U(t)}{a_{\star}(t)} - \frac{1}{\gamma}\right) \left(\log U(tx) - \log U(t)\right) \\ + \frac{1}{\gamma} \log \left(1 + x^{-\gamma} \frac{a_{\star}(t)}{U(t)} \frac{U(tx) - U(t)}{a_{\star}(t)} - (1 - x^{-\gamma})\right). \quad (A.44) $$

We start by controlling the first term in (A.44). To this end we use the convergence $a(t)/U(t) \to \gamma$ as $t \to \infty$ in order to write

$$ \frac{U(t)}{a_{\star}(t)} - \frac{1}{\gamma} = -\frac{1}{\gamma^2} \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) (1 + \eta(t)) $$

where $\eta(t) \to 0$ as $t \to \infty$. This results in

$$ \begin{aligned} & \left(\frac{U(t)}{a_{\star}(t)} - \frac{1}{\gamma}\right) \left(\log U(tx) - \log U(t)\right) \\ &= \gamma \left(\frac{U(t)}{a_{\star}(t)} - \frac{1}{\gamma}\right) \log(x) + \left(\frac{U(t)}{a_{\star}(t)} - \frac{1}{\gamma}\right) \left(\log U(tx) - \log U(t) - \gamma \log(x)\right) \\ &= -\frac{1}{\gamma} \log(x) \left(\frac{a_{\star}(t)}{U(t)} - \gamma\right) (1 + \eta(t)) \\ & \quad - \frac{1}{\gamma^2} \left(\frac{a_{\star}(t)}{U(t)} - \gamma\right) \left(\log U(tx) - \log U(t) - \gamma \log(x)\right) (1 + \eta(t)). \end{aligned} \quad (A.45) $$

The function $U$ is regularly varying with index $\gamma$, so

$$ \forall x > 0, \lim_{t \to \infty} \frac{\log U(tx) - \log U(t)}{\gamma} = \log(x). $$

This convergence makes it possible to apply Proposition B.2.17 p.382 in de Haan and Ferreira (2006) to the function $\log U$; combined with (A.45), this provides, for $t$ large enough,

$$ \left| \left( \frac{U(t)}{a_{\star}(t)} - \frac{1}{\gamma} \right) \left( \log U(tx) - \log U(t) \right) + \frac{1}{\gamma} \log(x) \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) \right| \le \varepsilon \left( \left| \frac{a(t)}{U(t)} - \gamma \right| + |A(t)| \right) x^{\delta} \quad (A.46) $$

for all $x > 1$ (recall that $|a_{\star}(t)/U(t) - \gamma| = O(|a(t)/U(t) - \gamma| + |A(t)|)$ by definition of $a_{\star}$).

We turn to the control of the second term in (A.44). When $\rho < 0$, note that (A.8) and (A.40)
imply

$$
\begin{align*}
x^{-\gamma} \frac{a_{\star}(t)}{U(t)} \frac{U(tx) - U(t)}{a_{\star}(t)} - (1 - x^{-\gamma}) &= \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) \frac{x^{-\gamma} - 1}{-\gamma} + \gamma x^{-\gamma} A_{\star}(t) \Psi_{\gamma, \rho}(x) \\
&\quad + x^{-\gamma} \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) \left( \frac{U(tx) - U(t)}{a_{\star}(t)} - \frac{x^{\gamma} - 1}{\gamma} \right) \\
&\quad + \gamma x^{-\gamma} \left( \frac{U(tx) - U(t)}{a_{\star}(t)} - \frac{x^{\gamma} - 1}{\gamma} - A_{\star}(t) \Psi_{\gamma, \rho}(x) \right) \\
&= \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) \frac{x^{-\gamma} - 1}{-\gamma} + \gamma x^{-\gamma} A_{\star}(t) \Psi_{\gamma, \rho}(x) \\
&\quad + o \left( \left| \frac{a(t)}{U(t)} - \gamma \right| + |A(t)| \right) \tag{A.47}
\end{align*}
$$

uniformly in $x > 1$ as $t \to \infty$. In particular

$$
\sup_{x>1} \left| x^{-\gamma} \frac{a_{\star}(t)}{U(t)} \frac{U(tx) - U(t)}{a_{\star}(t)} - (1-x^{-\gamma}) \right| = O \left( \left| \frac{a(t)}{U(t)} - \gamma \right| + |A(t)| \right) \to 0 \quad (A.48)
$$

as $t \to \infty$. Combine (A.47) and (A.48) with a Taylor expansion of $z \mapsto \log(1+z)$ around
$z=0$ to obtain

