# Non-compactly supported estimation of the diffusion function in ergodic scalar SDEs

Yichuan Huang

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Yichuan Huang. Non-compactly supported estimation of the diffusion function in ergodic scalar SDEs. 2025. hal-04901917v2

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# Non-compactly supported estimation of the diffusion function in ergodic scalar SDEs

Yichuan Huang

Laboratoire Modal'X, Université Paris Nanterre, Nanterre, France,
E-mail address: yhuang@parisnanterre.fr
URL: https://sites.google.com/view/yichuanhuang

**Abstract.** This paper introduces a projection-based estimator for the diffusion coefficient in a stochastic differential equation (SDE) using high-frequency observations of a single trajectory. The method extends existing approaches to non-compact estimation domains, allowing the use of projection spaces spanned by non-compactly supported functions such as Hermite and Laguerre bases. The improvement partly relies on a different decomposition of the squared increments that defines the approximate regression equation. We propose a data-driven model selection procedure and prove that it enables the estimator to automatically balance squared bias and variance. Numerical experiments confirm its effectiveness across various SDE settings.

## 1. Introduction

Consider the one-dimensional stochastic differential equation (SDE)

$$dX_t = b(X_t) dt + \sigma(X_t) dW_t, \quad t \ge 0, \quad X_0 \sim \mu, \quad (1.1)$$

where $(W_t)_{t \ge 0}$ is a standard Wiener process and $\mu$ is independent of $(W_t)_{t \ge 0}$. The drift $b$ and diffusion $\sigma$ are assumed to be Lipschitz continuous. We observe the process at discrete times $(X_{k\Delta_n})_{k=0,\dots,n}$ with sampling step $\Delta_n > 0$. Our goal is to construct a nonparametric estimator of the squared diffusion coefficient $\sigma^2(\cdot)$.

Two main approaches have been developed for nonparametric estimation of diffusion functions. The first uses kernel methods that estimate the product of the diffusion function and the transition density, then divide by a separate estimator of the transition density (see, e.g., Florens-Zmirou, 1993; Florens, 1998; Jiang and Knight, 1997; Bandi and Phillips, 2003; Jacod, 2000). The second uses projection estimators on finite-dimensional spaces of basis functions (see, e.g., Genon-Catalot et al., 1992; Hoffmann, 1999; Comte et al., 2007). Most projection-based procedures estimate only the truncated target $\sigma_A^2 := \sigma^2 1_A$ on a compact set $A \subset \mathbb{R}$. To address non-compact domains, Ella-Mintsa (2024) consider $N$ independent copies of (1.1) and adapt the strategy of Comte et al. (2007) by taking $A = A_N = [-\log N, \log N]$ and letting $N \to \infty$, thereby covering $\mathbb{R}$.

A methodological distinction concerns the asymptotic regime and stationarity assumptions. Much of the literature focuses on potentially non-stationary settings under the high-frequency framework $\Delta_n = 1/n$ with a fixed horizon $[0, 1]$ (see, e.g., Florens-Zmirou, 1993; Hoffmann, 1999; Ella-Mintsa, 2024). In that regime, consistent estimation of $\sigma^2$ is attainable. In contrast, we work under a stationary, ergodic setup and assume that the process is geometrically $\beta$-mixing. This allows us to

<footnote n="2010">2010 Mathematics Subject Classification. 60J60, 62G05, 62M20.</footnote>

<footnote>Key words and phrases. Diffusion Processes, mixing Processes, model selection, non parametric estimation.</footnote>

<footnote>This research was partially supported by the Laboratoire d'Excellence (Labex) MME-DII (ANR-11-LBX-0023) under the "Investissements d'Avenir" program.</footnote>
exploit the long-time behavior of (1.1) and to analyze estimators in a long-span framework where $T_n = n\Delta_n \to \infty$. Geometric $\beta$-mixing further provides concentration inequalities for empirical processes—an essential ingredient when working with non-compactly supported bases.

Our estimator is projection-based and uses non-compactly supported bases. Let $\{S_m : m \in \mathcal{M}_n\}$ be a family of finite-dimensional models spanned by functions that are orthonormal in $L^2(\mathbb{R}, dx)$. We define

$$ \hat{\sigma}_m^2 \in \arg \min_{g \in S_m} \frac{1}{n} \sum_{k=0}^{n-1} \left[ \frac{(X_{(k+1)\Delta_n} - X_{k\Delta_n})^2}{\Delta_n} - g(X_{k\Delta_n}) \right]^2. $$

For stability, we introduce a truncation of $\hat{\sigma}_m^2$, following a strategy similar to the one discussed by Comte and Genon-Catalot (2020a,b, 2021). We show that the truncated projection estimator is consistent on $A = \mathbb{R}/\mathbb{R}^+$. Moreover, we derive a risk bound that applies broadly to non-compactly supported bases (including Hermite and Laguerre families), and we propose a fully data-driven selection of $m$ via a penalty that achieves the optimal bias-variance trade-off.

The main contributions in the context of nonparametric diffusion function estimation from high-frequency observations on a large time interval are:

* A construction of mean-square estimators compatible with non-compactly supported bases on $\mathbb{R}$ or $\mathbb{R}^+$, such as Hermite or Laguerre, broadening applicability to unbounded intervals.

* A mechanism that accommodates a large class of bases (rather than fixed bases), imposing only mild conditions typically satisfied by most bases.

* An adaptive procedure applicable when only a single trajectory is available without imposing compactness of $A$, with a slight improvement over Comte and Genon-Catalot (2020a,b, 2021) in the selection step.

*Organization of the paper:* Section 2 details the assumptions on Model (1.1) and on the projection bases that enable our analysis. Section 3 presents the estimators and both non-adaptive and adaptive risk bounds, along with a discussion of the resulting convergence rates. Section 4 provides numerical experiments illustrating the theoretical claims. Finally, Section 5 contains the proofs of the main results.

*Notations.* Throughout the paper, we use the following conventions:

* For a function $f$ defined on a set $E$ and a subset $A \subset E$, we denote by $f_A$ the restriction of $f$ to $A$.

* For $m \in \mathbb{N}^*$ and $v = (v_0, \dots, v_{m-1})^\top \in \mathbb{R}^m$, $\|v\|_{2,m} := \left(\sum_{j=0}^{m-1} v_j^2\right)^{1/2}$. We denote by $(v^{(m)})_{m \ge 1}$ a sequence of vectors of dimension $m$, whose coordinates may vary with $m$. We write $v^{(m)} := (v_0^{(m)}, \dots, v_{m-1}^{(m)})^\top$. We denote by $e_r$ the $r$-th canonical basis vector of $\mathbb{R}^m$.

* $I_m$ denotes the $m \times m$ identity matrix, and $M^\top$ denotes the transpose of a matrix or vector $M$.

* For $M \in \mathbb{R}^{m \times n}$, $\|M\|_{op} := \sup_{\|x\|_{2,n}=1} \|Mx\|_{2,m} = \sqrt{\lambda_{\max}(M^\top M)} = \sigma_{\max}(M)$, where $\lambda_{\max}$ and $\sigma_{\max}$ denote the largest eigenvalue and the largest singular value of $M$, respectively.

* For symmetric matrices $A, B \in \mathbb{R}^{m \times m}$, we write $A \prec B$ (resp. $A \prec B$) if $B - A$ is positive semidefinite (resp. positive definite). Two useful facts are: (i) If $0 \prec A \prec B$, then $B^{-1} \prec A^{-1}$. (ii) $A \prec B \iff CAC^\top \prec CBC^\top$ for any full-rank matrix $C \in \mathbb{R}^{m \times m}$.
## 2. Framework and Assumptions

This section specifies the assumptions regarding the diffusion model and the general projection spaces used to estimate the diffusion coefficient. Several examples of projection spaces are provided. Additionally, the asymptotic sampling framework necessary for theoretical consistency and for deriving risk bounds is described.

### 2.1. Model assumptions.

In this section, we introduce the minimal set of assumptions on (1.1) required for our analysis. We emphasize that these conditions are not unique: any alternative assumptions leading to Assumptions 1–4 would also be admissible.

**Assumption 1.** (i) The drift $b$ belongs to $C^1(\mathbb{R})$, and there exists $L \ge 0$ such that $\forall x \in \mathbb{R}, |b'(x)| \le L$.

(ii) The diffusion function $\sigma$ belongs to $C^2(\mathbb{R})$, and there exists $\gamma \ge 0$ such that $\forall x \in \mathbb{R}, |\sigma'(x)| + |\sigma''(x)| \le \gamma (1 + |x|^\gamma)$.

Under Assumption 1, the SDE (1.1) admits a unique strong solution adapted to the natural filtration $\mathcal{F}_t = \sigma(X_0, W_s, s \le t)$. Moreover, both $b$ and $\sigma$ satisfy a linear growth condition: $\exists K > 0, \forall x \in \mathbb{R}, |b(x)| + |\sigma(x)| \le K(1 + |x|)$. Assumption 1(ii) will be used in the proof of Lemma 6.2.

**Assumption 2.** (i) The scale density $s(x) = \exp\{-2 \int_0^x b(u)/\sigma^2(u) du\}$ satisfies $\int_{-\infty}^\infty s(x) dx = \int_{-\infty}^\infty s(x) dx = +\infty$, and the speed density $m(x) = (\sigma^2(x)s(x))^{-1}$ is integrable with total mass $M = \int_{\mathbb{R}} m(x) dx < \infty$.

(ii) The initial law $\mu$ coincides with the invariant distribution, which has density $\pi(x) = M^{-1}m(x)$.

Assumption 2 guarantees the existence of a unique invariant probability measure $\pi$, and ensures that the process $(X_t)_{t \ge 0}$ is stationary and ergodic.

**Assumption 3.** (i) The process $(X_t)$ is geometrically $\beta$-mixing: there exist constants $K > 0$ and $\theta > 0$ such that

$$ \beta_X(t) \le Ke^{-\theta t}, \quad t \ge 0, \qquad (2.1) $$

where $\beta_X(t)$ denotes the $\beta$-mixing coefficient defined by $\beta_X(t) = \int_{\mathbb{R}} \pi(x) \|P_t(x, \cdot) - \pi(\cdot)\|_{TV} dx$.

(ii) The stationary distribution $\pi$ admits finite moments of all orders.

*Remark 2.1.* The exponential $\beta$-mixing property ensures that dependence between observations decays at an exponential rate, which allows the extension of many probabilistic tools designed for independent data (e.g., concentration inequalities) to the dependent setting; see Rio et al. (2017). This property is particularly useful for controlling variances and covariances of empirical processes.

The following technical result, adapted from Gloter (2000, Proposition A), will also be used.

**Proposition 2.2.** Let $f \in C^1(\mathbb{R})$ such that, for some $\gamma \ge 0$, $|f'(x)| \le c(1 + |x|^\gamma), \forall x \in \mathbb{R}$. Then for every integer $k$ such that $\mathbb{E}[|X_0|^{k(1+\gamma)}] < \infty$, it holds

$$ \mathbb{E} \left( \sup_{s \in [t, t+h]} |f(X_s) - f(X_t)|^k \right) \le c h^{k/2} (1 + \mathbb{E}[|X_0|^{k(1+\gamma)}]). $$

In particular, this bound applies to $f = b$ and $f = \sigma$ with $\gamma = 0$, under Assumption 1.

**Assumption 4.** The invariant density $\pi$ is bounded: $\|\pi\|_\infty < \infty$.

Assumption 4 will only be used in Section 3.2.1 and in the proof of Theorem 3.5.
Remark 2.3. For reference, recall the standard notions often imposed in the literature. A drift $b$ is called dissipative if $b(x)\text{sign}(x) \le -|x|^c, \forall x \in \mathbb{R}$, for some $c \ge 0$. The diffusion function $\sigma$ is called non-degenerate if there exists $\sigma_0 > 0$ such that $\sigma^2(x) \ge \sigma_0, \forall x \in \mathbb{R}$. It is well known (see, e.g., Kutoyants, 2013; Pardoux and Veretennikov, 2001; Veretennikov, 1988) that dissipativity of $b$ combined with non-degeneracy and boundedness of $\sigma$ implies both ergodicity and geometric $\beta$-mixing.

## 2.2. Approximation basis.

The goal is to estimate $\sigma_A^2 := \sigma^2 \mathbb{1}_A$ for a domain $A \subseteq \mathbb{R}$, which may be a compact interval, $\mathbb{R}^+$, or the entire real line. For this purpose, fix $m \in \mathbb{N}^+$ and consider functions $\varphi_0, \dots, \varphi_{m-1}$ that are orthogonal in $L^2(A, dx)$ and define

$$S_m = \text{span}\{\varphi_0, \dots, \varphi_{m-1}\}.$$

The adaptive procedure presented in this paper requires the following condition on $\{\varphi_j\}$:

**Assumption 5.** (i) For all $j, \int \varphi_j^2(x)\pi(x)dx < +\infty$, thus $S_m \subset L^2(A, \pi(x)dx)$, and the collection of models $S_m$ is nested, i.e., $m \le m' \Rightarrow S_m \subset S_{m'}$.

(ii) For each $m \in \mathbb{N}^+$, there exists a constant $c_\varphi > 0$ such that

$$L(m) \le c_\varphi^2 m \quad \text{with} \quad L(m) = \sup_{x \in A} \sum_{j=0}^{m-1} \varphi_j(x)^2.$$

Even though our analysis applies to a wide range of bases, we illustrate with two classical bases: the Laguerre basis on $\mathbb{R}_+$ and the Hermite basis on $\mathbb{R}$.

**Laguerre basis** ($A = \mathbb{R}_+$). For $j \ge 0$, the Laguerre polynomial and scaled Laguerre function are defined as

$$L_j(x) = \sum_{k=0}^{j} (-1)^k \binom{j}{k} \frac{x^k}{k!}, \quad \ell_j(x) = \sqrt{2} L_j(2x) e^{-x^2} \mathbf{1}_{x \ge 0}.$$

The family $\{\ell_j : j \ge 0\}$ forms an orthonormal basis of $L^2(\mathbb{R}_+)$ satisfying $\|\ell_j\|_\infty \le \sqrt{2}$ for every $j$ (see Abramowitz and Stegun (1968, §22.14.12)). Consequently, Assumption 5 holds with $L(m) = \sup_{x \ge 0} \sum_{j=0}^{m-1} \ell_j(x)^2 \le 2m$, given that the family is clearly nested; moreover, for all $j, \int \ell_j^2(x)\pi(x)dx \le 2$.

**Hermite basis** ($A = \mathbb{R}$). For $j \ge 0$, define the Hermite polynomial and Hermite function by

$$H_j(x) = (-1)^j e^{x^2} \frac{d^j}{dx^j} (e^{-x^2}), \quad h_j(x) = c_j H_j(x) e^{-x^2/2}, \quad c_j := (2^j j! \sqrt{\pi})^{-1/2}.$$

The sequence $\{h_j : j \ge 0\}$ forms an orthonormal and nested basis of $L^2(\mathbb{R})$. Classical bounds (Abramowitz and Stegun (1968, §22.14.17); Szegö (1962, p.242); Indritz (1961)) provide $\|h_j\|_\infty \le \Phi_0$ with $\Phi_0 = 1/\pi^{1/4} \approx 0.7511$. Thus, Assumption 5 holds with $L(m) = \sup_{x \in \mathbb{R}} \sum_{j=0}^{m-1} h_j(x)^2 \le \Phi_0^2 m$; moreover, for all $j, \int h_j^2(x)\pi(x)dx \le \Phi_0^2$.

Both polynomial families satisfy recursion relations useful for implementation:

$$(j+1)L_{j+1}(x) = (2j+1-x)L_j(x) - jL_{j-1}(x), \quad L_0(x) = 1, L_1(x) = 1-x,$$

and

$$H_0(x) = 1, H_1(x) = x, \quad H_{j+1}(x) = 2x H_j(x) - 2j H_{j-1}(x).$$
## 2.3. Asymptotic Framework.

Throughout the paper, the sampling interval is denoted by $\Delta = \Delta_n$, which may vary with $n$. For simplicity, we often omit writing this dependency explicitly. The asymptotic regime described below is introduced to derive non-asymptotic risk bounds in Sections 3.2 and 3.3. It should be emphasized that the validity of our risk bounds does not rely on letting $n \to \infty$: they remain valid for any finite $(n, \Delta)$.

**Assumption 6.** As $n \to \infty$: (i) $\Delta_n \to 0$ and $\Delta_n n^{3/5} \to +\infty$, (ii) $\Delta_n^2 = o(1/n)$.

In particular, while $\Delta_n \to 0$, the total observation horizon $T_n = n \Delta_n$ diverges, which allows us to exploit the mixing properties of the diffusion process in our proofs. The growth condition $\Delta_n n^{3/5} \to +\infty$ mainly ensures that the probability of certain undesirable events remains negligible (see the proof of Proposition 3.1), while the constraint $\Delta_n^2 = o(1/n)$ guarantees that discretization errors are dominated by the statistical error terms. Thus, if $\Delta_n \asymp n^{-\alpha} \log^2 n$, we require $\alpha \in (1/2, 3/5)$.

# 3. Estimation of the diffusion coefficient

In this section, we first describe estimators for the diffusion coefficient $\sigma^2$ on a fixed model space $S_m$ for a given $m \in \mathbb{N}^*$. The estimators are tailored to the challenging setting where $A$ is non-compact, while remaining applicable when $A$ is compact. We establish risk bounds for these estimators and introduce a sharp model-selection procedure, proving that it can automatically choose the dimension that balances bias and variance.

