An unexpected stochastic dominance: Pareto
distributions, dependence, and diversification
Yuyu Chen
Department of Economics, University of Melbourne, Australia, yuyu.chen@unimelb.edu.au

Paul Embrechts
RiskLab, Department of Mathematics & ETH Risk Center, ETH Zurich, Switzerland, embrechts@math.ethz.ch

Ruodu Wang
Department of Statistics and Actuarial Science, University of Waterloo, Canada, wang@uwaterloo.ca

We find the perhaps surprising inequality that the weighted average of independent and identically distributed Pareto random variables with infinite mean is larger than one such random variable in the sense of
first-order stochastic dominance. This result holds for more general models including super-Pareto distributions, negative dependence, and triggering events, and yields superadditivity of the risk measure Value-atRisk for these models.
Key words : Pareto distributions; diversification effect; risk pooling; first-order stochastic dominance

1. Introduction
Pareto distributions are arguably the most important class of heavy-tailed loss distributions, due
to their connection to regularly varying tails, Extreme Value Theory (EVT), and power laws in
economics and social networks; see, e.g., Embrechts et al. (1997), de Haan and Ferreira (2006) andGabaix (2009). In quantitative risk management, Pareto distributions are frequently used to model
losses from catastrophes such as earthquakes, hurricanes, and wildfires; see, e.g., Embrechts et al.
(1999). They are also widely used in economics for wealth distributions (e.g., Taleb (2020)) andmodeling the tails of financial asset losses and operational risks (e.g., McNeil et al. (2015)); applications of power laws in economics, finance, and insurance are treated in Ibragimov et al. (2015)
and Ibragimov and Prokhorov (2017). Andriani and McKelvey (2007) listed over 80 examples ofpower laws in diverse fields of applications. By the Pickands-Balkema-de Haan Theorem (Pickands(1975) and Balkema and de Haan (1974)), generalized Pareto distributions are the only possible
non-degenerate limiting distributions of the residual lifetime of random variables exceeding a highlevel.
In the realm of banking and insurance, distributions with infinite mean occur as a possible
mathematical model in several contexts. For instance, catastrophic losses, operational losses, large
1Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)2

insurance losses, and financial returns from technological innovations are often modelled by Pareto
distributions without finite mean; see Embrechts et al. (1997) in the context of extreme value
theory, Hofert and Wüthrich (2012) on nuclear power accidents, and the more recent Cheynel et al.
(2022) on modeling fraud. In risk management, such infinite-mean models often lead to intriguing
phenomena, such as the diversification disaster studied by Ibragimov et al. (2009, 2011).
Stochastic dominance relations are an important tool in economic decision theory, which allows
for the analysis of risk preferences for a group of decision makers (Hadar and Russell (1969)). They
have been studied in the forms of first and second degrees (Quirk and Saposnik (1962), Hadar andRussell (1969, 1971) and Rothschild and Stiglitz (1970)), larger integer degrees (Whitmore (1970)
and Caballé and Pomansky (1996)), and fractional degrees (Müller et al. (2017) and Huang et al.
(2020)), and they are widely applied in the expected utility and dual utility theory (Yaari (1987)),
behavioural decision models (Chew et al. (1987), Baucells and Heukamp (2006) and Schmidt andZank (2008)), and risk measures (Föllmer and Schied (2016)). See also Levy (1992, 2016) for thewide applicability of stochastic dominance relations in decision making, and Müller and Stoyan(2002) and Shaked and Shanthikumar (2007) for the mathematics of stochastic dominance.
The strongest form of commonly used stochastic dominance relations is first-order stochastic
dominance. For two random variables X and Y representing random losses, we say X is smaller
than Y in first-order stochastic dominance, denoted by X ≤st Y , if P(X ≤ x) ≥ P(Y ≤ x) for allx ∈ R. The relation X ≤st Y means that all decision makers with an increasing utility function will
prefer loss X to loss Y if their expected utilities are finite. In this paper, all terms like “increasing”
and “decreasing” are in the non-strict sense.
For iid random variables X1 , . . . , Xn following a Pareto distribution with infinite mean andPnweights θ1 , . . . , θn ≥ 0 with
i=1 θi = 1, one consequence of our main result, Theorem 1, is thestochastic dominance relation
X1 ≤st θ1 X1 + · · · + θn Xn ,(1)and (1) is not an equality except for the trivial case that only one of θ1 , . . . , θn is non-zero. As far
as we are aware, (1) is not known in the literature, even in the case that θ1 , . . . , θn are equal (i.e.,they are 1/n).
To appreciate the nature of (1), we first recall that for any identically distributed randomvariables X1 , . . . , Xn with finite mean, regardless of their distribution or dependence structure, forPnθ1 , . . . , θn > 0 with i=1 θn = 1, (1) can only hold if X1 = · · · = Xn (almost surely), in which case
we have the trivial equality X1 = θ1 X1 + · · · + θn Xn ; see Proposition 2. Therefore, the assumptionof infinite mean is very important for (1) to hold.
It is somewhat surprising that, for infinite-mean Pareto losses, inequality (1) holds for a verystrong form of risk comparison: For every decision maker with a risk preference favouring less loss