$$
\begin{align*}
& \frac{1}{\gamma} \log \left( 1 + x^{-\gamma} \frac{a_{\star}(t)}{U(t)} \frac{U(tx) - U(t)}{a_{\star}(t)} - (1 - x^{-\gamma}) \right) \\
&= \frac{1}{\gamma} \left\{ \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) \frac{x^{-\gamma} - 1}{-\gamma} + \gamma x^{-\gamma} A_{\star}(t) \Psi_{\gamma, \rho}(x) \right\} + o \left( \left| \frac{a(t)}{U(t)} - \gamma \right| + |A(t)| \right) \tag{A.49}
\end{align*}
$$

uniformly in $x > 1$ as $t \to \infty$. When $\rho = 0$, write instead the second term in (A.44) as

$$
\frac{1}{\gamma} \log \left( 1 + x^{-\gamma} \frac{a_{\star}(t)}{U(t)} \frac{U(tx) - U(t)}{a_{\star}(t)} - (1 - x^{-\gamma}) \right) = \frac{\log((tx)^{-\gamma}U(tx)) - \log(t^{-\gamma}U(t))}{\gamma}.
$$

By Theorems 2.1 and 3.1 in Fraga Alves et al. (2007),

$$
\forall x > 0, \lim_{t \to \infty} \frac{1}{A(t)} \left( \frac{U(tx)}{U(t)} - x^{\gamma} \right) = x^{\gamma} \log(x). \quad (A.50)
$$

Taking the logarithm in (A.50) produces the convergence

$$
\forall x > 0, \lim_{t \to \infty} \frac{\log((tx)^{-\gamma}U(tx)) - \log(t^{-\gamma}U(t))}{A(t)} = \log(x).
$$

Apply then Proposition B.2.17 p.382 in de Haan and Ferreira (2006) to obtain that

$$
|\log((tx)^{-\gamma}U(tx)) - \log(t^{-\gamma}U(t)) - A(t)\log(x)| \le \varepsilon|A(t)|x^{\delta}
$$

for all $x > 1$ when $t$ is large enough. In other words

$$
\left| \frac{1}{\gamma} \log \left( 1 + x^{-\gamma} \frac{a_{\star}(t)}{U(t)} \frac{U(tx) - U(t)}{a_{\star}(t)} - (1 - x^{-\gamma}) \right) - A(t) x^{-\gamma} \Psi_{\gamma, \rho}(x) \right| \\
= \left| \frac{\log((tx)^{-\gamma}U(tx)) - \log(t^{-\gamma}U(t))}{\gamma} - A(t) x^{-\gamma} \Psi_{\gamma, \rho}(x) \right| \le \varepsilon |A(t)| x^{\delta} \quad (A.51)
$$
for all $x > 1$ when $t$ is large enough, because $\Psi_{\gamma,\rho}(x) = \frac{1}{\gamma}x^{\gamma}\log(x)$ when $\rho = 0$ and $\gamma > 0$. Furthermore, Theorem 2.1 in Fraga Alves et al. (2007) guarantees that

$$ 
\begin{aligned}
\frac{a_{\star}(t)}{U(t)} - \gamma &= \frac{a(t)}{U(t)} \left( 1 - \frac{1}{\gamma}A(t) + o(|A(t)|) \right) - \gamma \\
&= \frac{a(t)}{U(t)} - \gamma - A(t) + o(|A(t)|) = o(|A(t)|). \qquad (A.52)
\end{aligned}
$$

Conclude, from a combination of (A.44), (A.46) and (A.49) when $\rho < 0$, that

$$ 
\left| \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \log(x) + \frac{1}{\gamma} \left( \frac{a_{\star}(t)}{U(t)} - \gamma \right) \left( \log(x) - \frac{x^{-\gamma}-1}{-\gamma} \right) - A_{\star}(t)x^{-\gamma} \Psi_{\gamma,\rho}(x) \right| \le \varepsilon \left( \left| \frac{a(t)}{U(t)} - \gamma \right| + |A(t)| \right) x^{\delta}
$$

when $t$ is large enough uniformly in $x > 1$, and likewise from a combination of (A.44), (A.46), (A.51) and (A.52) when $\rho = 0$ (since then $A_{\star} = A$). The result follows because when $\gamma > 0$,

$$ \int_{1}^{x} s^{\gamma_{-1}} \left( \int_{1}^{s} u^{-|\gamma|-1} du \right) ds = \int_{1}^{x} s^{-1} \left( \int_{1}^{s} u^{-\gamma-1} du \right) ds = \frac{1}{\gamma} \left( \log(x) - \frac{x^{-\gamma}-1}{-\gamma} \right). $$

The proof is complete.