## 3.1. Projection estimator on a fixed space $S_m$.

Fix $k \in \{1, \dots, n\}$ and define $U_{k\Delta} = (X_{(k+1)\Delta} - X_{k\Delta})^2/\Delta$. From (1.1), a regression-type relation holds:

$$U_{k\Delta} = \sigma(X_{k\Delta})^2 + V_{k\Delta} + R_{k\Delta}, \quad (3.1)$$

where $V_{k\Delta} := V_{k\Delta}^{(1)} + V_{k\Delta}^{(2)} + V_{k\Delta}^{(3)} + V_{k\Delta}^{(4)}$, with

$$V_{k\Delta}^{(1)} := \sigma^2(X_{k\Delta}) \left( \frac{(W_{(k+1)\Delta} - W_{k\Delta})^2}{\Delta} - 1 \right),$$

$$V_{k\Delta}^{(2)} := \frac{2}{\Delta} \sigma(X_{k\Delta}) \left[ (W_{(k+1)\Delta} - W_{k\Delta}) \int_{k\Delta}^{(k+1)\Delta} (\sigma(X_s) - \sigma(X_{k\Delta})) dW_s - \int_{k\Delta}^{(k+1)\Delta} (\sigma(X_s) - \sigma(X_{k\Delta})) ds \right],$$

$$V_{k\Delta}^{(3)} := \frac{2}{\Delta} \sigma(X_{k\Delta}) \left[ \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \sigma'(X_s) \sigma(X_s) dW_s \right],$$

$$V_{k\Delta}^{(4)} := 2b(X_{k\Delta}) \sigma(X_{k\Delta}) (W_{(k+1)\Delta} - W_{k\Delta}).$$
The remainder terms are $R_{k\Delta} := R_{k\Delta}^{(1)} + R_{k\Delta}^{(2)} + R_{k\Delta}^{(3)} + R_{k\Delta}^{(5)} + R_{k\Delta}^{(4)}$,

$$
\begin{align*}
R_{k\Delta}^{(1)} &:= \frac{1}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} b(X_s) \, ds \right)^2, \quad R_{k\Delta}^{(2)} := \frac{1}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} (\sigma(X_s) - \sigma(X_{k\Delta})) \, dW_s \right)^2, \\
R_{k\Delta}^{(3)} &:= \frac{2}{\Delta} \left( \int_{k\Delta}^{(k+1)\Delta} (b(X_s) - b(X_{k\Delta})) \, ds \right) \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s) \, dW_s, \\
R_{k\Delta}^{(4)} &:= 2b(X_{k\Delta}) \int_{k\Delta}^{(k+1)\Delta} (\sigma(X_s) - \sigma(X_{k\Delta})) \, dW_s, \\
R_{k\Delta}^{(5)} &:= \frac{2}{\Delta} \sigma(X_{k\Delta}) \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \psi(X_s) \, ds,
\end{align*}
$$

where

$$
\psi := \sigma' b + \frac{1}{2} \sigma'' \sigma^2. \tag{3.2}
$$

Here $V_{k\Delta}^{(1)}$ is the principal noise term. The decomposition (3.1) is sharper than that of Comte et al. (2007) in order to treat the empirical process $v_n := v_n^{(1)}(g) := \frac{1}{n} \sum_{k=1}^n g(X_{k\Delta}) V_{k\Delta}^{(1)}$ via Talagrand's inequality rather than a chaining argument. The remainders are additional noise-like and residual terms.

For integrable functions $f, g$ define $\langle f, g \rangle_n := \frac{1}{n} \sum_{k=1}^n f(X_{k\Delta})g(X_{k\Delta})$, $\|f\|_n^2 := \langle f, f \rangle_n$, $\langle f, g \rangle_\pi := \int f(x)g(x) \pi(x)dx$, and $\|f\|_\pi^2 := \langle f, f \rangle_\pi$.

Let $A \subset \mathbb{R}$ be the domain of interest, and let $\{\varphi_j\}_{j \ge 0} \subset L^2(A, dx)$ be an orthogonal basis of functions supported on $A$. For each $m \in \mathbb{N}^+$, define $S_m = \text{span}\{\varphi_0, \dots, \varphi_{m-1}\}$. For a fixed $m \in \mathbb{N}^*$, a projection estimator $\hat{\sigma}_m^2$ of $\sigma_A^2$ is defined as any solution of

$$
\hat{\sigma}_m^2 = \arg \min_{g \in S_m} \gamma_n(g), \quad \text{where} \quad \gamma_n(g) = \frac{1}{n} \sum_{k=1}^n \left( U_{k\Delta} - g(X_{k\Delta}) \right)^2. \quad (3.3)
$$

Introduce the matrices

$$
\hat{\Phi}_m = [\varphi_j(X_{i\Delta})]_{1\le i\le n, 0\le j\le m-1} \in \mathbb{R}^{n \times m}, \quad \hat{\Psi}_m = [\langle \varphi_j, \varphi_k \rangle_n]_{0\le j,k \le m-1} = \frac{1}{n} \hat{\Phi}_m^T \hat{\Phi}_m,
$$

and

$$
\Psi_m = \mathbb{E}[\hat{\Psi}_m] = [\langle \varphi_j, \varphi_k \rangle_{\pi}]_{0 \le j, k \le m-1} = \left[ \int_A \varphi_j(x) \varphi_k(x) \pi(x) dx \right]_{0 \le j, k \le m-1}.
$$

Let $U = (U_\Delta, \dots, U_{n\Delta})^T$. If $\hat{\Psi}_m$ is invertible, then

$$
\hat{\sigma}_m^2 = \sum_{j=0}^{m-1} \hat{a}_j^{(m)} \varphi_j, \quad \text{with} \quad \hat{a}^{(m)} = (\hat{\Phi}_m^T \hat{\Phi}_m)^{-1} \hat{\Phi}_m^T U = \frac{1}{n} \hat{\Psi}_m^{-1} \hat{\Phi}_m^T U. \quad (3.4)
$$

The invertibility of $\hat{\Psi}_m$ and $\Psi_m = \mathbb{E}[\hat{\Psi}_m]$ deserves careful consideration. These two matrices are
related through the event

$$
\Omega_m := \left\{ \sup_{g \in S_m, g \neq 0} \left| \frac{\|g\|_n^2}{\|g\|_\pi^2} - 1 \right| \le \frac{1}{2} \right\}, \quad (3.5)
$$

(see, for instance, (6.19)). In particular, $\Psi_m$ and $\hat{\Psi}_m$ are close in probability with respect to
the operator norm (Comte and Genon-Catalot, 2020a, Propositions 3-4) and (Comte and Genon-
Catalot, 2021, Proposition 6.1). For Laguerre and Hermite bases, (Comte and Genon-Catalot,
2020a, Lemma 3 and Proposition 8) show that $\Psi_m$ is invertible for all $m$, and that $\hat{\Psi}_m$ is almost surely invertible whenever $m \le n$. However, the conditioning of these matrices is not uniformly controlled.

It is important to note that, in general, $\Psi_m \ne I_m$, since the basis $\{\varphi_j\}_{j<m}$ is orthonormal in $L^2(A, dx)$ but not in $L^2(A, \pi(x) dx)$. Moreover,

$$ \|\Psi_m^{-1}\|_{\text{op}} = \frac{1}{\lambda_{\min}(\Psi_m)} = \left[ \min_{\|a\|_{2,m}=1} \int_A \left( \sum_{j=0}^{m-1} a_j \varphi_j(x) \right)^2 \pi(x) dx \right]^{-1}. $$

Writing $\pi_* := \inf_{x \in A} \pi(x)$, we obtain the crude bound $\|\Psi_m^{-1}\|_{\text{op}} \le 1/\pi_*$. Thus, if $\pi(x) \ge \pi_0 > 0$ for all $x \in A$ (which can only hold if $A$ is compact), then $\|\Psi_m^{-1}\|_{\text{op}} \le 1/\pi_0$. On the other hand, if $\pi_* = 0$ (for instance when $\pi$ has exponential tails on non-compact $A$), the operator norm $\|\Psi_m^{-1}\|_{\text{op}}$ may grow rapidly with $m$, unless one restricts attention to stability-preserving models. For this reason, we adopt the stability condition introduced in Cohen et al. (2013, 2019): the integer $m \ge 1$ is said to satisfy

$$ L(m) (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c n \Delta}{2 \log^2(n \Delta)} \quad \text{and} \quad m \le n \Delta, \qquad (3.6) $$

with $c = \theta(3 \log(\frac{3}{2}) - 1)/C_0$, where $C_0 \ge 49$ is a numerical constant and $\theta$ is as in (2.1).

We then evaluate the risk of $\hat{\sigma}_m^2$ through a truncated version defined by an empirical analogue of condition (3.6):

$$ \hat{\sigma}_m^2 = \hat{\sigma}_m^2 \mathbf{1}_{\Lambda_m}, \qquad (3.7) $$

where

$$ \Lambda_m := \left\{ L(m) \left( \|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1 \right) \le c \frac{n \Delta}{\log^2(n \Delta)} \right\}. \qquad (3.8) $$

### 3.2. Risk bounds for fixed m.

Let $m \in \mathbb{N}^*$ be such that it satisfies condition (3.6). The risk of the estimator defined in (3.7) can be decomposed as follows.

**Proposition 3.1.** Under Assumptions 1-3, 5, and 6(i), for every $m$ satisfying (3.6),

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_n^2] \le 7 \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \frac{128}{n} \text{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2}) + C\{\Delta^2 + \frac{1}{n}\}, \qquad (3.9) $$

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2] \le 45 \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \frac{640}{n} \text{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2}) + C' \{\Delta^2 + \frac{1}{n}\}, \qquad (3.10) $$

where

$$ \Psi_{m,\sigma^4} := \left[ \int \varphi_j(x) \varphi_k(x) \sigma^4(x) \pi(x) dx \right]_{0 \le j,k \le m-1}, $$

and $C, C'$ denote positive constants independent of $n, \Delta$, and $m$, but possibly depending on $b, \sigma$, and $\theta$.

The proof of Proposition 3.1 is given in Section 6.1.

The first term in (3.9) is the squared approximation error (bias) in $L^2(A, \pi(x)dx)$, while the second term is the variance. The remainder $C'\Delta^2 + C''/n$ arises mainly from the discretization error in (3.1) and from bounding the approximation error using the mixing property (2.1). Inequality (3.10) refers to a bound of risk expressed with respect to the $\pi$-norm and has the same structure, with
larger constants.

We now state a direct consequence of Proposition 3.1.

**Corollary 3.2.** Under the assumptions of Proposition 3.1, if $\|\sigma\|_{\infty,A} \le \Sigma$, then

$$ \mathrm{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2}) \le \Sigma^4 m \quad (3.11) $$

(see Comte and Genon-Catalot, 2021, Prop. 3.2). Hence,

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_n^2] \le 7 \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \frac{128 \Sigma^4 m}{n} + C \left\{ \Delta^2 + \frac{1}{n} \right\}, \quad (3.12) $$

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2] \le 45 \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \frac{640 \Sigma^4 m}{n} + C' \left\{ \Delta^2 + \frac{1}{n} \right\}. \quad (3.13) $$

In particular, for any fixed $m$, the term $\frac{\Sigma^4 m}{n} + \Delta^2 + \frac{1}{n}$ converges to 0 as $n \to \infty$ and $\Delta \to 0$ under Assumption 6. Moreover, the bias term $\inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2$ decreases with $m$ under Assumption 5(i).

### 3.2.1. Approximation spaces and rates.

This section illustrates how the risk bound of Proposition 3.1 can be applied. We evaluate the bias term using the following orthogonal-series smoothness class (with respect to $dx$):

$$ W_{\varphi}^{s}(A, R) = \left\{ h \in L^2(A, dx) : \sum_{j \ge 0} j^s \langle h, \varphi_j \rangle_{L^2(dx)}^2 \le R \right\}, \quad s > 0, R > 0, $$

where $\{\varphi_j\}_{j \ge 0}$ denotes an orthonormal basis of $L^2(A, dx)$ (Laguerre functions $\ell_j$ for $A = \mathbb{R}_+$, Hermite functions $h_j$ for $A = \mathbb{R}$). Under Assumption 4, for any $h \in L^2(A, dx)$ we have

$$ \inf_{g \in S_m} \|h - g\|_{\pi}^2 \le \|\pi\|_{\infty} \inf_{g \in S_m} \|h - g\|_{L^2(dx)}^2 = \|\pi\|_{\infty} \sum_{j \ge m} \langle h, \varphi_j \rangle_{L^2(dx)}^2. $$

If $h \in W_{\varphi}^s(A, R)$, then

$$ \sum_{j \ge m} \langle h, \varphi_j \rangle_{L^2(dx)}^2 = \sum_{j \ge m} j^{-s} (j^s \langle h, \varphi_j \rangle_{L^2(dx)}^2) \le m^{-s} \sum_{j \ge 0} j^s \langle h, \varphi_j \rangle_{L^2(dx)}^2 \le R m^{-s}. $$

Therefore, if $S_m = \mathrm{span}\{\varphi_0, \dots, \varphi_{m-1}\}$,

$$ \inf_{g \in S_m} \|\sigma_A^2 - g\|_{\pi}^2 \le \|\pi\|_{\infty} R m^{-s} \quad \text{whenever } \sigma_A^2 \in W_{\varphi}^s(A, R). \quad (3.14) $$

Combining (3.14) with Proposition 3.1 and using Assumption 6(ii) yields the following fully non-asymptotic bound.

**Corollary 3.3.** Under Assumptions 1-6, let $S_m = \mathrm{span}\{\varphi_0, \dots, \varphi_{m-1}\}$ be the first $m$ Laguerre (on $A = \mathbb{R}_+$) or Hermite (on $A = \mathbb{R}$) functions, orthonormal in $L^2(A, dx)$. Suppose $\sigma_A^2 \in W_{\varphi}^s(A, R)$ with $s > 0$. Then:

(i) For every $m$ satisfying (3.6),

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2] \le 45 \|\pi\|_{\infty} R m^{-s} + \frac{640 \mathrm{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2})}{n} + C'' \left\{ \Delta^2 + \frac{1}{n} \right\}, \quad (3.15) $$

where $C'' > 0$ is independent of $n$, $\Delta$, and $m$, but may depend on $b$, $\sigma$, and $\theta$.

(ii) If, in addition, $\|\sigma\|_{\infty,A} \le \Sigma$, then

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2] \le 45 \|\pi\|_{\infty} R m^{-s} + \frac{640 \Sigma^4}{n} + C'' \left\{ \Delta^2 + \frac{1}{n} \right\}. \quad (3.16) $$
The optimal dimension in (3.15) is typically non-explicit, as it depends on the trace term Tr(Ψ_m^{-1/2}Ψ_{m,σ^4}Ψ_m^{-1/2}). Moreover, minimization must be restricted to m satisfying (3.6). For example, if for some ρ ≥ 0 ||Ψ_m^{-1}||_{op} ≲ m^ρ, (which holds if π has polynomial tails, see Comte and Genon-Catalot, 2020a, Proposition 9), then (3.6) implies the admissible range

$$m \lesssim \left( \frac{n\Delta}{\log^2(n\Delta)} \right)^{1/(\rho+1)} \qquad (3.17)$$

If σ is bounded, we may apply (3.16). The oracle dimension balancing the bias term m^{-s} and the variance m/n is m^* ≍ n^{1/(s+1)}, yielding the rate n^{-s/(s+1)}, consistent with Comte and Genon-Catalot (2020a,b). This rate is attainable provided m^* lies within the admissible set (3.17).

As an illustration, if Δ_n ≍ n^{-α} log^2 n with α ∈ (1/2, 3/5) (see Assumption 6), a sufficient condition for the admissibility of m^* ≍ n^{1/(s+1)} is

$$n^{1/(s+1)} \lesssim \left( \frac{n^{1-\alpha}}{\log^2 n} \right)^{1/(\rho+1)} \iff s \geq \frac{\rho+1}{1-\alpha} - 1.$$

By contrast, if σ is bounded but the drift b is dissipative in the sense that b(x) sign(x) ≤ −|x|^c for some c ≥ 0, then the invariant density π typically has (super-)exponentially decaying tails, see Pardoux and Veretennikov (2001). In such non-compact settings, the bound ||Ψ_m^{-1}||_{op} ≲ m^ρ no longer holds (polynomial decay occurs only when c = −1), and ||Ψ_m^{-1}||_{op} may grow exponentially in m. As a result, the stability condition (3.6) forces m to grow at most logarithmically in nΔ, while the bias-variance optimal choice m^* is polynomial in n.

In this case, the risk bounds should be interpreted in a non-asymptotic sense: for any admissible m satisfying the stability condition, Proposition 3.1 provides a finite-sample bias-variance decomposition. The bias term inf_{g∈S_m} ||σ_A^2 - g||_π^2 decreases with m, while the variance scales as Σ^4m/n (when σ is bounded). The optimal trade-off must therefore be achieved within the restricted, stability-preserving class of models. The resulting bound is valid uniformly for all (n, Δ) without taking limits. This justifies the use of data-driven model selection (next section) to adaptively balance bias and variance in finite samples.