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3

over more loss and well defined for losses in (1), a diversified portfolio of such iid Pareto losses isless preferred to a non-diversified one. Flipping the sign, diversification is preferred if the Pareto
random variables are treated as profits or gains from, for instance, research and development.
We call such a stochastic dominance “unexpected” for both its surprising nature and the infinite
expectations involved.
The infinite mean assumption comes with a caveat: for a risk-averse expected utility decision
maker, due to the concavity of the utility function, (1) does not imply a preference for nondiversification, because losses in both sides of (1) have −∞ expected utility. It does, however,give superadditivity of the regulatory risk measure Value-at-Risk (VaR) in banking and insurance
sectors; that is, the weighted average of super-Pareto losses gives a larger VaR than the weighted
average of VaRs of individual super-Pareto losses. Different from the literature on VaR superadditivity for regularly varying distributions (e.g., Embrechts et al. (2009) and McNeil et al. (2015)),
the superadditivity of VaR implied by (1) holds for all probability levels, and this is not in an
asymptotic sense.
Observations similar to (1), although with less generality, occur in the literature in different
forms. Samuelson (1967) mentioned that having an infinite mean in portfolio diversification may
lead to a worse distribution; see also Fama and Miller (1972, p. 271) and Malinvaud (1972). Inequality (1) for n = 2 and the Pareto tail parameter α = 1/2 (see Section 2 for the parametrization) has an
explicit formula in Example 7 of Embrechts et al. (2002). Simple numerical examples are provided
by Embrechts and Puccetti (2010, Figure 5.2) and Bauer and Zanjani (2016, Table 2). Ibragimov(2005) showed that (1) holds for iid positive one-sided stable random variables with infinite mean.
Another relevant result of Ibragimov (2009) is that for iid random variables Z1 , . . . , Zn which follow
a convolution of symmetric stable distributions without finite mean, P(θ1 Z1 + · · · + θn Zn ≤ x) ≤P(Z1 ≤ x) for x > 0 but the opposite holds for x < 0 (and hence first-order stochastic dominancedoes not hold). The symmetry of distributions is essential for this inequality, and Z1 , . . . , Zn cantake negative values, unlike Pareto losses, which are positive, skewed and more suitable for themodeling of extreme losses.
In Section 2, we first introduce super-Pareto distributions, a class of infinite-mean distributions,and weak negative association, a notion of dependence weaker than negative association (Alamand Saxena (1981) and Joag-Dev and Proschan (1983)). The class of super-Pareto distributions
includes all infinite-mean Pareto distributions. Our main result, Theorem 1, shows that (1) holds if
X1 , . . . , Xn are weakly negatively associated super-Pareto random variables, and also in case they
are triggered by some events. Some discussions on this result are provided after its proof. Section
3 concludes.