<figure><img src="image_12.png" /></figure>

*Proof of Proposition 2.* We prove the result by suitably modifying the proof of Proposition 1(i). Fix $\delta, \varepsilon > 0$ such that the conclusion of Lemma A.7 holds. Write, with the notation of Lemma A.7,

$$ 
\begin{aligned}
\frac{\log U(tx) - \log U(t)}{a(t)/U(t)} - \int_{1}^{x} s^{\gamma_{-1}} ds \\
&= \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \int_{1}^{x} s^{\gamma_{-1}} ds + \left( \frac{a_{\star}(t)}{a(t)} - 1 \right) \int_{1}^{x} s^{\gamma_{-1}} ds \\
&\quad + \left( \frac{a_{\star}(t)}{a(t)} - 1 \right) \left( \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \int_{1}^{x} s^{\gamma_{-1}} ds \right).
\end{aligned}
\qquad (A.53)
$$

By Lemma A.7, one has, for $t$ large enough and all $x > 1$,

$$ \left| \left( \frac{a_{\star}(t)}{a(t)} - 1 \right) \left( \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \int_{1}^{x} s^{\gamma_{-1}} ds \right) \right| \le \frac{\varepsilon}{3} \left( \left| \frac{a(t)}{U(t)} - \gamma_{+} \right| + |A(t)| \right) x^{\delta}. \qquad (A.54) $$

Use Lemma A.7 again to obtain, for $t$ large enough and all $x > 1$,

$$ 
\left| \frac{\log U(tx) - \log U(t)}{a_{\star}(t)/U(t)} - \int_{1}^{x} s^{\gamma_{-1}} ds + \left( \frac{a_{\star}(t)}{U(t)} - \gamma_{+} \right) \int_{1}^{x} s^{\gamma_{-1}} \left( \int_{1}^{s} u^{-|\gamma|-1} du \right) ds - A_{\star}(t) x^{-\gamma_{+}} \Psi_{\gamma,\rho}(x) \right| \le \frac{\varepsilon}{3} \left( \left| \frac{a(t)}{U(t)} - \gamma_{+} \right| + |A(t)| \right) x^{\delta}. \qquad (A.55)
$$

Recall the convergence $a(t)/U(t) \to \gamma_+$ and, from the proof of Proposition 1(i), the notation $c(\gamma, \rho) = -1/\rho$ if $\rho < 0$, $-1/\gamma$ if $\rho = 0$ and $\gamma \neq 0$, and 0 if $\gamma = 0$ and $\rho = 0$, as well as the
notation $d(\rho) = 1/\rho$ if $\rho < 0$ and 1 if $\rho = 0$, so that $a_*(t)/a(t) - 1 = c(\gamma, \rho)A(t)(1 + o(|A(t)|))$ and $A_*(t) = d(\rho)A(t)(1 + o(|A(t)|))$ as $t \to \infty$. Then

$$ \frac{a_*(t)}{U(t)} - \gamma_+ = \frac{a(t)}{U(t)} - \gamma_+ + c(\gamma, \rho)\gamma_+ A(t) + o\left(\left|\frac{a(t)}{U(t)} - \gamma_+\right| + |A(t)|\right) $$

and, for $t$ large enough and all $x > 1$,

$$ \left| \left( \frac{a_*(t)}{a(t)} - 1 \right) \int_1^x s^{\gamma-1} ds - c(\gamma, \rho) A(t) \int_1^x s^{\gamma-1} ds \right| \le \frac{\varepsilon}{3} |A(t)| x^\delta. \quad (A.56) $$

Conclude that for $t$ large enough and all $x > 1$,

$$ \begin{aligned} & \left| \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} - \int_1^x s^{\gamma-1} ds \right. \\ & \qquad + \left( \frac{a(t)}{U(t)} - \gamma_+ \right) \int_1^x s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds - c(\gamma, \rho) A(t) \int_1^x s^{\gamma-1} ds \\ & \qquad + c(\gamma, \rho) \gamma_+ A(t) \int_1^x s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds - d(\rho) A(t) x^{-\gamma_+} \Psi_{\gamma, \rho}(x) \\ & \le \varepsilon \left( \left| \frac{a(t)}{U(t)} - \gamma_+ \right| + |A(t)| \right) x^\delta \end{aligned} $$

by combining (A.53), (A.54), (A.55) and (A.56). To conclude it is then sufficient to show that

$$ \begin{aligned} & c(\gamma, \rho) \left( \int_1^x s^{\gamma-1} ds - \gamma_+ \int_1^x s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds \right) \\ & + d(\rho) x^{-\gamma_+} \Psi_{\gamma, \rho}(x) = x^{-\gamma_+} \int_1^x s^{\gamma-1} \left( \int_1^s u^{\rho-1} du \right) ds. \end{aligned} \quad (A.57) $$