### 3.3. Model selection.

To choose the dimension m in a fully data-driven way, we restrict attention to (random) admissible models for which the least-squares projection remains numerically stable. Define the random model collection

$$\widehat{\mathcal{M}}_{n\Delta} := \left\{ m \in \mathbb{N}^* : c_{\varphi}^2 m (\|\widehat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le c \frac{n\Delta}{\log^2(n\Delta)} \right\}.$$

Its deterministic counterpart is

$$\mathcal{M}_{n\Delta} := \left\{ m \in \mathbb{N}^* : c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c}{2} \frac{n\Delta}{\log^2(n\Delta)} \right\}. \qquad (3.18)$$

In Comte and Genon-Catalot (2020a,b, 2021), admissible models are defined by the stronger condition c_{\varphi}^2 m (||Ψ_m^{-1}||_{op}^2 \vee 1) ≲ nΔ/ log^2(nΔ). Here we only require c_{\varphi}^2 m (||Ψ_m^{-1}||_{op} \vee 1) ≲ nΔ/ log^2(nΔ). which is strictly weaker. Consequently, both the deterministic set M_{nΔ} and the random set $\widehat{\mathcal{M}}_{nΔ}$ are strictly larger than in Comte and Genon-Catalot (2021). This enlargement has a direct practical impact: fewer models are discarded for stability reasons, thereby reducing the risk of overly conservative truncation of the optimal dimension. The improvement relies on the following observation.
Define the enlarged deterministic set

$$ \mathcal{M}_{n\Delta}^{+} := \left\{ m \in \mathbb{N}^{*} : c_{\varphi}^{2} m (\|\Psi_{m}^{-1}\|_{\text{op}} \vee 1) \le \frac{3\kappa}{2} \frac{n\Delta}{\log^{2}(n\Delta)} \right\}, \quad (3.19) $$

and the stability events $\Omega_{n\Delta} := \bigcap_{m \in \mathcal{M}_{n\Delta}^{+}} \Omega_m$, where $\Omega_m$ is defined in (3.5). Then the following lemma holds.

**Lemma 3.4.** On the event $\Omega_{n\Delta}$, $\mathcal{M}_{n\Delta} \subset \hat{\mathcal{M}}_{n\Delta} \subset \mathcal{M}_{n\Delta}^{+}$.

The proof of Lemma 3.4 is given in Section 6.6.

For each $m \in \hat{\mathcal{M}}_{n\Delta}$, let $\hat{\sigma}_{m}^{2}$ be the least-squares estimator defined in (3.3), and let $\tilde{\sigma}_{m}^{2}$ be its truncated version (3.7). By construction of $\hat{\mathcal{M}}_{n\Delta}$, the stability event $\Lambda_{m}$ holds for every $m \in \hat{\mathcal{M}}_{n\Delta}$; hence the truncation is inactive and $\hat{\sigma}_{m}^{2} = \tilde{\sigma}_{m}^{2}$ on this set. We select $m$ according to

$$ \hat{m} = \arg \min_{m \in \hat{\mathcal{M}}_{n\Delta}} \left\{ -\|\hat{\sigma}_m^2\|_n^2 + \kappa c_{\varphi}^2 \mathbb{E}[\sigma_A^4(X_0)] \frac{m \|\hat{\Psi}_m^{-1}\|_{\text{op}}}{n} \right\}. \quad (3.20) $$

The term $-\|\hat{\sigma}_m^2\|_n^2$ coincides with the empirical contrast $\gamma_n(\hat{\sigma}_m^2)$ up to an additive constant (see (3.3)). The numerical value of $\kappa$ derived theoretically is generally too large in practice; following Birgé and Massart (2007), one may calibrate $\kappa$ by a short pilot simulation to obtain a problem-specific value close to the minimal admissible one, and then keep this value fixed thereafter.

The next result is the analogue of the adaptive risk bound in Comte and Genon-Catalot (2021), adapted to the present volatility estimation setting and to our enlarged model collections.

**Theorem 3.5.** Suppose Assumptions 1–3 hold, as well as Assumptions 5 and 6(i). Let $\hat{m}$ be defined by (3.20). There exists a universal constant $\kappa_0 > 0$ such that, for all $\kappa \ge \kappa_0$, one can find constants $D, D' > 0$ (independent of $n, \Delta, m$) with

$$ \mathbb{E}[\|\hat{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2] \le D \inf_{m \in \mathcal{M}_{n\Delta}} \left\{ \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \kappa c_{\varphi}^2 \mathbb{E}[\sigma_A^4(X_0)] \frac{m \|\Psi_m^{-1}\|_{\text{op}}}{n} \right\} + D' \left(\Delta^2 + \frac{\log^6(n\Delta)}{n}\right), \quad (3.21) $$

$$ \mathbb{E}[\|\hat{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_{\pi}^2] \le D \inf_{m \in \mathcal{M}_{n\Delta}} \left\{ \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \kappa c_{\varphi}^2 \mathbb{E}[\sigma_A^4(X_0)] \frac{m \|\Psi_m^{-1}\|_{\text{op}}}{n} \right\} + D' \left(\Delta^2 + \frac{\log^6(n\Delta)}{n}\right). \quad (3.22) $$

The proof of Theorem 3.5 is given in Section 6.7.

The penalty in Theorem 3.5, $\text{pen}(m) := \kappa c_{\varphi}^2 \mathbb{E}[\sigma_A^4(X_0)] m \|\Psi_m^{-1}\|_{\text{op}}/n$, is an explicit upper bound on the fixed-$m$ variance term in Proposition 3.1. Indeed, by Comte and Genon-Catalot (2021, Proposition 3.2(iii)),

$$ \text{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2}) \le \mathbb{E}[\sigma_A^4(X_0)] L(m) \|\Psi_m^{-1}\|_{\text{op}} \le \mathbb{E}[\sigma_A^4(X_0)] c_{\varphi}^2 m \|\Psi_m^{-1}\|_{\text{op}}, $$

since $L(m) \le c_{\varphi}^2 m$. Assume the approximation error satisfies $\inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 \asymp m^{-s}$ for some $s > 0$, and $\|\Psi_m^{-1}\|_{\text{op}} \asymp m^{\rho}$, $\rho \ge 0$. By choosing $m_{\star} \asymp n^{1/(s+1+\rho)}$, the oracle criterion in (3.22) becomes

$$ \inf_{m \ge 1} \left\{ m^{-s} + m^{1+\rho}/n \right\} \asymp n^{-s/(s+1+\rho)}. $$

Here $\rho$ can be interpreted as a degree of ill-posedness. Admissibility ($m \in \mathcal{M}_{n\Delta}$) requires (up to constants) $m^{1+\rho} \lesssim n\Delta/\log^2(n\Delta)$. Evaluated at $m_{\star}$, this is equivalent to $n^{-s/(s+1+\rho)} \lesssim \Delta/\log^2(n\Delta)$. Under Assumption 6, one may take $\Delta_n \asymp n^{-\gamma} \log^2 n$ with any $\gamma \in (1/2, 3/5)$, so the admissibility constraint is non-binding whenever $\gamma \le s/(s+1+\rho)$, i.e. when $s > 1+\rho$. In
that regime, the adaptive risk in Theorem 3.5 achieves the rate $E[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_\pi^2] \lesssim n^{-s/(s+1+\rho)}$. If instead $\|\Psi_m^{-1}\|_{op}$ grows exponentially, admissible dimensions reduce to $M_{\max} \asymp \log(n\Delta)$ and the best attainable bias is only poly-logarithmic, so the overall rate deteriorates accordingly, although the penalty still enforces a bias-variance balance.

## 4. Simulation study

First, we consider the Ornstein-Uhlenbeck (OU) model $dX_t = -aX_t dt + \gamma dW_t$ with $a, \gamma > 0$ as a baseline; it admits a centered Gaussian invariant density with variance $\gamma^2/(2a)$. We simulate it exactly using

$$X_{(k+1)\Delta} = e^{-a\Delta} X_{k\Delta} + \gamma \sqrt{\frac{1 - e^{-2a\Delta}}{2a}} Z_k, \quad Z_k \sim \mathcal{N}(0, 1).$$

We then evaluate the efficiency of the adaptive estimator of the diffusion coefficient $\sigma^2$ on non-compact state spaces using four diffusion models:

(1) Hyperbolic diffusion on $\mathbb{R}$ (Hermite basis):

$$dX_t = -a X_t dt + \gamma \sqrt{1 + X_t^2} dW_t, \quad a = 2, \quad \gamma = \sqrt{\frac{1}{2}}.$$

The invariant density follows from the scale/speed formula:

$$\pi(x) \propto \frac{1}{\sigma^2(x)} \exp\left\{2 \int^x \frac{b(u)}{\sigma^2(u)} du\right\} \propto \frac{1}{1+x^2} (1+x^2)^{-a/\gamma^2} = (1+x^2)^{-(1+a/\gamma^2)}.$$

With $a = 2$ and $\gamma^2 = \frac{1}{2}$, we get $\pi(x) \propto (1+x^2)^{-5}$, i.e. polynomial tails $\pi(x) \asymp |x|^{-10}$ as $|x| \to \infty$. In particular, $\pi$ is bounded (Assumption 4) and has finite moments up to order 8 (since $E_\pi|X|^k < \infty$ iff $k < 4a + 1 = 9$), so not all moments exist, cf. Assumption 3(ii). For Gram conditioning, let $\rho \ge 5$, $\pi(x) \ge C(1+x^2)^{-\rho}$ for all $x \in \mathbb{R}$ and by Comte and Genon-Catalot (2020a, Prop.9), $\|\Psi_m^{-1}\|_{op} \lesssim m^\rho$. To check geometric mixing, set $Y_t := \operatorname{arsinh}(X_t)$. Then,

$$dY_t = -(a - \frac{\gamma^2}{2}) \tanh(Y_t) dt + \gamma dW_t,$$

so $Y$ has uniformly elliptic noise and a drift of the form $-c \tanh(\cdot)$ with $c = a - \gamma^2/2 > 0$, which implies Assumption 3(i); see, e.g., Pardoux and Veretennikov, 2001. Hence $X$ is geometrically $\beta$-mixing as well. Simulation is performed by Euler-Maruyama with 10 sub-steps per sampling interval $\Delta$.

(2) Lamperti-OU on $\mathbb{R}$ (Hermite basis). We prescribe $\sigma(x) = \frac{1}{1+x^2}$, and choose the drift via the Lamperti map $g'(x) = 1/\sigma(x) = 1+x^2$, namely

$$g(x) = x + \frac{x^3}{3}, \quad b(x) = \sigma(x) \left(-k g(x) + \frac{1}{2} \sigma'(x)\right), \quad k = 1.$$

Then $Y_t := g(X_t)$ solves $dY_t = -k Y_t dt + dW_t$, hence the stationary density is

$$\pi_X(x) \propto \pi_Y(g(x)) g'(x) \propto (1+x^2) \exp(-k g(x)^2),$$

which has super-exponential tails on $\mathbb{R}$. Assumptions 1-4 hold (note that $\sigma, \sigma', \sigma''$ are smooth and decay at infinity). Simulation is exact via the Lamperti map: simulate $Y$ exactly as OU and set $X = g^{-1}(Y)$ (invert $g$ numerically; $g$ is strictly increasing).
(3) Cox–Ingersoll–Ross (CIR) model on $\mathbb{R}_+$ (Laguerre basis):

$$dX_t = a(b - X_t) dt + \gamma \sqrt{X_t} dW_t, \quad a = 2, b = 2, \gamma = 1.$$

The stationary density is $\pi(x) = \frac{\lambda^k}{\Gamma(k)} x^{k-1} e^{-\lambda x}$, $k = \frac{2ab}{\gamma^2} = 8$, $\lambda = \frac{2a}{\gamma^2} = 4$, with exponential tail $e^{-\lambda x}$. The process is ergodic and geometrically mixing; the Feller condition $2ab \ge \gamma^2$ ensures strict positivity. While $\sigma(x) = \gamma\sqrt{x}$ is not $C^2$ at 0 on $\mathbb{R}$, the model is classical. Assumptions 2–4 hold. We include CIR to stress-test the Laguerre setup on a realistic positive diffusion. It is simulated exactly using the chi-square transition of the OU process : Let $U_t \in \mathbb{R}^d$ solve

$$dU_t = -\frac{a}{2} U_t dt + \frac{\gamma}{2} dW_t, \quad D = \frac{4ab}{\gamma^2} \text{ (here } D = 16\text{)},$$

with $W_t$ a $D$-dimensional Brownian motion. Then $X_t = \|U_t\|_{2,D}^2$ satisfies the CIR SDE above, and in stationarity one has $U_0 \sim N(0, \frac{\gamma^2}{4a} I_d)$.

(4) Power–Lamperti on $\mathbb{R}$ (Hermite basis). We take

$$\sigma^2(x) = (1 + |x|)^{-\beta}, \quad \beta = 1,$$

and construct the drift via the Lamperti transformation. Since $g'(x) = \sigma(x)^{-1} = (1 + |x|)^{\beta/2}$, we obtain for $p := 1 + \beta/2$:

$$g(x) = \operatorname{sign}(x) ((1 + |x|)^p - 1)/p, \quad b(x) = \sigma(x) \left(-k g(x) + \frac{1}{2} \sigma'(x)\right), \quad k = 1.$$

The stationary density of $X$ is

$$\pi_X(x) \propto g'(x) \exp(-k g(x)^2) \sim (1+|x|)^{\beta/2} \exp(-c|x|^{2+\beta}) \quad (|x| \to \infty),$$

which exhibits super-exponential tails. Moreover, Assumption 1 is not satisfied, since $x \mapsto (1 + |x|)^{-\beta/2}$ fails to be $C^2$ at 0. We deliberately include this example to test the robustness of the procedure under reduced smoothness of the diffusion coefficient. Simulation is exact via the Lamperti map, by simulating the OU process for $Y$ and applying the inverse transformation $X = g^{-1}(Y)$.

The estimator is defined in (3.3), and the model index $\hat{m}$ is selected by the penalized criterion (3.20) with a single tuning parameter $\kappa$. In line with our code and pilot calibrations, we fix $\kappa = 2 \times 10^{-4}$. To preserve numerical stability while retaining a rich model collection, we use the enlarged admissible set

$$\mathcal{M}_{n\Delta} := \left\{ m \in \mathbb{N}^* : c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le 10^{12} \frac{n\Delta}{\log^{2.05}(n\Delta)} \right\}.$$

For the asymptotic sampling framework, we take $\Delta_n = 10 n^{-0.52}$, which satisfies Assumption 6. For each model, we compute and plot 25 independent estimates (light green dashed curves) against the true $\sigma^2$ (solid red), and we report the selected dimension versus the maximal admissible dimension (means with standard deviations in parentheses), together with the MISE.

<table><caption>TABLE 4.1. MISE ×100 (mean; standard deviation) across $n$.</caption><thead><tr><th>n</th><th>Hyperbolic (Hermite)</th><th>Lamperti-OU (Hermite)</th><th>CIR (Laguerre)</th><th>Power-Lamperti &beta;=1 (Hermite)</th></tr></thead><tbody><tr><td>4,000</td><td>0.931 (0.324)</td><td>4.018 (0.641)</td><td>45.13 (10.24)</td><td>2.564 (0.632)</td></tr><tr><td>8,000</td><td>0.468 (0.149)</td><td>2.561 (0.557)</td><td>22.54 (5.116)</td><td>1.479 (0.273)</td></tr><tr><td>16,000</td><td>0.224 (0.077)</td><td>1.210 (0.205)</td><td>12.71 (3.434)</td><td>0.858 (0.151)</td></tr><tr><td>32,000</td><td>0.141 (0.092)</td><td>0.698 (0.145)</td><td>6.295 (1.740)</td><td>0.578 (0.111)</td></tr><tr><td>100,000</td><td>0.039 (0.012)</td><td>0.232 (0.037)</td><td>2.200 (0.636)</td><td>0.295 (0.032)</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>(A) Example (1): Hyperbolic (Hermite) with $n = 10\,000$, $\Delta = 10\,n^{-0.52}$</figcaption></figure>

Dimension: $\hat{m} = 7.6$ (sd = 1.3); $M_{max} = 10.4$ (sd = 0.9)

MISE: 0.000502 (sd = 0.0002108)

<figure><img src="image_4.png" /><figcaption>(C) Example (3): CIR (Laguerre) with $n = 500\,000$, $\Delta = 10\,n^{-0.52}$</figcaption></figure>

Dimension: $\hat{m} = 6.2$ (sd = 0.4); $M_{max} = 8.0$ (sd = 0.2)

MISE: 0.004935 (sd = 0.001604)

<figure><img src="image_5.png" /><figcaption>(B) Example (2): Lamperti-OU (Hermite) with $n = 500\,000$, $\Delta = 10\,n^{-0.52}$</figcaption></figure>

Dimension: $\hat{m} = 6.5$ (sd = 0.9); $M_{max} = 8.0$ (sd = 0.0)

MISE: 0.000549 (sd = 0.000174)

<figure><img src="image_6.png" /><figcaption>(D) Example (4): Power-Lamperti (Hermite) with $n = 500\,000$, $\Delta = 10\,n^{-0.52}$</figcaption></figure>

Dimension: $\hat{m} = 7.0$ (sd = 0.0); $M_{max} = 10.0$ (sd = 0.2)

MISE: 0.001955 (sd = 8.416 × 10⁻⁵)

<caption>FIGURE 4.1. Adaptive estimation of $\sigma^2$: 25 light-green estimates (dashed) vs. truth (red) for four models.</caption>

**Observations.** The log-log plot in Fig. 4.2 and Table 4.1 shows a clear monotone decrease of the MISE with $n$ for all models. Empirical slopes (in log-log) are close to -1 for OU (-0.988) and Hyperbolic (-0.967), and remain high for Lamperti-OU (-0.899) and CIR (-0.935), while Power-Lamperti (with a cusp at 0) is slower at -0.668. These patterns are consistent with our theory: when $\sigma^2$ is very smooth and the basis is well adapted (OU/Hyperbolic with Hermite), the bias decreases fast and the overall behavior is close to the “parametric” $n^{-1}$ benchmark in integrated risk; reduced smoothness (Power-Lamperti) and boundary/singularity effects (CIR near 0 with $\sigma^2(x) = \gamma^2 x$) increase the error level and slightly flatten the slope. Across models, the absolute MISE level is smallest for Hyperbolic and Lamperti-OU (super-/polynomial tails on $\mathbb{R}$ with Hermite), and largest for CIR on $\mathbb{R}_+$ (Laguerre), reflecting both the tail/heteroscedasticity of $\sigma^4$ and boundary behavior. The model selection step consistently picks moderate dimensions ($\hat{m} \approx 6-8$) that stay below the maximal admissible $M_{max}$ (computed with the enlarged 10¹² cutoff), indicating numerical stability and confirming that our penalty calibration (here $\kappa = 2 \times 10^{-4}$) balances bias and variance effectively. Finally, since we use $\Delta_n = cn^{-\alpha}$ with $\alpha \in (0.5, 0.6)$, the
<figure><img src="image_7.png" /><figcaption>FIGURE 4.2. Log-log plot of MISE versus *n* for five models (Hermite/Laguerre bases).</figcaption></figure>

discretization remainder $\Delta_n^2 = O(n^{-2\alpha})$ is negligible compared to the statistical terms over the range of *n* considered, so the empirical slopes mainly reflect the estimation error predicted by our non-asymptotic bounds.