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We fix some notation. Throughout, random variables are defined on an atomless probability space(Ω, F , P). Denote by N the set of all positive integers and R+ the set of non-negative real numbers.
For n ∈ N, let [n] = {1, . . . , n}. Denote by ∆n the standard simplex, that is, ∆n = {(θ1 , . . . , θn ) ∈Pn[0, 1]n : i=1 θi = 1}. For x, y ∈ R, write x ∧ y = min{x, y }, x ∨ y = max{x, y }, and x+ = max{x, 0}.
d

We write X = Y if X and Y have the same distribution. We always assume n ≥ 2. Equalities andinequalities are interpreted component-wise when applied to vectors. For any random variable X,
its essential infimum is given by zX = inf {z ∈ R : P(X > z) > 0}.

2. Stochastic dominance for super-Pareto risks
2.1. Super-Pareto distributions and weak negative association
We first introduce the Pareto distribution and the super-Pareto distribution. A common parameterization of Pareto distributions is given by, for θ, α > 0, the cumulative distribution function
 αθ
Pα,θ (x) = 1 −, x ≥ θ.
xAs θ is a scale parameter, it suffices to study Pα,1 , which we write as Pareto(α). The mean of Pα,θ isinfinite if and only if the tail parameter α is in (0, 1]. We say that the Pα,θ distribution is extremely
heavy-tailed if α ≤ 1.
dDefinition 1. A random variable X is super-Pareto (or has a super-Pareto distribution) if X =f (Y ) for some increasing, convex, and non-constant function f and Y ∼ Pareto(1). Moreover, X isregular if f (0) = 0 and f (1) > 0.
All extremely heavy-tailed Pareto distributions are super-Pareto and regular. By definition, thesuper-Pareto property is preserved under increasing, convex, and non-constant transforms, including location-scale transforms. Intuitively, increasing convex transforms, such as x 7→ xβ for β > 1
and x 7→ exp(x), generally make the tail of a random variable heavier. Thus, super-Pareto risks
have heavier tails than (or equivalent to) Pareto(1) risks. It is straightforward to check that any
super-Pareto random variable has infinite mean.
Some examples of the super-Pareto family include the generalized Pareto distribution when ξ ≥ 1,
specified byxP(X ≤ x) = 1 − 1 + ξ
β

−1/ξ
,

x ≥ 0,

where β > 0, and the Burr distribution when α, τ ∈ (0, 1], specified byα1P(X ≤ x) = 1 −, x ≥ 0.
xτ + 1
Special cases of the Burr family are the paralogistic (α = τ ) and the log-logistic (α = 1) distributions.

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The next proposition gives an equivalent formulation for super-Pareto distributions, which will
become useful in proving some results.
Proposition 1. A random variable X with essential infimum zX ∈ R is super-Pareto if and only
if the function g : x 7→ 1/P(X > x) is strictly increasing and concave on [zX , ∞). If further X isregular, then zX > 0 and g(x) ≤ x/zX for x ≥ zX .
Next, we introduce a new notion of negative dependence.
Definition 2. A set S ⊆ Rk , k ∈ N is decreasing if x ∈ S implies y ∈ S for all y ≤ x. Randomvariables X1 , . . . , Xn are weakly negatively associated if for any i ∈ [n], decreasing set S ⊆ Rn−1 , andx ∈ R with P(Xi ≤ x) > 0,P(X−i ∈ S | Xi ≤ x) ≤ P(X−i ∈ S),(2)

where X−i = (X1 , . . . , Xi−1 , Xi+1 , . . . , Xn ).
Weak negative association includes independence as a special case. Intuitively, (2) means thatknowing Xi is small implies that X−i is less likely to be small, thus a concept of negative dependence. Moreover, (2) implies, by flipping signs,
P(X−i > x | Xi > x) ≤ P(X−i > x)(3)