A simple calculation proves the identity

$$ \int_1^x s^{\gamma-1} ds - \gamma_+ \int_1^x s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds = x^{-\gamma_+} \int_1^x s^{\gamma-1} ds $$

so that (A.57) is in fact equivalent to (A.12), which was shown to be correct in the proof of Proposition 1(i). Conclude that (A.57) holds, thus completing the proof. □

We may now proceed to the statement of Lemma A.8, which is nothing but a joint convergence result for $(M_n^{(1)}, M_n^{(2)}, X_{[n\tau_n]:n})$. It is the key to the proof of the joint convergence Lemma A.9 about $\widehat{ES}_n(\tau_n)$, the corresponding intermediate empirical quantile $X_{[n\tau_n]:n}$ and the moment estimators when $\gamma < 0$, which is an essential ingredient in the proofs of Theorems 4 and 5. Let us highlight that the present lemma is not a consequence of earlier results by Dekkers et al. (1989) and Chapter 3.5 in de Haan and Ferreira (2006), because it does not feature the restriction that $\gamma \neq \rho$.
**Lemma A.8** (Joint weak convergence of the building blocks for the moment estimator). Suppose that the tail quantile function $U$ of $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $0 < U(\infty) = q(1) \le \infty$. Let $k = k(n)$ be a sequence of integers with $k \to \infty$, $k/n \to 0$, $\sqrt{k}A(n/k) \to \lambda \in \mathbb{R}$ and $\sqrt{k}(a(n/k)/U(n/k) - \gamma_+) \to \mu \in \mathbb{R}$. Let further $(Y_i)$ be a sequence of independent copies of a unit Pareto random variable $Y$. Define

$$ \mathfrak{M}_n^{(1)} = \frac{1}{k} \sum_{i=1}^{k} \log \frac{U(Y_{n-i+1:n})}{U(Y_{n-k:n})} \quad \text{and} \quad \mathfrak{M}_n^{(2)} = \frac{1}{k} \sum_{i=1}^{k} \log^2 \frac{U(Y_{n-i+1:n})}{U(Y_{n-k:n})}. $$

Then

$$ \sqrt{k} \left( \frac{U(Y_{n-k:n})}{a(Y_{n-k:n})} \mathfrak{M}_n^{(1)} - \frac{1}{1-\gamma_-} \left\{ \frac{U(Y_{n-k:n})}{a(Y_{n-k:n})} \right\}^2 \mathfrak{M}_n^{(2)} - \frac{2}{(1-\gamma_-)(1-2\gamma_-)} \frac{U(Y_{n-k:n}) - U(n/k)}{a(n/k)} \right) \xrightarrow{d} \mathcal{N} \left( \lambda(B_A^{(1)}(\gamma, \rho), B_A^{(2)}(\gamma, \rho), 0) + \mu(B_{a/U}^{(1)}(\gamma), B_{a/U}^{(2)}(\gamma), 0), \mathbf{V} \right) $$

where the bias terms in the limiting distribution are

$$ B_A^{(1)}(\gamma, \rho) = \frac{1}{(1 + |\gamma|)(1 - \gamma_--\rho)} = \begin{cases} \frac{1}{(1 + \gamma)(1 - \rho)} & \text{if } \gamma \ge 0, \\ \frac{1}{(1 - \gamma)(1 - \gamma - \rho)} & \text{if } \gamma < 0, \end{cases} $$

$$ \begin{aligned} B_A^{(2)}(\gamma, \rho) &= \frac{2(3 + 2\gamma_+ - 4\gamma_- - 2\rho - \gamma_+\rho)}{(1 + \gamma_+)(1 + |\gamma|)(1 - 2\gamma_-)(1 - \gamma_- - \rho)(1 - 2\gamma_- - \rho)} \\ &= \begin{cases} \frac{2(3 + 2\gamma - 2\rho - \gamma\rho)}{(1 + \gamma)^2(1 - \rho)^2} & \text{if } \gamma \ge 0, \\ \frac{2(3 - 4\gamma - 2\rho)}{(1 - \gamma)(1 - 2\gamma)(1 - \gamma - \rho)(1 - 2\gamma - \rho)} & \text{if } \gamma < 0, \end{cases} \end{aligned} $$