5. Concluding remarks

We proposed a projection-based estimator of the diffusion coefficient $\sigma^2$ for ergodic scalar SDEs observed at high frequency over an increasing time horizon, with an emphasis on non-compact state spaces. The method (i) exploits non-compactly supported bases (Hermite on $\mathbb{R}$, Laguerre on $\mathbb{R}_+$), (ii) relies on a refined regression decomposition that isolates the leading noise term, and (iii) relaxes the classical stability restriction in order to accommodate richer model collections than in compact-domain settings. For fixed $m$, we derived non-asymptotic bias-variance bounds; with a suitable penalty, the fully data-driven choice of $m$ achieves an automatic squared bias-variance compromise. Simulations on several benchmark diffusions support the theoretical findings and confirm the practical stability of the procedure.

Several open directions remain. Establishing minimax lower bounds on non-compact domains would allow a rigorous assessment of the adaptive rates and clarify the exact role of $\|\Psi_m^{-1}\|_{op}$ in determining the best achievable risk. The constraint (3.6) may be overly restrictive in certain situations, especially under exponential tails. A systematic theoretical treatment of enlarged model collections, as used in simulations, remains to be developed. Exploring bases adapted to the invariant measure, reweighted Hermite/Laguerre systems, or other orthogonal constructions could reduce bias and improve conditioning in heavy-tailed or heteroscedastic regimes.

Finally, one should stress that the penalty ideally ought to reflect the true variance term, which is of order $m/n$: either through the exact trace term $\text{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2})/n$, or through its natural upper bound $\Sigma^4 m/n$ when $\sigma$ is bounded. While our choice of penalty provides a rather stable implementation, further refinements in this direction could yield sharper theoretical results and improved empirical performance.
# 6. Proofs

## 6.1. *Proof of Proposition 3.1.*

Throughout the proof, $C_{b,\sigma} > 0$ denotes a generic constant depending only on $b$ and $\sigma$, and which may vary from line to line. Let $m \in \mathbb{N}^*$ be such that it satisfies condition (3.6). Recall the regression decomposition (see (3.1))

$$ U_{k\Delta} = \sigma^2(X_{k\Delta}) + \underbrace{V_{k\Delta}^{(1)}}_{\text{principal noise}} + \underbrace{V_{k\Delta}^{(2)} + V_{k\Delta}^{(3)} + V_{k\Delta}^{(4)}}_{=:\bar{V}_{k\Delta}} + \underbrace{R_{k\Delta}^{(1)} + \cdots + R_{k\Delta}^{(5)}}_{=:\bar{R}_{k\Delta}} \\ V := (V_{k\Delta})_{1\le k \le n} = (V_{\Delta}, \dots, V_{n\Delta})^{\top}, \quad R := (R_{k\Delta})_{1\le k \le n} = (R_{\Delta}, \dots, R_{n\Delta})^{\top}. $$

For $g \in S_m$ define the empirical sums

$$ v_n^{(1)}(g) := \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) V_{k\Delta}^{(1)}, \quad \bar{v}_n(g) := \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) \bar{V}_{k\Delta}, \quad R_n(g) := \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) R_{k\Delta}. $$

We will use the following lemmas (proved in Sections 6.2, 6.3, 6.4, 6.5).

**Lemma 6.1.** Let $\Omega_m$ a $\Lambda_m$ be the events defined in 3.5 and (3.8). Under Assumptions 1-3, for every $m$ satisfying (3.6) and every $n$ with $\log(n\Delta) \ge 1$,

$$ \mathbb{P}(\Lambda_m^c) \le \mathbb{P}(\Omega_m^c) \le C_{\text{mix}} (n\Delta)^{-6}, \quad C_{\text{mix}} := \max\left(\frac{\theta}{7}, 4\right), $$

where $\theta$ is the mixing constant in (2.1).

**Lemma 6.2.** Under Assumptions 1-3, there exists a finite constant $C_{b,\sigma} > 0$ depending only on $b, \sigma$ and on the order-8 Burkholder-Davis-Gundy (BDG) Inequality constant such that, such that for every $k \in \mathbb{N}$, $i \in \{2, 3, 4\}$, $j \in \{1, 2, 3, 4, 5\}$

$$ \mathbb{E}[(V_{k\Delta}^{(i)})^2] \le C_{b,\sigma} \Delta, \quad \mathbb{E}[(R_{k\Delta}^{(j)})^2] \le C_{b,\sigma} \Delta^2, \quad (6.1) $$

$$ \mathbb{E}[(V_{k\Delta}^{(i)})^4] \le C_{b,\sigma} \Delta^2, \quad \mathbb{E}[(R_{k\Delta}^{(j)})^4] \le C_{b,\sigma} \Delta^4. \quad (6.2) $$

**Lemma 6.3.** Under Assumptions 1-3, for every $m \in \mathbb{N}^*$:

(i) For any $g \in S_m$ with $\|g\|_{\pi} = 1$,

$$ \mathbb{E}[(v_n^{(1)}(g))^2] = \frac{2}{n} \int g(x)^2 \sigma^4(x) \pi(x) dx. \quad (6.3) $$

Consequently,

$$ \mathbb{E} \left[ \sup_{\substack{g \in S_m \\ \|g\|_{\pi} = 1}} (v_n^{(1)}(g))^2 \right] = \frac{2}{n} \mathrm{Tr} \left( \Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2} \right). \quad (6.4) $$

(ii) Suppose, in addition, the stability condition (3.6) holds with constant $c$ and that $L(m) \le c_{\varphi}^2 m$ from Assumption 5(ii). Let $C_{b,\sigma} > 0$ be the constant (depending only on $b, \sigma$) such that, for every $k$, $\mathbb{E}[(V_{k\Delta}^{(i)})^2] \le C_{b,\sigma} \Delta$, $i \in \{2, 3, 4\}$ (cf. Lemma 6.2). Then, for all $n$ with $\log(n\Delta) \ge 1$,

$$ \mathbb{E} \left[ \sup_{\substack{g \in S_m \\ \|g\|_{\pi} = 1}} (\bar{v}_n(g))^2 \right] \le \frac{c C_{b,\sigma}}{2} \Delta^2. \quad (6.5) $$
**Lemma 6.4.** Under Assumptions 1–3, there exists $C_{b,\sigma} > 0$ depending only on $b, \sigma$ such that, for all $m, n, \Delta$,

$$ \mathbb{E}\left[\left\|\frac{1}{n}\hat{\Phi}_m^\top(V+R)\right\|_{2,m}^4\right] \le C_{b,\sigma} m L(m)^2 \left(\frac{2C_{\text{BDG}}(4)}{n^2} + \Delta^4\right), \quad (6.6) $$

where $C_{\text{BDG}}(4) > 0$ is a universal constant from the order-4 BDG inequality.

**6.1.1.** *Proof of (3.9).* By the definition of $\tilde{\sigma}_m^2 = \hat{\sigma}_m^2 \mathbf{1}_{\Lambda_m}$,

$$ \|\tilde{\sigma}_m^2 - \sigma_A^2\|_n^2 \le \|\sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m^c} + \|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c} + \|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m}. $$

We bound the three expectations separately.

**Step 1.** Term on $\Lambda_m^c$. By Hölder's inequality and Lemma 6.4,

$$ \mathbb{E}[\|\sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m^c}] \le (\mathbb{E}\sigma^8(X_0))^{1/2} (\mathbb{P}(\Lambda_m^c))^{1/2} \le (\mathbb{E}\sigma^8(X_0))^{1/2} \sqrt{C_{\text{mix}}} (n\Delta)^{-3}. $$

If Assumption 6 holds, then $n\Delta \to \infty$ and $(n\Delta)^{-3} \le n^{-1}$ for all large $n$, whence

$$ \mathbb{E}[\|\sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m^c}] \le C_1 \left(\frac{1}{n\Delta}\right)^3 \le \frac{C_1}{n}, \quad C_1 := (\mathbb{E}\sigma^8(X_0))^{1/2} \sqrt{C_{\text{mix}}}. \quad (6.7) $$

**Step 2.** Term on $\Lambda_m \cap \Omega_m$. Let $f \in S_m$ be arbitrary, $\gamma_n(\hat{\sigma}_m^2) \le \gamma_n(f)$. Since for all $g$,

$$ \begin{aligned} \gamma_n(g) &= \frac{1}{n} \sum_{k=1}^{n} \left( \sigma^2(X_{k\Delta}) + V_{k\Delta}^{(1)} + \bar{V}_{k\Delta} + R_{k\Delta} - g(X_{k\Delta}) \right)^2 \\ &= \frac{1}{n} \sum_{k=1}^{n} \left( \sigma^2(X_{k\Delta}) - g(X_{k\Delta}) \right)^2 + \frac{1}{n} \sum_{k=1}^{n} \left( V_{k\Delta}^{(1)} + \bar{V}_{k\Delta} + R_{k\Delta} \right)^2 \\ &\quad + \frac{2}{n} \sum_{k=1}^{n} (\sigma^2 - g)(X_{k\Delta}) (V_{k\Delta}^{(1)} + \bar{V}_{k\Delta} + R_{k\Delta}), \end{aligned} \quad (6.8) $$

one has $\|\hat{\sigma}_m^2 - \sigma^2\|_n^2 \le \|f - \sigma^2\|_n^2 + \frac{2}{n} \sum_{k=1}^n (\hat{\sigma}_m^2 - f)(X_{k\Delta}) (V_{k\Delta}^{(1)} + \bar{V}_{k\Delta} + R_{k\Delta})$. It implies, in particular, since $\hat{\sigma}_m^2$ and $f$ are A-supported,

$$ \|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \le \|f - \sigma_A^2\|_n^2 + \frac{2}{n} \sum_{k=1}^n (\hat{\sigma}_m^2 - f)(X_{k\Delta}) (V_{k\Delta}^{(1)} + \bar{V}_{k\Delta} + R_{k\Delta}). $$

By the elementary inequality $2ab \le \eta a^2 + \eta^{-1}b^2$, we get

$$ \begin{aligned} \|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \le \|f - \sigma_A^2\|_n^2 + 16 \sup_{\|g\|_{\pi}=1} (v_n^{(1)}(g))^2 + 16 \sup_{\|g\|_{\pi}=1} (\bar{v}_n(g))^2 + \frac{1}{8} \|\hat{\sigma}_m^2 - f\|_{\pi}^2 \\ + \frac{8}{n} \sum_{k=1}^n R_{k\Delta}^2 + \frac{1}{8} \|\hat{\sigma}_m^2 - f\|_n^2. \end{aligned} \quad (6.9) $$
Using the norm equivalence $\| \cdot \|_{\pi}^{2} \le 2 \| \cdot \|_{n}^{2}$ on $\Omega_{m}$, we obtain from (6.9)

$$
\begin{align*}
\|\hat{\sigma}_{m}^{2} - \sigma_{A}^{2}\|_{n}^{2} &\le \|f - \sigma_{A}^{2}\|_{n}^{2} + 16 \sup_{\|g\|_{\pi}=1} (v_{n}^{(1)}(g))^{2} + 16 \sup_{\|g\|_{\pi}=1} (\bar{v}_{n}(g))^{2} + \frac{1}{4}\|\hat{\sigma}_{m}^{2} - f\|_{n}^{2} \\
&\quad + \frac{8}{n}\sum_{k=1}^{n} R_{k\Delta}^{2} + \frac{1}{4}\|f - \sigma_{A}^{2}\|_{n}^{2} + \frac{1}{4}\|\hat{\sigma}_{m}^{2} - \sigma_{A}^{2}\|_{n}^{2} \\
&\le \|f - \sigma_{A}^{2}\|_{n}^{2} + 16 \sup_{\|g\|_{\pi}=1} (v_{n}^{(1)}(g))^{2} + 16 \sup_{\|g\|_{\pi}=1} (\bar{v}_{n}(g))^{2} \\
&\quad + \frac{8}{n}\sum_{k=1}^{n} R_{k\Delta}^{2} + \frac{3}{4}\|f - \sigma_{A}^{2}\|_{n}^{2} + \frac{3}{4}\|\hat{\sigma}_{m}^{2} - \sigma_{A}^{2}\|_{n}^{2} \\
&= \frac{7}{4}\|f - \sigma_{A}^{2}\|_{n}^{2} + 16 \sup_{\|g\|_{\pi}=1} (v_{n}^{(1)}(g))^{2} + 16 \sup_{\|g\|_{\pi}=1} (\bar{v}_{n}(g))^{2} + \frac{8}{n}\sum_{k=1}^{n} R_{k\Delta}^{2} + \frac{3}{4}\|\hat{\sigma}_{m}^{2} - \sigma_{A}^{2}\|_{n}^{2}.
\end{align*}
$$

Thus, $\|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \le 7 \|f - \sigma_A^2\|_n^2 + 64 \sup_{\|g\|_{\pi}=1} (v_n^{(1)}(g))^2 + 64 \sup_{\|g\|_{\pi}=1} (\bar{v}_n(g))^2 + 32 \sum_{k=1}^n R_{k\Delta}^2 / n$.
Taking expectations and applying Lemma 6.3 and Lemma 6.2 yields

$$
\begin{align}
\mathbb{E}[\|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m}] \le & 7 \|f - \sigma_A^2\|_{\pi}^2 + \frac{128}{n} \operatorname{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2}) \nonumber \\
& + 32 \mathfrak{C} C_{b,\sigma} \Delta^2 + 800 C_{b,\sigma} \Delta^2. \tag{6.10}
\end{align}
$$

Since $f \in S_m$ is arbitrary, we may replace $\|f - \sigma_A^2\|_{\pi}^2$ by $\inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2$.

**Step 3. Term on $\Lambda_m \cap \Omega_m^c$.** Let $\Pi_m \sigma_A^2$ denote the (empirical) least-squares projection of $\sigma_A^2$ on $S_m$,

$$
\Pi_m \sigma_A^2(x) := \sum_{j=0}^{m-1} \tilde{c}_j^{(m)} \varphi_j(x), \quad \tilde{c}^{(m)} = \frac{1}{n} \hat{\Psi}_m^{-1} \hat{\Phi}_m^{\top} \boldsymbol{\sigma}^2, \quad \boldsymbol{\sigma}^2 := (\sigma^2(X_{\Delta}), \dots, \sigma^2(X_{n\Delta}))^{\top}.
$$

Then, using $\hat{\sigma}_m^2 - \Pi_m \sigma_A^2 = \frac{1}{n} \sum_{j=0}^{m-1} \tilde{c}_j \varphi_j$ with $\tilde{c}^{(m)} = \frac{1}{n} \hat{\Psi}_m^{-1} \hat{\Phi}_m^{\top} (V+R)$

$$
\begin{align}
\|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_n^2 &= \frac{1}{n^2} (V+R)^{\top} \hat{\Phi}_m \hat{\Psi}_m^{-1} \hat{\Phi}_m^{\top} (V+R) \le \|\hat{\Psi}_m^{-1}\|_{\text{op}} \|\frac{1}{n} \hat{\Phi}_m^{\top} (V+R)\|_{2,m}^2 \nonumber \\
&\le \|\hat{\Psi}_m^{-1}\|_{\text{op}} \mathbb{E} \left[ \left\| \frac{1}{n} \hat{\Phi}_m^{\top} (V+R) \right\|_{2,m}^4 \right]^{1/2} (\mathbb{P}(\Omega_m^c))^{1/2} \tag{6.11}
\end{align}
$$

Since on $\Lambda_m$, $\|\hat{\Psi}_m^{-1}\|_{\text{op}} \le c n\Delta / (\log^2(n\Delta)L(m))$, Lemma 6.4, Lemma 6.1 and (6.11) give,

$$
\begin{align*}
\mathbb{E}[\|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}] &\le \frac{c n \Delta}{\log^2(n\Delta)} \cdot \frac{1}{L(m)} \cdot \left(C_{b,\sigma}^{(4)} m L(m)^2 \left(\frac{2C_{\text{BDG}}(4)}{n^2} + \Delta^4\right)\right)^{1/2} \cdot \sqrt{C_{\text{mix}}} (n\Delta)^{-3} \\
&= \frac{c \sqrt{C_{\text{mix}} C_{b,\sigma}}}{\log^2(n\Delta)} \sqrt{m} \left(\frac{2C_{\text{BDG}}(4)}{n^2} + \Delta^4\right)^{1/2} (n\Delta)^{-2}.
\end{align*}
$$

Since $m \le n\Delta$ we deduce

$$
\mathbb{E}[\|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}] \le c \sqrt{C_{\text{mix}} C_{b,\sigma}} \left( \frac{\sqrt{2C_{\text{BDG}}(4)}}{n} + \Delta^2 \right). \quad (6.12)
$$

Moreover,

$$
\mathbb{E}[\|\Pi_m \sigma_A^2 - \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}] \le \mathbb{E}[\|\sigma_A^2\|_n^2 \mathbf{1}_{\Omega_m^c}] \le (\mathbb{E}\sigma^8(X_0))^{1/2} \sqrt{C_{\text{mix}}} (n\Delta)^{-3}. \quad (6.13)
$$
Combining (6.12)–(6.13), we obtain a bound of the form

$$ \mathbb{E}[\|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}] \le \frac{C_2}{n} + C_3 \Delta^2, \quad (6.14) $$

for explicit constants

$$ C_2 := c \sqrt{2C_{\text{BDG}}(4)C_{\text{mix}}C_{b,\sigma}} + \sqrt{C_{\text{mix}}} (\mathbb{E}\sigma^8(X_0))^{1/2}, \quad C_3 := c \sqrt{C_{\text{mix}}C_{b,\sigma}}. $$

**Conclusion.** Combining (6.7), (6.10) and (6.14), we obtain the stated fixed-$m$ risk bounds of Proposition 3.1 with explicit constants. More precisely, there exist finite constants

$$ C := (32c C_{b,\sigma} + 32 C_{b,\sigma} + C_3 \lor C_1 + C_2) $$

such that

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_n^2] \le 7 \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \frac{64}{n} \text{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma^4} \Psi_m^{-1/2}) + C\{\Delta^2 + \frac{1}{n}\}. $$

This completes the proof of (3.9).