for i ∈ [n], x ∈ R, and x ∈ Rn−1 .
Weak negative association is weaker than the popular notion of negative association (Alam andSaxena (1981) and Joag-Dev and Proschan (1983)), hence the name. Examples satisfying negative
association, such as normal distributions with non-positive correlations, are studied by Joag-Dev
and Proschan (1983). It is also implied by negative regression dependence (Lehmann (1966) andBlock et al. (1985)), and it implies negative orthant dependence (Block et al. (1982)); see Remark
3 for more details on these notions of dependence.
In most results, we consider weakly negatively associated and identically distributed (WNAID)
super-Pareto random variables. This setting includes the iid Pareto(α) model for α ∈ (0, 1].
2.2. The main result
For random variables X and Y , we write X <st Y if P(X > x) < P(Y > x) for all x ∈ R satisfying
P(X > x) ∈ (0, 1), and this will be referred to as strict stochastic dominance. Note that this condition is stronger than a different notion of strict stochastic dominance defined by X ≤st Y andX 6≥st Y . The following theorem is our main result.
dTheorem 1. Suppose that X1 , . . . , Xn are WNAID super-Pareto, (θ1 , . . . , θn ) ∈ ∆n , and X = X1 .

Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

6(i) Stochastic dominance holds:
X ≤st

n
X

θi Xi ,

(4)i=1

and strict stochastic dominance X <stPn

i=1 θi Xi holds if θi > 0 for at least two i ∈ [n].

(ii) If X is regular, then for any events A1 , . . . , An independent of (X1 , . . . , Xn ) and event A
Pn
independent of X satisfying P(A) = i=1 θi P(Ai ), we have
X1A ≤stnX

θi Xi 1Ai .

(5)

i=1

We will say that a diversification penalty exists if (4) or (5) holds. To interpret Theorem 1, theleft-hand side of (4) can be regarded as the loss of an agent who keeps their own risk, and the righthand side is the loss of an agent who shares risks with other agents. By pooling among super-Pareto
losses, agents expect to suffer less loss when their own loss occurs. However, every agent in the pool
will have a higher frequency of bearing losses. Hence, diversification has two competing effects on
the loss portfolio: It increases the frequency of losses and decreases the sizes of individual losses.
The above results show that the combined effects of diversification of super-Pareto losses lead to aPnhigher default probability at any capital reserve level, that is, P( i=1 θi Xi > x) > P(X > x) for allx > 1.
The model in Theorem 1 (ii) reflects that catastrophic losses are large losses but usually occur
with very small probabilities. Such losses are usually modelled by X1A , where X is a positive
random loss, and A is an event with a small probability, indicating the occurrence of a catastrophe,such as an earthquake or a flood. Note that in (ii), the events A1 , . . . , An are arbitrary, meaningthat X1 , . . . , Xn may be caused by either the same or different triggering events. In particular, if
A1 , . . . , An are the same, then A can also be chosen as the same event, and in this case (5) follows
Pnfrom (4). More generally, the condition P(A) = i=1 θi P(Ai ) makes it fair to compare the two sidesof (5); for instance, if X has a finite mean instead of being super-Pareto, then both sides of (5)would have the same mean. Although our setting mainly concerns the losses of an agent, it is also
applicable to the setting of investment decisions. For instance, X1 , . . . , Xn can represent profits
from technology innovations modelled by A1 , . . . , An ; negatively dependent profits may arise ininnovation races (see e.g., Shapiro and Varian (1999)).
An immediate consequence of Theorem 1 is that if super-Pareto random variables X1 , . . . , Xn
are independent and comparable in first-order stochastic dominance, for (θ1 , . . . , θn ) ∈ ∆n , we havePn
Xi∗ ≤st i=1 θi Xi if Xi∗ ≤st Xi for all i ∈ [n].
To better understand the result in Theorem 1, we stress that (4) and (5) cannot be expected
if X1 , . . . , Xn have finite mean, regardless of their dependence structure, as summarized in thefollowing proposition.