$$ B_{a/U}^{(1)}(\gamma) = -\frac{1}{(1 - \gamma_-)(1 - \gamma_- + |\gamma|)} = \begin{cases} -\frac{1}{1 + \gamma} & \text{if } \gamma \ge 0, \\ -\frac{1}{(1 - \gamma)(1 - 2\gamma)} & \text{if } \gamma < 0, \end{cases} $$

$$ \begin{aligned} \text{and } B_{a/U}^{(2)}(\gamma) &= -\frac{2(3 + 2\gamma_+)}{(1 - \gamma_-)(1 - \gamma_- + |\gamma|)(1 - 2\gamma_- + |\gamma|)} \\ &= \begin{cases} -\frac{2(3 + 2\gamma)}{(1 + \gamma)^2} & \text{if } \gamma \ge 0, \\ -\frac{6}{(1 - \gamma)(1 - 2\gamma)(1 - 3\gamma)} & \text{if } \gamma < 0, \end{cases} \end{aligned} $$
and the covariance matrix $V$ is

$$V = \begin{pmatrix} \frac{1}{(1 - \gamma_{-})^2(1 - 2\gamma_{-})} & \frac{4}{(1 - \gamma_{-})^2(1 - 2\gamma_{-})(1 - 3\gamma_{-})} & 0 \\ \frac{4}{(1 - \gamma_{-})^2(1 - 2\gamma_{-})(1 - 3\gamma_{-})} & \frac{4(5 - 11\gamma_{-})}{(1 - \gamma_{-})^2(1 - 2\gamma_{-})^2(1 - 3\gamma_{-})(1 - 4\gamma_{-})} & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$

*Proof of Lemma A.8.* Combine Corollary 2.2.2 p.41 in de Haan and Ferreira (2006), Proposition 1(i) and the assumption $\sqrt{k}A(n/k) = O(1)$ to get

$$\sqrt{k} \frac{U(Y_{n-k:n}) - U(n/k)}{a(n/k)} = \sqrt{k} \int_{1}^{\frac{k}{n} Y_{n-k:n}} s^{\gamma-1} ds + o_{\mathbb{P}}(1) \xrightarrow{d} \mathcal{N}(0, 1). \quad (A.58)$$

Write then

$$\frac{a(Y_{n-k:n})}{U(Y_{n-k:n})} - \frac{a(n/k)}{U(n/k)} = \frac{a(n/k)}{U(Y_{n-k:n})} \left( \frac{a(Y_{n-k:n})}{a(n/k)} - 1 - \frac{a(n/k)}{U(n/k)} \frac{U(Y_{n-k:n}) - U(n/k)}{a(n/k)} \right).$$

It follows from the fact that $\frac{k}{n}Y_{n-k:n} = 1 + o_{\mathbb{P}}(1)$ and the regular variation property of $U$ with index $\gamma_+$ (use Corollary 1.2.10.1 on p.23 in de Haan and Ferreira (2006) when $\gamma > 0$, Lemma 1.2.9.3 on p.22 in de Haan and Ferreira (2006) when $\gamma = 0$, and the convergence of $U(t)$ to $U(\infty) \in (0, \infty)$ when $\gamma < 0$) that $a(n/k)/U(Y_{n-k:n}) \xrightarrow{\mathbb{P}} \gamma_+$. Recalling (A.21) and combining Lemma A.5 for the function $a$ with (A.58) yields

$$\frac{a(Y_{n-k:n})}{U(Y_{n-k:n})} - \frac{a(n/k)}{U(n/k)} = (\gamma_+ + o_{\mathbb{P}}(1)) \left( \left( \frac{k}{n} Y_{n-k:n} \right)^{\gamma} - 1 - \gamma_+ \int_{1}^{\frac{k}{n} Y_{n-k:n}} s^{\gamma-1} ds + o_{\mathbb{P}} \left( \frac{1}{\sqrt{k}} \right) \right).$$

Consequently, from (A.58) again,

$$\sqrt{k} \left( \frac{a(Y_{n-k:n})}{U(Y_{n-k:n})} - \gamma_+ \right) = \sqrt{k} \left( \frac{a(n/k)}{U(n/k)} - \gamma_+ \right) + o_{\mathbb{P}}(1) \xrightarrow{\mathbb{P}} \mu. \quad (A.59)$$