### 6.1.2. *Proof of (3.10).*

We decompose the risk into three pieces

$$ \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2] = (I) + (II) + (III), \quad (6.15) $$

where

$$ (I) := \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m^c}], \quad (II) := \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}], \quad (III) := \mathbb{E}[\|\tilde{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}]. $$

**Control of (III).** For any $g \in S_m$, we have using $(x+y)^2 \le (1+1/\theta)x^2 + (1+\theta)y^2$ with $\theta = 4$,

$$ \begin{aligned} \|\hat{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} &\le \frac{5}{4} \|\hat{\sigma}_m^2 - g\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} + 5 \|g - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} \\ &\le \frac{5}{2} \|\hat{\sigma}_m^2 - g\|_{n}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} + 5 \|g - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} \end{aligned} $$

by using the definition of $\Omega_m$. We insert $\sigma_A^2$ again and get:

$$ \|\hat{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} \le 5 \|\hat{\sigma}_m^2 - \sigma_A^2\|_{n}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} + 5 \|\sigma_A^2 - g\|_{n}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} + 5 \|g - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m}. $$

Therefore, taking infimum over $g \in S_m$ and applying (3.9) yield

$$ \mathbb{E} \left( \|\hat{\sigma}_m^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m} \right) \le 45 \inf_{g \in S_m} \left( \|g - \sigma_A^2\|_{\pi}^2 \right) + 5 \times 128 \frac{\text{Tr} \left[ \Psi_m^{-1/2} \Psi_{m,\sigma^2} \Psi_m^{-1/2} \right]}{n} + 5C \left\{ \Delta^2 + \frac{1}{n} \right\}, \quad (6.16) $$

**Control of (I).** By construction $\tilde{\sigma}_m^2 = 0$ on $\Lambda_m^c$, hence

$$ (I) = \mathbb{E}[\|\sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m^c}] \le \|\sigma_A^2\|_{\pi}^2 \mathbb{P}(\Lambda_m^c) = \mathbb{E}|\sigma^2(X_0)|\mathbb{P}(\Lambda_m^c). $$

By Lemma 6.1, $\mathbb{P}(\Lambda_m^c) \le \mathbb{P}(\Omega_m^c) \le C_{\text{mix}}(n\Delta)^{-6}$, and hence $(I) \le C_2 n^{-1}$ for large $n$ (under Assumption 6).

**Control of (II).** Write

$$ (II) \le 2 \mathbb{E}[\|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}] + 2 \mathbb{E}[\|\Pi_m \sigma_A^2 - \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Omega_m^c}] =: 2(II)_1 + 2(II)_2, $$

where $\Pi_m$ is the empirical least-squares projector onto $S_m$.

For $(II)_2$, by Hölder's inequality, $(II)_2 \le \|\sigma_A^2\|_{\pi}^2 \mathbb{P}(\Omega_m^c) \le \mathbb{E}|\sigma^2(X_0)|\mathbb{P}(\Omega_m^c) \le C(b, \sigma, \theta) n^{-1}$.
For (II)$_1$, Let $d^{(m)} = \frac{1}{n} \hat{\Psi}_m^{-1} \hat{\Phi}_m^\top (V + R)$ be the coefficient vector of $\hat{\sigma}_m^2 - \Pi_m \sigma_A^2 = \sum_{j=0}^{m-1} d_j \varphi_j$ in the basis $\{\varphi_j\}_{j=0}^{m-1}$. Then, by definition,

$$ \|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_{\pi}^2 = \int \left( \sum_{j=0}^{m-1} d_j \varphi_j(x) \right)^2 \pi(x) dx = (d^{(m)})^\top \Psi_m d^{(m)}. \quad (6.17) $$

Moreover, for any $a \in \mathbb{R}^m$, by the pointwise Cauchy-Schwarz inequality,

$$ \left(\sum_{j=0}^{m-1} a_j \varphi_j(x)\right)^2 \le \left(\sum_{j=0}^{m-1} a_j^2\right) \left(\sum_{j=0}^{m-1} \varphi_j(x)^2\right) \le \|a\|_{2,m}^2 L(m), $$

and since $\pi$ is a probability density, $a^\top \Psi_m a = \int (\sum_j a_j \varphi_j)^2 \pi(m) \|a\|_{2,m}^2 \mathrm{d}m$. Hence

$$ \|\Psi_m\|_{\mathrm{op}} = \sup_{\|a\|_{2,m}=1} a^\top \Psi_m a \le L(m). \quad (6.18) $$

Combining (6.17) and (6.18) yields

$$ \|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_{\pi}^2 \le \|\Psi_m\|_{\mathrm{op}} \|d^{(m)}\|_{2,m}^2 \le L(m) \|d^{(m)}\|_{2,m}^2 = L(m) \|\hat{\Psi}_m^{-1}\|_{\mathrm{op}}^2 \|\frac{1}{n}\hat{\Phi}_m^\top (V + R)\|_{2,m}^2. $$

On the stability event $\Lambda_m$, $\|\hat{\Psi}_m^{-1}\|_{\mathrm{op}} \le c \frac{n\Delta}{L(m) \log^2(n\Delta)}$. Therefore, by Cauchy-Schwarz and Lemma 6.4,

$$ \begin{aligned} (II)_1 &= \mathbb{E}[\|\hat{\sigma}_m^2 - \Pi_m \sigma_A^2\|_{\pi}^2 \mathbf{1}_{\Lambda_m \cap \Omega_m^c}] \\ &\le L(m) \frac{c^2 n^2 \Delta^2}{L(m)^2 \log^4(n\Delta)} \left( \mathbb{E}\left\|\frac{1}{n}\hat{\Phi}_m^\top (V + R)\right\|_{2,m}^4 \right)^{1/2} \mathbb{P}(\Omega_m^c)^{1/2} \\ &\le C(b, \sigma) \frac{n^2 \Delta^2}{L(m) \log^4(n\Delta)} \left(m L(m)^2 \left(\frac{2}{n^2} + \Delta^4\right)\right)^{1/2} (n\Delta)^{-3} \\ &\le C(b, \sigma) \left(\frac{C_{\mathrm{BDG}}(4)}{n} + \Delta^2\right), \end{aligned} $$

using $m \le n\Delta$, $L(m) \ge 1$ and $\log(n\Delta) \ge 1$. Together with the bound on (II)$_2$, this gives $(II) \le C(b, \sigma)(\frac{1}{n} + \Delta^2)$.

**Conclusion.** Combining (6.16) with the bounds for (I) and (II) in (6.15) yields (3.10).

## 6.2. Proof of Lemma 6.1.

For $g = \sum_{j=0}^{m-1} x_j \varphi_j \in S_m$, we have $\|g\|_{\pi}^2 = \vec{x}^\top \Psi_m \vec{x}$, $\|g\|_n^2 = \vec{x}^\top \hat{\Psi}_m \vec{x}$. Thus

$$ \sup_{\|g\|_{\pi}=1} \left| \frac{\|g\|_n^2}{\|g\|_{\pi}^2} - 1 \right| = \sup_{\substack{\vec{x} \in \mathbb{R}^m \\ \|\Psi_m^{1/2} \vec{x}\|_2=1}} \left| \vec{x}^\top \hat{\Psi}_m \vec{x} - \vec{x}^\top \Psi_m \vec{x} \right| = \left\| \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} - \mathrm{Id}_m \right\|_{\mathrm{op}}. \quad (6.19) $$

Therefore, $\Omega_m = \left\{ \|\Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} - \mathrm{Id}_m\|_{\mathrm{op}} < \frac{1}{2} \right\}$. Then, on $\Omega_m$, $\frac{1}{2} I_m \asymp \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} \asymp \frac{3}{2} I_m$. In particular, by congruence and inversion preserving the Loewner order,

$$ \hat{\Psi}_m \succeq \frac{1}{2} \Psi_m \quad \text{and} \quad \|\hat{\Psi}_m^{-1}\|_{\mathrm{op}} \le 2 \|\Psi_m^{-1}\|_{\mathrm{op}}. $$

Combining this with (3.6) yields

$$ L(m)(\|\hat{\Psi}_m^{-1}\|_{\mathrm{op}} \vee 1) \le 2 L(m)(\|\Psi_m^{-1}\|_{\mathrm{op}} \vee 1) \le \frac{c n \Delta}{\log^2(n\Delta)}, $$
i.e. $\Omega_m \subset \Lambda_m$. Hence $\mathbb{P}(\Lambda_m^c) \le \mathbb{P}(\Omega_m^c)$.

To prove $\mathbb{P}(\Omega_m^c) \le C(n\Delta)^{-6}$, we use Berbee's coupling lemma (see, e.g., Berbee (1979)) combined with the blocking method of Viennet (1997, Prop. 5.1). Let $(Y_i)_{i \ge 1}$ be a sequence with $\beta$-mixing coefficients $\beta(k)$. For any $q \ge 1$, there exists a coupled sequence $(Y_i^*)_{i \ge 1}$ such that:

* Each block of length $q$ has the same distribution as the original block;

* The coupled blocks are independent;

* $\mathbb{P}[(Y_1, \dots, Y_q) \ne (Y_1^*, \dots, Y_q^*)] \le \beta(q)$.

Assume $n = 2p_nq_n$ for integers $p_n, q_n$. Apply Berbee's lemma to $(X_{i\Delta})_{1 \le i \le n}$ with block length $q_n$. Define the blocks:

$$ \vec{X}_{\ell,1} = (X_{(2\ell q_n+1)\Delta}, \dots, X_{(2\ell+1)q_n\Delta})^{\top}, \quad \vec{X}_{\ell,2} = (X_{[(2\ell+1)q_n+1]\Delta}, \dots, X_{(2\ell+2)q_n\Delta})^{\top}, $$

and their coupled versions $(\vec{X}_{\ell,\delta}^*)$, $\delta = 1, 2$, such that: $\mathbb{P}[\vec{X}_{\ell,\delta} \ne \vec{X}_{\ell,\delta}^*] \le \beta_X(q_n\Delta) \le e^{-\theta q_n\Delta}$, and for each fixed $\delta$, the coupled blocks $\vec{X}_{0,\delta}^*, \dots, \vec{X}_{p_n-1,\delta}^*$ are independent. Let $\Omega^* := \{X_{i\Delta} = X_{i\Delta}^*, i = 1, \dots, n\}$. Then:

$$ \mathbb{P}(\Omega_m^c) \le \mathbb{P}(\Omega_m^c \cap \Omega^*) + 2p_n e^{-\theta q_n \Delta}. $$

Choosing $q_n\Delta := 7 \log(n\Delta)/\theta$, $p_n = n/2q_n = n\Delta\theta/14 \log(n\Delta)$, yields:

$$ 2p_n e^{-\theta q_n \Delta} = \frac{\theta n \Delta}{7 \log(n\Delta)} \cdot \frac{1}{(n\Delta)^7} = \frac{\theta}{7 \log(n\Delta)} \cdot \frac{1}{(n\Delta)^6}. \quad (6.20) $$

Set: $\mathbf{S}_m := \Psi_m^{-1/2}\hat{\Psi}_m\Psi_m^{-1/2} - \mathrm{Id}_m = \frac{1}{n}\sum_{i=1}^n \{\mathbf{K}_m(X_{i\Delta}) - \mathbb{E}[\mathbf{K}_m(X_{i\Delta})]\}$, where:

$$ \mathbf{K}_m(X) = \Psi_m^{-1/2} \tilde{\mathbf{K}}_m(X) \Psi_m^{-1/2}, \quad \tilde{\mathbf{K}}_m(X) = [\varphi_j(X)\varphi_k(X)]_{0 \le j,k \le m-1}. $$

By definition, $\mathbb{E}[\mathbf{K}_m(X_{i\Delta})] = \mathrm{Id}_m$. Write: $\mathbf{S}_m = \frac{1}{2}(\mathbf{S}_{m,1} + \mathbf{S}_{m,2})$, where $\mathbf{S}_{m,1}$ is the average over the $\vec{X}_{\ell,1}$-blocks and $\mathbf{S}_{m,2}$ over the $\vec{X}_{\ell,2}$-blocks. On $\Omega^*$, replacing the variables by their coupled versions, we have:

$$ \mathbb{P}(\|\mathbf{S}_m\|_{\mathrm{op}} > 1/2, \Omega^*) \le \mathbb{P}(\|\mathbf{S}_{m,1}^*\|_{\mathrm{op}} > 1/2) + \mathbb{P}(\|\mathbf{S}_{m,2}^*\|_{\mathrm{op}} > 1/2), $$

where $\mathbf{S}_{m,\delta}^*$ are the corresponding sums over independent coupled blocks. Focus on $\mathbf{S}_{m,1}^*$. Write:

$$ \mathbf{S}_{m,1}^* = \sum_{\ell=0}^{p_n-1} \mathbf{G}_{\ell}^* - \mathrm{Id}_m, \quad \mathbf{G}_{\ell}^* := \frac{1}{p_n} \cdot \frac{1}{q_n} \sum_{r=1}^{q_n} \mathbf{K}_m(X_{(2\ell q_n+r)\Delta}^*). $$

Since $\|\tilde{\mathbf{K}}_m(x)\|_{\mathrm{op}} \le \sum_{j=0}^{m-1} \varphi_j(x)^2$, $\|\mathbf{K}_m(x)\|_{\mathrm{op}} \le \|\Psi_m^{-1/2}\|_{\mathrm{op}}^2 L(m) = L(m) \|\Psi_m^{-1}\|_{\mathrm{op}}$, and hence

$$ \|\mathbf{G}_{\ell}^*\|_{\mathrm{op}} \le \frac{1}{p_n q_n} \sum_{r=1}^{q_n} \|\mathbf{K}_m(X_{(2\ell q_n+r)\Delta}^*)\|_{\mathrm{op}} \le \frac{L(m) \|\Psi_m^{-1}\|_{\mathrm{op}}}{p_n}. $$

Applying the matrix Chernoff inequality (see Lemma A.1, we get for $c := 3 \log(\frac{3}{2}) - 1 > 0$:

$$ \mathbb{P} \left( \| \mathbf{S}_{m,1}^* \|_{\mathrm{op}} > \frac{1}{2} \right) \le 2m \exp \left( - \frac{c p_n}{L(m) \| \Psi_m^{-1} \|_{\mathrm{op}}} \right). $$

By (3.6), $c p_n/L(m) \|\Psi_m^{-1}\|_{\mathrm{op}} \ge 2cp_n \log^2(n\Delta)/c n\Delta = 2C_0 \log(n\Delta)/14 \ge 7 \log(n\Delta)$, so

$$ \mathbb{P} \left( \| \mathbf{S}_{m,1}^* \|_{\mathrm{op}} > \frac{1}{2} \right) \le 2(n\Delta)^{-6}. \quad (6.21) $$

The same bound holds for $\mathbf{S}_{m,2}^*$. Combining the bounds (6.20) and (6.21) completes the proof of Lemma 6.1.
6.3. *Proof of Lemma 6.2.* Fix $k \in \mathbb{N}$ and note,

$$G := \int_{k\Delta}^{(k+1)\Delta} b(X_s) \, ds, \quad H := \int_{k\Delta}^{(k+1)\Delta} \sigma(X_s) \, dW_s,$$

$$I := \int_{k\Delta}^{(k+1)\Delta} (\sigma(X_s) - \sigma(X_{k\Delta})) \, dW_s, \quad J := \int_{k\Delta}^{(k+1)\Delta} (\sigma(X_s) - \sigma(X_{k\Delta})) \, ds,$$

$$K := \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \sigma'(X_s) \sigma(X_s) \, dW_s,$$

$$L := \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s) \psi(X_s) \, ds, \quad \psi := \sigma'b + \frac{1}{2}\sigma''\sigma^2,$$

$$M := \int_{k\Delta}^{(k+1)\Delta} (b(X_s) - b(X_{k\Delta})) \, ds, \quad N := W_{(k+1)\Delta} - W_{k\Delta}.$$

Throughout we use three routine tools (with constants depending only on the order $p$):

* 1. Hölder's inequality in For $p \ge 1$, $|\int_a^b f|^p \le (b-a)^{p-1} \int_a^b |f|^p$.