Now, combining Proposition 2 with (A.17), (A.59) and the regular variation property of $A$, we find

$$\begin{aligned} & \sqrt{k} \left( \frac{U(Y_{n-k:n})}{a(Y_{n-k:n})} \mathfrak{M}_n^{(1)} - \frac{1}{1 - \gamma_{-}} \right) \\ & = \sqrt{k} \left( \frac{1}{k} \sum_{i=1}^{k} \int_{1}^{Y_{n-i+1:n}/Y_{n-k:n}} s^{\gamma_{-1}-1} ds - \frac{1}{1 - \gamma_{-}} \right) \\ & - \sqrt{k} \left( \frac{a(n/k)}{U(n/k)} - \gamma_{+} \right) \mathbb{E} \left( \int_{1}^{Y} s^{\gamma_{-1}-1} \left( \int_{1}^{s} u^{-|\gamma|-1} du \right) ds \right) \\ & + \sqrt{k} A(n/k) \mathbb{E} \left( Y^{-\gamma_{+}} \int_{1}^{Y} s^{\gamma_{-1}-1} \left( \int_{1}^{s} u^{\rho_{-1}-1} du \right) ds \right) + o_{\mathbb{P}}(1). \end{aligned} \quad (A.60)$$
Likewise, writing

$$ 
\begin{aligned}
& \left( \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} \right)^2 - \left( \int_1^x s^{\gamma-1} ds \right)^2 \\
&= 2 \left( \int_1^x s^{\gamma-1} ds \right) \left( \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} - \int_1^x s^{\gamma-1} ds \right) \\
&\quad + \left( \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} - \int_1^x s^{\gamma-1} ds \right)^2
\end{aligned}
$$

we find

$$ 
\begin{aligned}
& \sqrt{k} \left( \left\{ \frac{U(Y_{n-k:n})}{a(Y_{n-k:n})} \right\}^2 \mathfrak{M}_n^{(2)} - \frac{2}{(1-\gamma_-)(1-2\gamma_-)} \right) \\
&= \sqrt{k} \left( \frac{1}{k} \sum_{i=1}^k \left( \int_1^{Y_{n-i+1:n}/Y_{n-k:n}} s^{\gamma-1} ds \right)^2 - \frac{2}{(1-\gamma_-)(1-2\gamma_-)} \right) \\
&\quad - 2\sqrt{k} \left( \frac{a(n/k)}{U(n/k)} - \gamma_+ \right) \mathbb{E} \left( \left( \int_1^Y s^{\gamma-1} ds \right) \left\{ \int_1^Y s^{\gamma-1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds \right\} \right) \\
&\quad + 2\sqrt{k} A(n/k) \mathbb{E} \left( \left( \int_1^Y s^{\gamma-1} ds \right) Y^{-\gamma_+} \left\{ \int_1^Y s^{\gamma-1} \left( \int_1^s u^{\rho-1} du \right) ds \right\} \right) + o_P(1). \quad (A.61)
\end{aligned}
$$

Obviously

$$ 
\begin{aligned}
& \sqrt{k} \left( \frac{1}{k} \sum_{i=1}^k \int_1^{Y_{n-i+1:n}/Y_{n-k:n}} s^{\gamma-1} ds - \frac{1}{1-\gamma_-}, \right. \\
& \left. \frac{1}{k} \sum_{i=1}^k \left( \int_1^{Y_{n-i+1:n}/Y_{n-k:n}} s^{\gamma-1} ds \right)^2 - \frac{2}{(1-\gamma_-)(1-2\gamma_-)}, \int_1^{\frac{k}{n}Y_{n-k:n}} s^{\gamma-1} ds \right) \\
& \xrightarrow{d} \mathcal{N}(\mathbf{0}_3, \mathbf{V}) \qquad (A.62)
\end{aligned}
$$

by using (A.16) combined with the Cramér-Wold device, the fact that $\int_1^Y s^{\gamma-1} ds = \log Y$ is unit exponential when $\gamma \ge 0$, and the straightforward identity

$$ \forall \alpha, \beta > 0 \text{ with } (\alpha + \beta)\gamma < 1, \text{ Cov}(Y^{\alpha\gamma}, Y^{\beta\gamma}) = \frac{\alpha\beta\gamma^2}{(1 - \alpha\gamma)(1 - \beta\gamma)(1 - (\alpha + \beta)\gamma)}, $$

as well as the standard central limit theorem, and Corollary 2.2.2 p.41 in de Haan and Ferreira (2006) for the asymptotic normality of the third component. From (A.62), it follows that it suffices to compute the bias terms in order to complete the proof. Using (A.19) twice, we get

$$ \mathbb{E} \left( \int_{1}^{Y} s^{\gamma-1} \left( \int_{1}^{s} u^{-|\gamma|-1} du \right) ds \right) = \frac{1}{(1-\gamma_{-})(1-\gamma_{-} + |\gamma|)} $$

and

$$ 
\begin{aligned}
& \mathbb{E} \left( Y^{-\gamma_+} \int_1^Y s^{\gamma-1} \left( \int_1^s u^{\rho-1} du \right) ds \right) = \frac{1}{(1+\gamma_+)(1+\gamma_+ - \gamma)(1+\gamma_+ - \gamma - \rho)} \\
& = \frac{1}{(1+|\gamma|)(1-\gamma_- - \rho)}.
\end{aligned}
$$
Using (A.20), we find after straightforward calculations that

$$
\begin{align*}
& \mathbb{E} \left( \left( \int_1^Y s^{\gamma_--1} ds \right) \left\{ \int_1^Y s^{\gamma_--1} \left( \int_1^s u^{-|\gamma|-1} du \right) ds \right\} \right) \\
&= \frac{3 - 7\gamma_- + 6\gamma_-^2 + 2|\gamma|}{(1 - \gamma_-)^2 (1 - 2\gamma_-) (1 - \gamma_- + |\gamma|) (1 - 2\gamma_- + |\gamma|)} \\
&= \frac{3 + 2\gamma_+}{(1 - \gamma_-) (1 - \gamma_- + |\gamma|) (1 - 2\gamma_- + |\gamma|)}
\end{align*}
$$

and

$$
\begin{align*}
& \mathbb{E} \left( \left( \int_1^Y s^{\gamma_--1} ds \right) Y^{-\gamma_+} \left\{ \int_1^Y s^{\gamma_--1} \left( \int_1^s u^{\rho-1} du \right) ds \right\} \right) \\
&= \frac{3 + 2\gamma_+ - 7\gamma_- + 4\gamma_-^2 - 2\rho - \gamma_+\rho + 2\gamma_-\rho}{(1 + \gamma_+)^2 (1 - \gamma_-)^2 (1 - 2\gamma_-)(1 - \gamma_- - \rho)(1 - 2\gamma_- - \rho)} \\
&= \frac{3 + 2\gamma_+ - 4\gamma_- - 2\rho - \gamma_+\rho}{(1 + \gamma_+)(1 + |\gamma|)(1 - 2\gamma_-)(1 - \gamma_- - \rho)(1 - 2\gamma_- - \rho)}.
\end{align*}
$$

The proof is complete.

<figure><img src="image_13.png" /></figure>

**Lemma A.9** (Joint weak convergence of $\widehat{\text{ES}}_n(\tau_n)$ and the building blocks for the moment estimator in the short tail case). Suppose that $X$ satisfies condition $C_2(\gamma, a, \rho, A)$ with $\gamma < 0$ and $0 < U(\infty) = q(1) < \infty$. Let $\tau_n \uparrow 1$ be such that $n(1 - \tau_n) \to \infty$, $\sqrt{n(1 - \tau_n)}A((1 - \tau_n)^{-1}) \to \lambda \in \mathbb{R}$ and $\sqrt{n(1 - \tau_n)} a((1 - \tau_n)^{-1})/q(\tau_n) \to \mu \in \mathbb{R}$.

1. One has

$$
\begin{align*}
& \sqrt{n(1-\tau_n)} \frac{\widehat{\text{ES}}_n(\tau_n) - \text{ES}(\tau_n)}{a((1-\tau_n)^{-1})} \\
&= \sqrt{n(1-\tau_n)} \left( \frac{X_{[n\tau_n]:n} M_n^{(1)}}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma} \right) + \sqrt{n(1-\tau_n)} \frac{X_{[n\tau_n]:n} - q(\tau_n)}{a((1-\tau_n)^{-1})} \\
&\quad - \frac{\lambda}{(1-\gamma)(1-\gamma-\rho)} + \frac{\mu}{(1-\gamma)(1-2\gamma)} + o_{\mathbb{P}}(1).
\end{align*}
$$

2. The following joint weak convergence result holds:

$$
\sqrt{n(1-\tau_n)} \left( \frac{\widehat{\text{ES}}_n(\tau_n) - \text{ES}(\tau_n)}{a((1-\tau_n)^{-1})}, \frac{X_{[n\tau_n]:n} M_n^{(1)}}{a((1-\tau_n)^{-1})} - \frac{1}{1-\gamma}, \hat{\gamma}_{n,-}^{\text{Mom}} - \gamma, \frac{X_{[n\tau_n]:n} - q(\tau_n)}{a((1-\tau_n)^{-1})} \right) \\
\xrightarrow{d} \mathcal{N} \left( \left( 0, \frac{\lambda}{(1-\gamma)(1-\gamma-\rho)} - \frac{\mu}{(1-\gamma)(1-2\gamma)}, \frac{\lambda(1-\gamma)(1-2\gamma)}{(1-\gamma-\rho)(1-2\gamma-\rho)} - \frac{\mu(1-\gamma)}{1-3\gamma}, 0 \right), \mathbf{C} \right)
$$
where $\mathbf{C} = \mathbf{C}(\gamma)$ is the $4 \times 4$ positive semidefinite symmetric matrix whose entries are