* 2. BDG inequality For $p \ge 2$, $\mathbb{E}|\int_a^b \xi_s \, dW_s|^p \le C_{\text{BDG}}(p) \mathbb{E}(|\xi_s|^2 \, ds)^{p/2}$.

* 3. Increment bound Proposition 2.2: For $f = b$ or $f = \sigma$ and $h > 0$,

$$\mathbb{E}\left[\sup_{u \in [t, t+h]} |f(X_u) - f(X_t)|^p\right] \le C_{\text{inc}}(b, \sigma, p) h^{p/2}. \quad (6.22)$$

By the standing assumptions on $(b, \sigma)$, all moments below are finite and stationary; we bundle harmless constants into $C_{b,\sigma} > 0$ that may change from line to line (but is independent of $k$ and $\Delta$). **Step 1: Eighth moments of the elementary blocks.** Using Hölder, BDG and (6.22) with $p=8$, we obtain

$$\mathbb{E}|G|^8 \le \Delta^7 \int_{k\Delta}^{(k+1)\Delta} \mathbb{E}|b(X_s)|^8 ds = \Delta^8 \mathbb{E}|b(X_0)|^8 \le C_{b,\sigma} \Delta^8,$$

$$\mathbb{E}|H|^8 \le C_{\text{BDG}}(8) \mathbb{E}\left(\int_{k\Delta}^{(k+1)\Delta} \sigma^2 ds\right)^4 \le C_{b,\sigma} \Delta^4,$$

$$\mathbb{E}|I|^8 \le C_{\text{BDG}}(8) \Delta^3 \int_{k\Delta}^{(k+1)\Delta} \mathbb{E}|\sigma(X_s) - \sigma(X_{k\Delta})|^8 ds \le C_{b,\sigma} \Delta^8,$$

$$\mathbb{E}|J|^8 \le \Delta^7 \int_{k\Delta}^{(k+1)\Delta} \mathbb{E}|\sigma(X_s) - \sigma(X_{k\Delta})|^8 ds \le C_{b,\sigma} \Delta^{12},$$

$$\mathbb{E}|K|^8 \le C_{\text{BDG}}(8) \Delta^3 \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s)^8 \mathbb{E}|\sigma'(X_s)\sigma(X_s)|^8 ds \le C_{b,\sigma} \Delta^{12},$$

$$\mathbb{E}|L|^8 \le \Delta^7 \int_{k\Delta}^{(k+1)\Delta} ((k+1)\Delta - s)^8 \mathbb{E}|\psi(X_s)|^8 ds \le C_{b,\sigma} \Delta^{16},$$

$$\mathbb{E}|M|^8 \le \Delta^7 \int_{k\Delta}^{(k+1)\Delta} \mathbb{E}|b(X_s) - b(X_{k\Delta})|^8 ds \le C_{b,\sigma} \Delta^{12},$$

$$\mathbb{E}|N|^8 = 105 \Delta^4 \quad (\text{Gaussian moment}).$$

**Step 2: Fourth moments of $V_{k\Delta}^{(i)}$ and $R_{k\Delta}^{(i)}$.** We repeatedly use, for random variables $A, B$, $\mathbb{E}|AB|^4 \le (\mathbb{E}|A|^8)^{1/2} (\mathbb{E}|B|^8)^{1/2}$, $|a+b|^4 \le 8 (|a|^4+|b|^4)$.
(i) $V_{k\Delta}^{(2)} = \frac{2}{\Delta}\sigma(X_{k\Delta})(NI - J): \mathbb{E}[(V_{k\Delta}^{(2)})^4] \le \frac{16}{\Delta^4}(\mathbb{E}\sigma^8)^{1/2}(8\mathbb{E}|NI|^4 + 8\mathbb{E}|J|^4)^{1/2} \le C_{b,\sigma}\Delta^2$, since $\mathbb{E}|NI|^4 \le (\mathbb{E}N^8)^{1/2}(\mathbb{E}I^8)^{1/2} \le C_{b,\sigma}\Delta^6$ and $(\mathbb{E}|J|^8)^{1/2} \le C_{b,\sigma}\Delta^6$.

(ii) $V_{k\Delta}^{(3)} = \frac{2}{\Delta}\sigma(X_{k\Delta}) K: \mathbb{E}[(V_{k\Delta}^{(3)})^4] \le \frac{16}{\Delta^4}(\mathbb{E}\sigma^8)^{1/2}(\mathbb{E}|K|^8)^{1/2} \le C_{b,\sigma}\Delta^2$.

(iii) $V_{k\Delta}^{(4)} = 2b(X_{k\Delta})\sigma(X_{k\Delta})N: \mathbb{E}[(V_{k\Delta}^{(4)})^4] = 16\mathbb{E}[b^4\sigma^4N^4] \le 16(\mathbb{E}b^8\sigma^8)^{1/2}(\mathbb{E}N^8)^{1/2} \le C_{b,\sigma}\Delta^2$.

(iv) $R_{k\Delta}^{(1)} = \Delta^{-1}G^2: \mathbb{E}[(R_{k\Delta}^{(1)})^4] = \Delta^{-4}\mathbb{E}|G|^8 \le C_{b,\sigma}\Delta^4$.

(v) $R_{k\Delta}^{(2)} = \Delta^{-1}I^2: \mathbb{E}[(R_{k\Delta}^{(2)})^4] = \Delta^{-4}\mathbb{E}|I|^8 \le C_{b,\sigma}\Delta^4$.

(vi) $R_{k\Delta}^{(3)} = \frac{2}{\Delta}MH: \mathbb{E}[(R_{k\Delta}^{(3)})^4] \le \frac{16}{\Delta^4}(\mathbb{E}M^8)^{1/2}(\mathbb{E}H^8)^{1/2} \le C_{b,\sigma}\Delta^4$.

(vii) $R_{k\Delta}^{(4)} = 2b(X_{k\Delta})I: \mathbb{E}[(R_{k\Delta}^{(4)})^4] \le 16(\mathbb{E}b^8)^{1/2}(\mathbb{E}I^8)^{1/2} \le C_{b,\sigma}\Delta^4$.

(viii) $R_{k\Delta}^{(5)} = \frac{2}{\Delta}\sigma(X_{k\Delta})L: \mathbb{E}[(R_{k\Delta}^{(5)})^4] \le \frac{16}{\Delta^4}(\mathbb{E}\sigma^8)^{1/2}(\mathbb{E}|L|^8)^{1/2} \le C_{b,\sigma}\Delta^4$.

**Conclusion.** Collecting (i)–(viii) yields the fourth-moment bounds claimed in (6.2). The second-moment bounds (6.1) then follow by Cauchy-Schwarz.

## 6.4. *Proof of Lemma 6.3.*

*Proof of (6.3).*

Fix $g \in S_m$ with $\|g\|_{\pi} = 1$. Write $Z_k := \frac{(W_{(k+1)\Delta} - W_{k\Delta})^2}{\Delta} - 1$ so that $V_{k\Delta}^{(1)} = \sigma^2(X_{k\Delta}) Z_k$. Since $Z_k$ is independent of $\mathcal{F}_{k\Delta}$, centered, and $\mathbb{E}[Z_k^2] = 2$, we have

$$ \mathbb{E}[(v_n^{(1)}(g))^2] = \mathbb{E}\left[\mathrm{Var}\left(\frac{1}{n}\sum_{k=1}^n g(X_{k\Delta}) \sigma^2(X_{k\Delta}) Z_k \middle| (X_{u\Delta})_{u \le n}\right)\right] = \frac{1}{n^2} \sum_{k=1}^n 2 \mathbb{E}[g^2(X_{k\Delta}) \sigma^4(X_{k\Delta})], $$

and by stationarity this equals $(2/n) \mathbb{E}[g^2(X_0)\sigma^4(X_0)]$, which is (6.3).

*Proof of (6.4).* Let $\{\varphi_0, \dots, \varphi_{m-1}\}$ be the fixed basis of $S_m$ and

$$ \varphi(x) := (\varphi_0(x), \dots, \varphi_{m-1}(x))^{\top}. $$

For $g = \sum_j a_j\varphi_j$ and $h = \sum_k b_k\varphi_k$, we have $\langle g, h \rangle_{\pi} = a^{\top}\Psi_m b$. Define $f_{\ell} := \sum_{j=0}^{m-1} [\Psi_m^{-1/2}]_{\ell j} \varphi_j$, $0 \le \ell \le m-1$. Then, for $\ell, r$,

$$ \langle f_{\ell}, f_{r} \rangle_{\pi} = e_{\ell}^{\top} \Psi_{m}^{-1/2} \left( \int \varphi(x) \varphi(x)^{\top} \pi(x) dx \right) \Psi_{m}^{-1/2} e_{r} = e_{\ell}^{\top} I_{m} e_{r} = \delta_{\ell r}, $$

so $\{f_0, \dots, f_{m-1}\}$ is $\pi$-orthonormal. Any $g \in S_m$ with $\|g\|_{\pi} = 1$ can be written as $g = \sum_{\ell=0}^{m-1} \theta_{\ell} f_{\ell}$ with $\sum_{\ell} \theta_{\ell}^2 = 1$. Thus, $\sup_{\|g\|_{\pi}=1} (v_n^{(1)}(g))^2 = \sum_{\ell=0}^{m-1} (v_n^{(1)}(f_{\ell}))^2$. Using (6.3), write $f_{\ell} = \sum_j [\Psi_m^{-1/2}]_{\ell j} \varphi_j$,

$$ \begin{align*} \mathbb{E}\left[\sup_{\|g\|_{\pi}=1} (v_n^{(1)}(g))^2\right] &= \frac{2}{n} \sum_{\ell=0}^{m-1} \int f_{\ell}(x)^2 \sigma^4(x) \pi(x) dx \\ &= \frac{2}{n} \sum_{\ell=0}^{m-1} \sum_{j,k=0}^{m-1} [\Psi_m^{-1/2}]_{\ell j} [\Psi_m^{-1/2}]_{\ell k} \int \varphi_j \varphi_k \sigma^4 \pi \\ &= \frac{2}{n} \mathrm{Tr}(\Psi_m^{-1/2} \Psi_{m,\sigma} \Psi_m^{-1/2}). \end{align*} $$

*Proof of (6.5).*

For $g \in S_m$ with $\|g\|_{\pi} = 1$ write $g(x) = a^{\top}\varphi(x)$ with $a^{\top}\Psi_m a = 1$. Define the random vector $Z :=$
$\frac{1}{n} \sum_{k=1}^{n} \varphi(X_{k\Delta}) \bar{V}_{k\Delta} \in \mathbb{R}^m$. Then $\bar{v}_n(g) = a^\top Z$ and $\sup_{\|g\|_\pi=1} (\bar{v}_n(g))^2 = \sup_{a^\top \Psi_m a=1} (a^\top Z)^2 = Z^\top \Psi_m^{-1} Z$. Then,

$$ \mathbb{E}\left[\sup_{\|g\|_\pi=1} (\bar{v}_n(g))^2\right] = \mathbb{E}[Z^\top \Psi_m^{-1} Z] \le \|\Psi_m^{-1}\|_{\text{op}} \mathbb{E}[\|Z\|_{2,m}^2]. \quad (6.23) $$

Next, $\|Z\|_{2,m}^2 = \left\| \frac{1}{n} \sum_{k=1}^{n} \varphi(X_{k\Delta}) \bar{V}_{k\Delta} \right\|_{2,m}^2 \le \frac{1}{n} \sum_{k=1}^{n} \|\varphi(X_{k\Delta})\|_{2,m}^2 \bar{V}_{k\Delta}^2$, hence, by stationarity and $\|\varphi(x)\|_2^2 = \sum_{j=0}^{m-1} \varphi_j(x)^2 \le L(m)$,

$$ \mathbb{E}[\|Z\|_{2,m}^2] \le \frac{L(m)}{n} \mathbb{E}[\bar{V}_{k\Delta}^2] \le \frac{L(m)}{n} \sum_{i=2}^{4} \mathbb{E}[(V_{k\Delta}^{(i)})^2] \le \frac{L(m)}{n} C_{b,\sigma} \Delta. \quad (6.24) $$

Combining (6.23) and (6.24) gives $\mathbb{E}[\sup_{\|g\|_\pi=1} (\bar{v}_n(g))^2] \le \|\Psi_m^{-1}\|_{\text{op}} \frac{L(m)}{n} C_{b,\sigma} \Delta$. By the stability condition (3.6), $\|\Psi_m^{-1}\|_{\text{op}} \frac{L(m)}{n} \le \frac{c}{2} \frac{\Delta}{\log^2(n\Delta)} \le \frac{c}{2} \Delta$ whenever $\log(n\Delta) \ge 1$. Hence (6.5).

### 6.5. Proof of Lemma 6.4.

Set $E := V + R$ and note $\left\| \frac{1}{n} \hat{\Phi}_m^\top E \right\|_{2,m}^4 = \frac{1}{n^4} \left( \sum_{j=0}^{m-1} \left( \sum_{i=1}^n \varphi_j(X_{i\Delta}) E_{i\Delta} \right)^2 \right)^2$. By Hölder's inequality (with exponents (2,2) on the index $j$),

$$ \left( \sum_{j=0}^{m-1} a_j^2 \right)^2 \le m \sum_{j=0}^{m-1} a_j^4, \quad a_j \in \mathbb{R}, \quad (6.25) $$

hence

$$ \left\| \frac{1}{n} \hat{\Phi}_m^\top E \right\|_{2,m}^4 \le \frac{m}{n^4} \sum_{j=0}^{m-1} \left( \sum_{i=1}^n \varphi_j(X_{i\Delta}) E_{i\Delta} \right)^4. \quad (6.26) $$

Using $(a+b)^4 \le 8(a^4+b^4)$, we split $E = V + R$ and obtain

$$ \left\| \frac{1}{n} \hat{\Phi}_m^\top E \right\|_{2,m}^4 \le \frac{8m}{n^4} \sum_{j=0}^{m-1} \left[ \left( \sum_{i=1}^n \varphi_j(X_{i\Delta}) V_{i\Delta} \right)^4 + \left( \sum_{i=1}^n \varphi_j(X_{i\Delta}) R_{i\Delta} \right)^4 \right]. $$

Taking expectations and treating the two contributions separately:

$$ \begin{gathered} \text{(I) The R-term. Since } \left( \sum_{i=1}^n |a_i| \right)^4 \le n^3 \sum_{i=1}^n |a_i|^4, \text{ so} \\ \mathbb{E}\left[ \left( \sum_{i=1}^n \varphi_j(X_{i\Delta}) R_{i\Delta} \right)^4 \right] \le n^3 \sum_{i=1}^n \mathbb{E}[\varphi_j^4(X_{i\Delta}) R_{i\Delta}^4]. \end{gathered} $$

By stationarity, $\sum_{i=1}^n \mathbb{E}[\varphi_j^4(X_{i\Delta}) R_{i\Delta}^4] = n \mathbb{E}[\varphi_j^4(X_0) R_0^4]$. Summing over $j$ and using $\sum_{j=0}^{m-1} \varphi_j(x)^4 \le (\sum_{j=0}^{m-1} \varphi_j(x)^2)^2 \le L(m)^2$,

$$ \sum_{j=0}^{m-1} \mathbb{E}[\varphi_j^4(X_0) R_0^4] = \mathbb{E}\left[\left(\sum_{j=0}^{m-1} \varphi_j^4(X_0)\right) R_0^4\right] \le L(m)^2 \mathbb{E}[R_0^4]. $$

Therefore, by Lemma 6.2,

$$ \begin{aligned} \frac{8m}{n^4} \sum_{j=0}^{m-1} \mathbb{E} \left[ \left( \sum_{i=1}^n \varphi_j(X_{i\Delta}) R_{i\Delta} \right)^4 \right] &\le \frac{8m}{n^4} n^3 \cdot n L(m)^2 \mathbb{E}[R_0^4] = 8m L(m)^2 \mathbb{E}[R_0^4] \\ &\le 8 C_{b,\sigma} m L(m)^2 \Delta^4. \end{aligned} \quad (6.27) $$
(II) The V-term. Fix $j$ and set $D_i^{(j)} := \varphi_j(X_{i\Delta}) V_{i\Delta}$. Since $\mathbb{E}[V_{i\Delta} | \mathcal{F}_{i\Delta}] = 0$ (each $V_{i\Delta}^{(r)}$ has zero conditional mean) and $\varphi_j(X_{i\Delta})$ is $\mathcal{F}_{i\Delta}$-measurable, $(D_i^{(j)})_{1\le i\le n}$ is a martingale difference sequence. By fourth-moment BDG inequality, there exists a universal $C_{\text{BDG}}(4) > 0$ such that

$$ \mathbb{E}\left[\left(\sum_{i=1}^{n} D_i^{(j)}\right)^4\right] \le C_{\text{BDG}}(4) \left\{ \left(\sum_{i=1}^{n} \mathbb{E}[(D_i^{(j)})^2]\right)^2 + \sum_{i=1}^{n} \mathbb{E}[(D_i^{(j)})^4] \right\}. $$

By stationarity, $\sum_{i=1}^n \mathbb{E}[(D_i^{(j)})^2] = n \mathbb{E}[\varphi_j^2(X_0) V_0^2]$, $\sum_{i=1}^n \mathbb{E}[(D_i^{(j)})^4] = n \mathbb{E}[\varphi_j^4(X_0) V_0^4]$. Applying Hölder's inequality, $(\mathbb{E}[\varphi_j^2(X_0) V_0^2])^2 \le \mathbb{E}[\varphi_j^4(X_0)] \mathbb{E}[V_0^4]$. Summing over $j$ and invoking $\sum_{j=0}^{m-1} \varphi_j^4(x) \le L(m)^2$ as above, we obtain

$$ \sum_{j=0}^{m-1} (\mathbb{E}[\varphi_j^2(X_0) V_0^2])^2 \le L(m)^2 \mathbb{E}[V_0^4], \quad \sum_{j=0}^{m-1} \mathbb{E}[\varphi_j^4(X_0) V_0^4] \le L(m)^2 \mathbb{E}[V_0^4]. $$

Hence

$$ \begin{aligned} \frac{8m}{n^4} \sum_{j=0}^{m-1} \mathbb{E} \left[ \left( \sum_{i=1}^{n} \varphi_j(X_{i\Delta}) V_{i\Delta} \right)^4 \right] &\le \frac{8m}{n^4} C_{\text{BDG}}(4) \sum_{j=0}^{m-1} \left\{ n^2 (\mathbb{E}[\varphi_j^2(X_0) V_0^2])^2 + n \mathbb{E}[\varphi_j^4(X_0) V_0^4] \right\} \\ &\le \frac{8m}{n^4} C_{\text{BDG}}(4) \left\{ n^2 L(m)^2 \mathbb{E}[V_0^4] + n L(m)^2 \mathbb{E}[V_0^4] \right\} \\ &\le \frac{8m C_{\text{BDG}}(4) L(m)^2 \mathbb{E}[V_0^4]}{n^2} \left( 1 + \frac{1}{n} \right) \le \frac{16m C_{\text{BDG}}(4) L(m)^2 \mathbb{E}[V_0^4]}{n^2}. \end{aligned} $$

By Lemma 6.2,

$$ \frac{8m}{n^4} \sum_{j=0}^{m-1} \mathbb{E} \left[ \left( \sum_{i=1}^{n} \varphi_j(X_{i\Delta}) V_{i\Delta} \right)^4 \right] \le \frac{16 C_{\text{BDG}}(4) C_{b,\sigma}}{n^2} m L(m)^2. \quad (6.28) $$

Finally, taking expectations in (6.26) and combining (6.27)–(6.28) yields (6.6).

### 6.6. Proof of Lemma 3.4.

Fix $m \in \mathcal{M}_{n\Delta}^+$. On $\Omega_m$, from (6.19), $\frac{1}{2} I_m \asymp \Psi_m^{-1/2} \hat{\Psi}_m \Psi_m^{-1/2} \asymp \frac{3}{2} I_m$, thus $\frac{1}{2} \Psi_m \asymp \hat{\Psi}_m \asymp \frac{3}{2} \Psi_m$, and, $\frac{2}{3} \Psi_m^{-1} \asymp \hat{\Psi}_m^{-1} \asymp 2 \Psi_m^{-1}$. Taking operator norms,

$$ \frac{2}{3} \|\Psi_m^{-1}\|_{\text{op}} \le \|\hat{\Psi}_m^{-1}\|_{\text{op}} \le 2 \|\Psi_m^{-1}\|_{\text{op}}. \quad (6.29) $$

We now prove the two inclusions. (i) $\mathcal{M}_{n\Delta} \subset \hat{\mathcal{M}}_{n\Delta}$. Let $m \in \mathcal{M}_{n\Delta}$. Then $c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le \frac{c}{2} \frac{n\Delta}{\log^2(n\Delta)}$. Using the right-hand inequality in (6.29) and $(2x \vee 1) \le 2(x \vee 1)$,

$$ c_{\varphi}^2 m (\|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le c_{\varphi}^2 m (2 \|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le 2 c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le c \frac{n\Delta}{\log^2(n\Delta)}. $$

Hence $m \in \hat{\mathcal{M}}_{n\Delta}$.

(ii) $\hat{\mathcal{M}}_{n\Delta} \subset \mathcal{M}_{n\Delta}^+$. Let $m \in \hat{\mathcal{M}}_{n\Delta}$. Then $c_{\varphi}^2 m (\|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le c \frac{n\Delta}{\log^2(n\Delta)}$. Using the left-hand inequality in (6.29) and $(\frac{3}{2}x \vee 1) \le \frac{3}{2}(x \vee 1)$,

$$ c_{\varphi}^2 m (\|\Psi_m^{-1}\|_{\text{op}} \vee 1) \le c_{\varphi}^2 m (\frac{3}{2} \|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le \frac{3}{2} c_{\varphi}^2 m (\|\hat{\Psi}_m^{-1}\|_{\text{op}} \vee 1) \le \frac{3c}{2} \frac{n\Delta}{\log^2(n\Delta)}. $$

Hence $m \in \mathcal{M}_{n\Delta}^+$.

Combining (i) and (ii) for all $m \in \mathcal{M}_{n\Delta}^+$ on $\Omega_m$ yields the claimed set inclusion on $\Omega_{n\Delta}$.
## 6.7. Proof of Theorem 3.5.

For $m, m' \in \mathbb{N}^*$, let $B_{m,m'}^{\pi} := \{g \in S_m + S_{m'} : \|g\|_{\pi} = 1\}$. Define for $m, m' \in \mathbb{N}^*$:

$$p(m, m') := \kappa_0 c_{\varphi}^2 \mathbb{E}[\sigma^4(X_0)] \frac{(m \vee m') \|\Psi_{m \vee m'}^{-1}\|_{\text{op}}}{n}, \quad (6.30)$$

with $\kappa_0 > 0$ given by Lemma 6.5 below.

**Lemma 6.5.** Under the assumptions of Theorem 3.5, there exist $\kappa_0 > 0$ and $C_{\text{Tal}} > 0$ (independent of $m, n, \Delta$) such that, for all $m \in \mathcal{M}_{n\Delta}$ and all $\hat{m} \in \hat{\mathcal{M}}_{n\Delta}$,

$$\mathbb{E}\left[(\sup_{g \in B_{m, \hat{m}}^{\pi}} (v_n^{(1)}(g))^2 - p(m, \hat{m}))_+ 1_{\Omega_{n\Delta}}\right] \le \frac{C_{\text{Tal}} \log^6(n\Delta)}{n}.$$

**Lemma 6.6.** Under the assumptions of Theorem 3.5, there exist constants $c_1, c_2 > 0$ (independent of $m, n, \Delta$) such that for all $m \in \mathcal{M}_{n\Delta}$ and $\hat{m} \in \hat{\mathcal{M}}_{n\Delta}$,

$$\mathbb{E}[\widehat{\text{pen}}(m) 1_{\Omega_{n\Delta}}] \le c_1 \text{pen}(m) + \frac{c_2}{n}, \quad \mathbb{E}[(\text{pen}(\hat{m}) - \widehat{\text{pen}}(\hat{m}))_+ 1_{\Omega_{n\Delta}}] \le \frac{c_2}{n}.$$

$$\text{where } \text{pen}(m) := \kappa c_{\varphi}^2 \mathbb{E}[\sigma^4(X_0)] \frac{m \|\Psi_{\hat{m}}^{-1}\|_{\text{op}}}{n}.$$

(Lemma 6.6 follows from Comte and Genon-Catalot (2021, Sec. 6.11); Lemma 6.5 is proved in Section 6.8.)

$$\text{We split } \mathbb{E}[\|\tilde{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2] = \underbrace{\mathbb{E}[\|\tilde{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2 1_{\Omega_{n\Delta}}]}_{=: \mathcal{I}_1} + \underbrace{\mathbb{E}[\|\tilde{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2 1_{\Omega_{n\Delta}^c}]}_{=: \mathcal{I}_2}. \quad (6.31)$$

**Step 1: Control of $\mathcal{I}_2$.** We use the elementary decomposition

$$\|\tilde{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2 \le 2 \|\tilde{\sigma}_{\hat{m}}^2 - \Pi_{\hat{m}} \sigma_A^2\|_n^2 + 2 \|\sigma_A^2\|_n^2.$$

For the first term,

$$\|\tilde{\sigma}_{\hat{m}}^2 - \Pi_{\hat{m}} \sigma_A^2\|_n^2 = \frac{1}{n^2}(V+R)^{\top} \hat{\Phi}_{\hat{m}} \hat{\Psi}_{\hat{m}}^{-1} \hat{\Phi}_{\hat{m}}^{\top} (V+R) \le \|\hat{\Psi}_{\hat{m}}^{-1}\|_{\text{op}} \|\frac{1}{n} \hat{\Phi}_{\hat{m}}^{\top} (V+R)\|_{2, \hat{m}}^2.$$

By the definition of $\hat{\mathcal{M}}_{n\Delta}$, $\|\hat{\Psi}_{\hat{m}}^{-1}\|_{\text{op}} \le c\Delta/c_{\varphi}^2 m_{\hat{m}} \log^2(n\Delta) \le c\Delta/c_{\varphi}^2 \log^2(n\Delta)$, and by Lemma 6.4 (with $m = \hat{m}$) and $L(m) \le c_{\varphi}^2 m$,

$$\mathbb{E}\left[\left\|\frac{1}{n}\hat{\Phi}_{\hat{m}}^{\top}(V+R)\right\|_{2,\hat{m}}^4\right] \le C_{b,\sigma} \hat{m} (c_{\varphi}^2 \hat{m})^2 \left(2C_{\text{BDG}}(4)/n^2 + \Delta^4\right).$$

Therefore, by Cauchy-Schwarz's inequality,

$$\begin{aligned} \mathbb{E}[\|\tilde{\sigma}_{\hat{m}}^2 - \Pi_{\hat{m}} \sigma_A^2\|_n^2 1_{\Omega_{n\Delta}^c}] &\le \frac{c}{c_{\varphi}^2} \frac{n\Delta}{\log^2(n\Delta)} \left(C_{b,\sigma} \hat{m} (c_{\varphi}^2 \hat{m})^2 \left(\frac{2C_{\text{BDG}}(4)}{n^2} + \Delta^4\right)\right)^{1/2} \cdot \sqrt{C_{\text{mix}}} (n\Delta)^{-3} \\ &\le \frac{c c_{\varphi} \sqrt{C_{b,\sigma} C_{\text{mix}}}}{\log^2(n\Delta)} \hat{m}^{3/2} \left(\frac{\sqrt{2C_{\text{BDG}}(4)}}{n} + \Delta^2\right) (n\Delta)^{-2}. \end{aligned}$$

Using $\hat{m} \le n\Delta$ yields

$$\mathbb{E}[\|\tilde{\sigma}_{\hat{m}}^2 - \Pi_{\hat{m}} \sigma_A^2\|_n^2 1_{\Omega_{n\Delta}^c}] \le \frac{c c_{\varphi} \sqrt{C_{b,\sigma} C_{\text{mix}}}}{\log^2(n\Delta)} \left(\frac{\sqrt{2C_{\text{BDG}}(4)}}{n} + \Delta^2\right) \frac{1}{\sqrt{n\Delta}}. \quad (6.32)$$

For the second term, by Cauchy-Schwarz's inequality,

$$\mathbb{E}[\|\sigma_A^2\|_n^2 1_{\Omega_{n\Delta}^c}] \le (\mathbb{E}[\sigma^8(X_0)])^{1/2} \sqrt{C_{\text{mix}}} (n\Delta)^{-3}. \quad (6.33)$$
Combining (6.32)-(6.33), there exist finite constants $C_2(b, \sigma, \theta, c_\varphi)$ and $C_3(b, \sigma, \theta)$ such that

$$ \mathcal{I}_2 \leq C_2 \left( \frac{1}{n} + \Delta^2 \right) + \frac{C_3}{n}. \quad (6.34) $$

**Step 2: Control of $\mathcal{I}_1$.** By the penalized selection rule (3.20), we have for all $m \in \tilde{\mathcal{M}}_{n\Delta}$,

$$ - \|\hat{\sigma}_{\hat{m}}^2\|_n^2 + \widehat{\text{pen}}(\hat{m}) \leq - \|\hat{\sigma}_m^2\|_n^2 + \widehat{\text{pen}}(m). $$

Here, the quantity $-\|\hat{\sigma}_m^2\|_n^2$ is, up to an additive constant independent of $m$, $\gamma_n(\hat{\sigma}_m)$, thus

$$ \gamma_n(\hat{\sigma}_m^2) + \widehat{\text{pen}}(\hat{m}) \leq \gamma_n(\hat{\sigma}_m^2) + \widehat{\text{pen}}(m). \quad (6.35) $$

On the event $\Omega_{n\Delta}$, Lemma 3.4 ensures $\mathcal{M}_{n\Delta} \subset \tilde{\mathcal{M}}_{n\Delta}$, hence (6.35) holds for all $m \in \mathcal{M}_{n\Delta}$. Since $\hat{\sigma}_m^2$ minimizes $\gamma_n(\cdot)$ over $S_m$, we have for any $g \in S_m$,

$$ \gamma_n(\hat{\sigma}_{\hat{m}}^2) + \widehat{\text{pen}}(\hat{m}) \leq \gamma_n(\hat{\sigma}_m^2) + \widehat{\text{pen}}(m) \leq \gamma_n(g) + \widehat{\text{pen}}(m) $$

Expanding $\gamma_n$ as in the proof of Proposition 3.1 (see (6.9)), we obtain on $\Omega_{n\Delta}$:

$$ \|\hat{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2 \leq \frac{7}{4}\|g - \sigma_A^2\|_n^2 + \frac{3}{4}\|\hat{\sigma}_{\hat{m}}^2 - \sigma_A^2\|_n^2 \\ + 8\left(\sup_{t \in B_{\pi_{m,\hat{m}}}^\pi}(v_n^{(1)}(t))^2 - p(m, \hat{m})\right)_+ + 8\sup_{t \in B_{\pi_{m,\hat{m}}}^\pi}(\bar{v}_n(t))^2 + \frac{8}{n}\sum_{k=1}^n R_{k\Delta}^2 \\ + \widehat{\text{pen}}(m) + 8p(m, \hat{m}) - \widehat{\text{pen}}(\hat{m}). $$

Multiplying by 4 and taking expectations over $\Omega_{n\Delta}$,

$$ \begin{aligned} \mathcal{I}_1 \leq {}& 7 \mathbb{E}[\|g - \sigma_A^2\|_n^2 \mathbb{1}_{\Omega_{n\Delta}}] + 32 \mathbb{E}\left[(\sup_{t \in B_{\pi_{m,\hat{m}}}^\pi}(v_n^{(1)}(t))^2 - p(m, \hat{m}))_+ \mathbb{1}_{\Omega_{n\Delta}}\right] \\ & + 32 \mathbb{E}\left[\sup_{t \in B_{\pi_{m,\hat{m}}}^\pi}(\bar{v}_n(t))^2\right] + \frac{32}{n} \sum_{k=1}^n \mathbb{E}[R_{k\Delta}^2] + 4 \mathbb{E}[\widehat{\text{pen}}(m) \mathbb{1}_{\Omega_{n\Delta}}] + 32 \mathbb{E}[p(m, \hat{m}) \mathbb{1}_{\Omega_{n\Delta}}] \\ & - 4 \mathbb{E}[\widehat{\text{pen}}(\hat{m}) \mathbb{1}_{\Omega_{n\Delta}}]. \end{aligned} \quad (6.36) $$

On $\Omega_{n\Delta}$, $\frac{1}{2}\|\cdot\|_{\pi}^2 \leq \|\cdot\|_n^2 \leq \frac{3}{2}\|\cdot\|_{\pi}^2$. Thus, for any $g \in S_m$,

$$ \mathbb{E}[\|g - \sigma_A^2\|_n^2 \mathbb{1}_{\Omega_{n\Delta}}] \leq \frac{3}{2} \|g - \sigma_A^2\|_{\pi}^2. \quad (6.37) $$

By Lemma 6.5, there exists $\kappa_0 = 12 > 0$ such that

$$ \mathbb{E}\left[(\sup_{t \in B_{\pi_{m,\hat{m}}}^{\pi}} (v_n^{(1)}(t))^2 - p(m, \hat{m}))_+ \mathbb{1}_{\Omega_{n\Delta}}\right] \leq \frac{C_{\text{Tal}} \log^6(n\Delta)}{n}. \quad (6.38) $$

Applying Lemma 6.3(ii) with $m = M_{n\Delta}^+$ and noting that $B_{m,\hat{m}}^\pi \subset S_{M_{n\Delta}^+}$ on $\Omega_{n\Delta}$,

$$ \mathbb{E}\left[\sup_{t \in B_{\pi_{m,\hat{m}}}^{\pi}} (\bar{v}_n(t))^2 \mathbb{1}_{\Omega_{n\Delta}}\right] \leq \frac{c C_{b,\sigma}}{2} \Delta^2. \quad (6.39) $$

By Lemma 6.2,

$$ \frac{1}{n} \sum_{k=1}^{n} \mathbb{E}[R_{k\Delta}^2] \leq C_{b,\sigma} \Delta^2. \quad (6.40) $$

Let $\text{pen}(m) := \kappa c_\varphi^2 \mathbb{E}[\sigma^4(X_0)] \frac{m\|\Psi_m^{-1}\|_{\text{op}}}{n}$. If $\kappa \geq 32\kappa_0$, then from (6.30),

$$ 32p(m, \hat{m}) \leq \text{pen}(m) + \text{pen}(\hat{m}). \quad (6.41) $$

By Lemma 6.6,

$$ -\mathbb{E}[\widehat{\text{pen}}(\hat{m}) \mathbb{1}_{\Omega_{n\Delta}}] \leq -\frac{1}{2} \mathbb{E}[\text{pen}(\hat{m})] + \frac{c_2}{n}. \quad (6.42) $$
Collecting (6.36)–(6.42). Using (6.37)–(6.42) in (6.36),

$$
\begin{align*}
\mathcal{I}_1 &\le \frac{21}{2} \|g - \sigma_A^2\|_{\pi}^2 + \frac{32 C_{\text{Tal}} \log^6(n\Delta)}{n} + 16 c C_{b,\sigma} \Delta^2 + 32 C_{b,\sigma} \Delta^2 + 4c_1 \text{pen}(m) + \frac{8c_2}{n} \\
&\qquad + \text{pen}(m) + \text{pen}(\hat{m}) - \frac{1}{2} \text{pen}(\hat{m}) \\
&\le \frac{21}{2} \|g - \sigma_A^2\|_{\pi}^2 + (4c_1+1) \text{pen}(m) + (16 c C_{b,\sigma} + 32 C_{b,\sigma}) \Delta^2 + \frac{32 C_{\text{Tal}} \log^6(n\Delta)}{n} + 8c_2.
\end{align*}
$$

Since $g \in S_m$ and $m \in \mathcal{M}_{n\Delta}$ are arbitrary, we finally obtain

$$
\mathcal{I}_1 \le D_1 \inf_{m \in \mathcal{M}_{n\Delta}} \left\{ \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \text{pen}(m) \right\} + D_2 \Delta^2 + \frac{D_3 \log^6(n\Delta)}{n}, \quad (6.43)
$$

with the explicit constants

$$
D_1 = \max\left(\frac{21}{2}, 4c_1+1\right), \quad D_2 = 16c C_{b,\sigma} + 32C_{b,\sigma}, \quad D_3 = 32C_{\text{Tal}} + 8c_2.
$$

**Step 3: Conclusion.** Combining (6.31), (6.34), and (6.43), we arrive at

$$
\mathbb{E}[\|\hat{\sigma}_m^2 - \sigma_A^2\|_n^2] \le D \inf_{m \in \mathcal{M}_{n\Delta}} \left\{ \inf_{g \in S_m} \|g - \sigma_A^2\|_{\pi}^2 + \kappa c_{\varphi}^2 \mathbb{E}[\sigma^4(X_0)] \frac{m \|\Psi_m^{-1}\|_{\text{op}}}{n} \right\} + D' \left(\Delta^2 + \frac{\log^6(n\Delta)}{n}\right),
$$

for some explicit $D, D' > 0$ depending only on $(b, \sigma, \theta, c_\varphi)$ which proves (3.21). We proceed as in Section 6.1.2 to derive the $L^2(\pi)$-risk bound (3.22).