$$C_{11} = \frac{2}{(1-\gamma)(1-2\gamma)}, \quad C_{12} = \frac{1+2\gamma}{(1-\gamma)(1-2\gamma)}, \quad C_{13} = \frac{2\gamma}{1-3\gamma}, \quad C_{14} = \frac{1}{1-\gamma},$$

$$C_{22} = \frac{1 + \gamma + 2\gamma^2}{(1 - \gamma)(1 - 2\gamma)}, \quad C_{23} = \frac{2\gamma}{1 - 3\gamma}, \quad C_{24} = \frac{\gamma}{1 - \gamma},$$

$$C_{33} = \frac{(1 - \gamma)^2(1 - 2\gamma)(1 - \gamma + 6\gamma^2)}{(1 - 3\gamma)(1 - 4\gamma)}, \quad C_{34} = 0 \quad \text{and} \quad C_{44} = 1.$$

This results in particular corrects Corollary 3.5.6 p.108 in de Haan and Ferreira (2006), in which the reported asymptotic variance of $\hat{\gamma}_{n,-}^{\text{Mom}} - \gamma$ is unfortunately incorrect.

*Proof of Lemma A.9.* (i) From Propositions 1(i) and 2, and with the notation therein, we have, for any $\delta, \varepsilon > 0$ sufficiently small (in particular, we choose $\delta < 1$), that for $t$ large enough and all $x > 1$,

$$\left| \frac{U(tx) - U(t)}{a(t)} - \frac{\log U(tx) - \log U(t)}{a(t)/U(t)} - \frac{a(t)}{U(t)} \frac{1}{2} \left( \frac{x^\gamma - 1}{\gamma} \right)^2 \right| \le \varepsilon \left( \frac{a(t)}{U(t)} + |A(t)| \right) x^\delta. \quad (\text{A.63})$$

Besides, and as in the proof of Proposition A.1, it suffices to work in the case when $k = n(1 - \tau_n)$ is a sequence of integers, with then

$$\frac{\widehat{\text{ES}}_n(\tau_n) - X_{[n\tau_n]:n}}{a((1-\tau_n)^{-1})} \quad \text{replaced by} \quad \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a(n/k)}.$$

Let then $(Y_i)$ be a sequence of independent unit Pareto random variables. Obviously

$$\sqrt{k} \left( \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a(n/k)}, \frac{X_{n-k:n} M_n^{(1)}}{a(n/k)} \right) \\ \stackrel{\text{d}}{=} \sqrt{k} \frac{a(Y_{n-k:n})}{a(n/k)} \left( \frac{1}{k} \sum_{i=1}^{k} \frac{U(Y_{n-i+1:n}) - U(Y_{n-k:n})}{a(Y_{n-k:n})}, \frac{1}{k} \sum_{i=1}^{k} \frac{\log U(Y_{n-i+1:n}) - \log U(Y_{n-k:n})}{(a/U)(Y_{n-k:n})} \right)$$

so that

$$\sqrt{k} \left( \frac{1}{k} \sum_{i=1}^{k} \frac{X_{n-i+1:n} - X_{n-k:n}}{a(n/k)} - \frac{X_{n-k:n} M_n^{(1)}}{a(n/k)} \right) \\ \stackrel{\text{d}}{=} \frac{a(Y_{n-k:n})}{a(n/k)} \left( \sqrt{k} \frac{a(Y_{n-k:n})}{U(Y_{n-k:n})} \times \frac{1}{k} \sum_{i=1}^{k} \frac{1}{2} \left( \frac{(Y_{n-i+1:n}/Y_{n-k:n})^\gamma - 1}{\gamma} \right)^2 + o_P(1) \right) \\ \stackrel{\text{d}}{=} \frac{\mu}{(1-\gamma)(1-2\gamma)} + o_P(1)$$

by Equations (A.17), (A.18) and the uniform inequality (A.63), along with the convergence $\frac{k}{n}Y_{n-k:n} \xrightarrow{\mathbb{P}} 1$ combined with the regular variation properties of $a$, $U$ and $A$ (see de Haan