6.8. *Proof of Lemma* **6.5.**

Let $N_k := W_{(k+1)\Delta} - W_{k\Delta}$ and $\chi_k := N_k^2/\Delta - 1$ (so $\mathbb{E}[\chi_k] = 0$, $\mathbb{E}[\chi_k^2] = 2$). Fix the truncation levels

$$
k_n = l_n := \frac{n^{1/8}}{\log(n\Delta)} \quad (\text{for } n\Delta \ge e). \tag{6.44}
$$

Define

$$
\begin{align*}
\chi_k^c &:= \left(\frac{N_k^2}{\Delta} - 1\right) \mathbf{1}_{\{|\frac{N_k^2}{\Delta} - 1| \le k_n\}} - \mathbb{E}\left[\left(\frac{N_k^2}{\Delta} - 1\right) \mathbf{1}_{\{|\frac{N_k^2}{\Delta} - 1| \le k_n\}}\right], & \chi_k^e &:= \chi_k - \chi_k^c, \\
\sigma^{2,c}(x) &:= \sigma^2(x) \mathbf{1}_{\{|\sigma^2(x)| \le l_n\}}, & \sigma^{2,e}(x) &:= \sigma^2(x) - \sigma^{2,c}(x).
\end{align*}
$$

For $g \in S_m + S_{m'}$ with $\|g\|_{\pi} = 1$ write

$$
v_n^{(1)}(g) = \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) \sigma^2(X_{k\Delta}) \chi_k = \sum_{r=1}^{3} v_{n,r}^{(1)}(g), \quad (6.45)
$$

with

$$
\begin{align*}
v_{n,1}^{(1)}(g) &= \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) \sigma^{2,c}(X_{k\Delta}) \chi_k^c, & v_{n,2}^{(1)}(g) &= \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) \sigma^{2,e}(X_{k\Delta}) \chi_k^e, \\
v_{n,3}^{(1)}(g) &= \frac{1}{n} \sum_{k=1}^{n} g(X_{k\Delta}) \sigma^2(X_{k\Delta}) \chi_k^e.
\end{align*}
$$

Let $B_{m,m'}^{\pi} := \{g \in S_m + S_{m'} : \|g\|_{\pi} = 1\}$, one has

$$
\left( \sup_{g \in B_{m, \hat{m}}^{\pi}} (v_n^{(1)}(g))^2 - p(m, \hat{m}) \right)_+ \le \left( 3 \sup_{g \in B_{m, \hat{m}}^{\pi}} (v_{n,1}^{(1)}(g))^2 - p(m, \hat{m}) \right)_+ + 3 \sup_{g \in B_{m, \hat{m}}^{\pi}} (v_{n,2}^{(1)}(g))^2 \\
+ 3 \sup_{g \in B_{m, \hat{m}}^{\pi}} (v_{n,3}^{(1)}(g))^2. \tag{6.46}
$$
For $g(x) = a^\top \varphi(x)$ in $S_m + S_{\hat{m}}$, $a^\top \Psi_{m\lor\hat{m}} a = 1$ gives

$$ \sup_x |g(x)| \le \|\Psi_{m\lor\hat{m}}^{-1}\|_{\text{op}}^{1/2} L(m\lor\hat{m})^{1/2} \le c_\varphi \sqrt{(m\lor\hat{m})\|\Psi_{m\lor\hat{m}}^{-1}\|_{\text{op}}}, \quad (6.47) $$

by Assumption 5(ii). We also use $\mathbb{E}[(\chi_k^c)^2] \le 2$, and for all integers $r \ge 1$,

$$ \mathbb{E}[|\sigma(X_0)|^{2r}] \le C_{b,\sigma}(r) < \infty \quad (\text{Assumptions 1-3}). \quad (6.48) $$

Work on the event $\Omega_{n\Delta}$, so that for every $\hat{m} \in \mathcal{M}_{n\Delta}^+$,

$$ c_{\varphi}^2(m \lor \hat{m}) (\|\Psi_{m \lor \hat{m}}^{-1}\|_{\text{op}} \lor 1) \le \frac{3c}{2} \frac{n\Delta}{\log^2(n\Delta)}. \quad (6.49) $$

Using (6.47), (6.48), (6.49) and the tail bounds $\mathbf{1}_{\{|y|>k\}} \le y^8/l^8$ and $\mathbf{1}_{\{|y|>k\}} \le |y|^{16}/k^{16}$,

$$ \begin{aligned} \mathbb{E}\left[\sup_{g \in B_{m, \hat{m}}^{\pi}} (v_{n,2}^{(1)}(g))^2 \mathbf{1}_{\Omega_{n\Delta}}\right] &\le \frac{2}{n} c_{\varphi}^2(m \lor \hat{m}) \|\Psi_{m \lor \hat{m}}^{-1}\|_{\text{op}} \frac{\mathbb{E}[|\sigma(X_0)|^{10}]}{l_n^8} \\ &\le \frac{3c \Delta}{\log^2(n\Delta)} \cdot \frac{C_{b,\sigma}}{l_n^8} \le \frac{\log^6(n\Delta)C_2}{n}. \end{aligned} \quad (6.50) $$

Similarly, letting $C_\chi := \mathbb{E}[|\chi_1|^{10}] < \infty$,

$$ \begin{aligned} \mathbb{E}\left[\sup_{g \in B_{m, \hat{m}}^{\pi}} (v_{n,3}^{(1)}(g))^2 \mathbf{1}_{\Omega_{n\Delta}}\right] &\le \frac{2}{n} c_{\varphi}^2(m \lor \hat{m}) \|\Psi_{m \lor \hat{m}}^{-1}\|_{\text{op}} \mathbb{E}[\sigma^4(X_0)] \frac{C_\chi}{k_n^8} \\ &\le \frac{3c \Delta}{\log^2(n\Delta)} \mathbb{E}[\sigma^4(X_0)] \frac{C_\chi}{k_n^8} \le \frac{C_3 \log^6(n\Delta)}{n}. \end{aligned} \quad (6.51) $$

Let $n = 2p_nq_n$ with

$$ q_n := \left\lceil \frac{7 \log(n\Delta)}{\theta} \right\rceil, \quad p_n := \frac{n}{2q_n}. \quad (6.52) $$

By Berbee's lemma there exist independent blocks $Y_\ell^*$ such that, with $\Omega^* := \{Y_\ell^* = Y_\ell, \forall \ell\}$,

$$ \mathbb{P}((\Omega^*)^c) \le 2p_n \beta_X(q_n\Delta) \le \frac{\theta}{7\log(n\Delta)} (n\Delta)^{-6}. \quad (6.53) $$

On $\Omega^*$ define (for odd blocks; even ones are identical) $v_{n,1}^{*,1}(g) := \frac{1}{p_n} \sum_{\ell=0}^{p_n-1} (f^{(g)}(Y_\ell^*) - \mathbb{E}[f^{(g)}(Y_\ell^*)])$, with

$$ f^{(g)}((z_r, x_r)_{r=1}^{q_n}) := \frac{1}{2q_n} \sum_{r=1}^{q_n} \left( z_r \mathbf{1}_{\{|z_r| \le k_n\}} - \mathbb{E}[z_r \mathbf{1}_{\{|z_r| \le k_n\}}] \right) g(x_r) \sigma^{2,c}(x_r). $$

For the class $\mathcal{F} = \{f^{(g)} : g \in B_{m,m'}^\pi\}$, one checks

$$ M := \sup_{g \in \mathcal{F}} \|f\|_{\infty} \le c_{\varphi} k_n l_n \sqrt{(m \lor m') \|\Psi_{m \lor m'}^{-1}\|_{\text{op}}}, \quad (6.54) $$

$$ H^2 := \mathbb{E}\left[\sup_{f \in \mathcal{F}} |v_n(f)|\right]^2 \le \frac{2 \mathbb{E}[\sigma^4(X_0)] c_{\varphi}^2 (m \lor m') \|\Psi_{m \lor m'}^{-1}\|_{\text{op}}}{n}, \quad (6.55) $$

$$ v := \sup_{f \in \mathcal{F}} \frac{1}{p_n} \sum_{\ell=1}^{p_n} \mathrm{Var}(f(Y_{\ell}^{*})) \le \frac{2 c_{\varphi}}{q_n} \mathbb{E}[\sigma^8(X_0)]^{1/2} \sqrt{(m \lor m') \|\Psi_{m \lor m'}^{-1}\|_{\text{op}}}. \quad (6.56) $$

Talagrand's inequality (Lemma A.2) yields

$$ \mathbb{E}\left[(\sup_{g \in B_{m,m'}^\pi} (v_{n,1}^{*,1}(g))^2 - 6H^2)_+\right] \le \frac{4}{b} \left( \frac{v}{p_n} e^{-b \frac{p_n H^2}{v}} + \frac{49 M^2}{6 C(1)^2 p_n^2} e^{-\frac{\sqrt{2} b C(1)}{7} \frac{p_n H}{M}} \right). \quad (6.57) $$
Write $Z := (m \vee m') \|\Psi_{m \vee m'}^{-1}\|_{\text{op}}$. From (6.55)-(6.56), $p_n H^2/v \asymp \sqrt{Z}$ (independent of $n$). Hence
$$ \frac{4}{b} \cdot \frac{v}{p_n} e^{-b \frac{p_n H^2}{v}} \le \frac{C_A}{n} \sqrt{Z} e^{-c_A \sqrt{Z}}, $$
for constants $C_A, c_A > 0$ depending only on $c_\varphi, \theta, \mathbb{E}[\sigma^4], \mathbb{E}[\sigma^8]$.

For the second term, (6.54) gives $M \le c_\varphi k_n l_n \sqrt{Z}$ and

$$ \frac{p_n H}{M} \ge c_B \frac{\sqrt{n}}{q_n k_n l_n} \stackrel{(6.52),(6.44)}{\ge} c'_B n^{1/4} \log(n\Delta), $$

for constants $c_B, c'_B > 0$. Therefore

$$ \frac{4}{b} \cdot \frac{49 M^2}{6 C(1)^2 p_n^2} \exp\left(-\frac{\sqrt{2} b C(1)}{7} \frac{p_n H}{M}\right) \le \frac{C_B}{n} U_n \exp\left(-c_B'' n^{1/4} \log(n\Delta)\right), $$

where (using (6.49))

$$ U_n := \sup_{m' \in \mathcal{M}_{n\Delta}^{+}} \frac{M^2}{p_n^2} \cdot n \le \frac{k_n^4 q_n^2}{n} \cdot \frac{n\Delta}{\log^2(n\Delta)} \stackrel{(6.52),(6.44)}{\lesssim} \frac{\Delta}{n^{1/2} \log^4(n\Delta)}. $$

We must sum over all $m' \in \mathcal{M}_{n\Delta}^{+}$, with $|\mathcal{M}_{n\Delta}^{+}| \le n\Delta$. First, using $Z \ge c_\pi m'$ (e.g. $Z \ge m'/\|\pi\|_\infty$) and $\sum_{m' \ge 1} \sqrt{m'} e^{-c\sqrt{m'}} < \infty$,

$$ \sum_{m' \in \mathcal{M}_{n\Delta}^{+}} \frac{1}{n} \sqrt{Z} e^{-c_A \sqrt{Z}} \le \frac{C_1}{n}. \qquad (6.58) $$

Second,

$$ \sum_{m' \in \mathcal{M}_{n\Delta}^{+}} \frac{1}{n} U_n e^{-c_B'' n^{1/4} \log(n\Delta)} \le \frac{n\Delta}{n} \cdot \frac{C\Delta}{n^{1/2} \log^4(n\Delta)} e^{-c_B'' n^{1/4} \log(n\Delta)} \le \frac{C_2}{n}. \qquad (6.59) $$

Set $p(m, m') := 6H^2 = 12 \mathbb{E}[\sigma^4(X_0)] c_\varphi^2(m \vee m') \|\Psi_{m \vee m'}^{-1}\|_{\text{op}}/n$, combining (6.57), (6.58) and (6.59), we obtain

$$ \mathbb{E}\left[\left(\sup_{g \in B_{m, \hat{m}}^{\pi}} (v_{n,1}^{*,1})^2 - p(m, \hat{m})\right) + \mathbf{1}_{\Omega_{n\Delta}}\right] \lesssim \frac{1}{n}. $$

Putting everything together yields the statement of Lemma 6.5.

## Acknowledgements

I am deeply grateful to my PhD advisors, FABIENNE COMTE and NICOLAS MARIE, for their guidance and support. I also thank the referees and the editor of ALEA for their constructive comments, which substantially improved the paper.

## Appendix A. Auxiliary probabilistic inequalities

This appendix gathers the two technical bounds that underlie the risk analysis in Section 6. We simply quote the results; the interested reader can consult the given references for proofs.

**Lemma A.1 (Matrix Chernoff Tropp et al., 2015).** Let $X_1, \dots, X_N$ be independent, self-adjoint $d \times d$ random matrices satisfying $0 \preceq X_i \preceq L$ Id. Set $Y := \sum_{i=1}^N X_i$ and write

$$ \mu_{\min} := \lambda_{\min}(\mathbb{E}Y), \quad \mu_{\max} := \lambda_{\max}(\mathbb{E}Y). $$

For every $\varepsilon \in (0, 1]$,

$$ \mathbb{P}(\lambda_{\min}(Y) \le (1 - \varepsilon)\mu_{\min}) \le d \left[ \frac{e^{-\varepsilon}}{(1-\varepsilon)^{1-\varepsilon}} \right]^{\mu_{\min}/L}, $$

$$ \mathbb{P}(\lambda_{\max}(Y) \ge (1 + \varepsilon)\mu_{\max}) \le d \left[ \frac{e^{\varepsilon}}{(1+\varepsilon)^{1+\varepsilon}} \right]^{\mu_{\max}/L}. $$
**Lemma A.2** (Talagrand's Deviation Inequality Klein and Rio, 2005). Let $n$ be a positive integer and let $\mathcal{F}$ be a countable collection of measurable functions. Suppose $X_1, \dots, X_n$ are independent real-valued random variables and for each $f \in \mathcal{F}$ define

$$v_n(f) = \frac{1}{n} \sum_{i=1}^{n} (f(X_i) - \mathbb{E}[f(X_i)]).$$

Assume there exist positive constants $M, H$, and $v$ such that:

(1) $\sup_{f \in \mathcal{F}} \|f\|_{\infty} \le M,$

(2) $\mathbb{E}[\sup_{f \in \mathcal{F}} |v_n(f)|] \le H,$

(3) $\sup_{f \in \mathcal{F}} \frac{1}{n} \sum_{i=1}^{n} \mathrm{Var}(f(X_i)) \le v.$

Define $C(\alpha) := (\sqrt{1+\alpha}-1) \wedge 1$ for each $\alpha > 0$, and set $b := \frac{1}{6}$. Then, for all $\alpha > 0$,

$$\mathbb{E}\left[(\sup_{f \in \mathcal{F}} |v_n(f)|^2 - 2(1+2\alpha)H^2)_{+}\right] \le \frac{4}{b}\left(\frac{v}{n}\exp(-b\alpha \frac{nH^2}{v})\right. \\ \left. + \frac{49M^2}{6C(\alpha)^2 n^2}\exp\left(-\frac{\sqrt{2}bC(\alpha)\sqrt{\alpha}}{7}\frac{nH}{M}\right)\right).$$

## References

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