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Proposition 2. For θ1 , . . . , θn > 0 with

7

Pn

i=1 θn = 1 and identically distributed random variables

X, X1 , . . . , Xn with finite mean and any dependence structure, (4) holds if and only if X1 = · · · = Xnalmost surely.
Proposition 2 implies, in particular, that (4) never holds for WNAID non-degenerate randomvariables X, X1 , . . . , Xn with finite mean. Even if X, X1 , . . . , Xn have an infinite mean, we are notaware of any other distributions for which (4) and (5) hold other than the super-Pareto distributions
studied in this paper.
Risk measures are popular tools to quantify the risk of a financial portfolio; see Artzner et al.
(1999) and Föllmer and Schied (2016). Value-at-Risk (VaR) is one of the most widely used classesof risk measures in financial regulation. For a random variable X with distribution function FX ,VaR at level p ∈ (0, 1) is defined as the (left) quantile of X at p, that is,
VaRp (X) = FX−1 (p) = inf {t ∈ R : FX (t) ≥ p}.
For any random variable X, VaRp (X) is always finite, making it suitable for assessing losses with
infinite mean. Moreover, for two random variables X and Y , X ≤st Y (resp. X <st Y ) if andonly if VaRp (X) ≤ VaRp (Y ) (resp. VaRp (X) < VaRp (Y )) for all p ∈ (0, 1). Noting the equality
Pn
VaRp (X1 ) = i=1 θi VaRp (Xi ) for iid random variables, Theorem 1 gives superadditivity:
!
n
n
X
X
VaRp
θi Xi >
θi VaRp (Xi ).
(6)
i=1

i=1

Inequality (6) and its non-strict version can be intuitively seen as diversification penalty for VaRp .
2.3. Proof of Theorem 1
We first show in step (a) that the result in (i) holds for two independent Pareto(1) losses. An
induction argument yields the result for the sums of n such losses. We then extend the result to
WNAID Pareto losses in step (b), and further to super-Pareto losses in step (c). Finally, step (d)proves the result in (ii) by applying (i) and analysis on the combination of indicator functions,where the regularity assumption of X is needed.
n(a) Let Y, Y1 , . . . , Yn ∼ Pareto(1) such that Y1 , . . . , Yn are independent and ∆+n = ∆n ∩ (0, 1) . It

suffices to showP(θ1 Y1 + · · · + θn Yn ≥ x) >1
for all x ∈ (1, ∞) and (θ1 , . . . , θn ) ∈ ∆+
n.
x

(7)

Let δ = (x − 1 + θn )/θn . We will use the following fact: For (y1 , . . . , yn ) ∈ (1, ∞)n , if yn ≥ δ, thenθ1 y1 + · · · + θn yn ≥ (θ1 + · · · + θn−1 ) + θn δ = 1 − θn + x − 1 + θn = x.

Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

8{ Y2 ≥ δ }δy2{ θ1Y1 + θ2Y2 ≥ x ; Y2 < δ }1
1Figure 1y1

An illustration of {θ1 Y1 + θ2 Y2 ≥ x} = {Y2 ≥ δ} ∪ {θ1 Y1 + θ2 Y2 ≥ x; Y2 < δ}, where x = 5, θ1 = 2/5,
θ2 = 3/5, and δ = 23/3

We first show the case of n = 2. For all x ∈ (1, ∞) and (θ1 , θ2 ) ∈ ∆+2 (see Figure 1),
P(θ1 Y1 + θ2 Y2 ≥ x) = P(Y2 ≥ δ) + P(θ1 Y1 + θ2 Y2 ≥ x; Y2 < δ)
≥ P(Y2 ≥ δ) + P(θ1 Y1 + θ2 ≥ x; Y2 < δ)


θ1
1
1
1−
= +
δ x − θ2
δ


1 θ1
1
> +
1−
δ
x
δ
1
1
= (x + δ − θ2 δ − 1 + θ2 ) = .
xδ
x

(8)

Next, let n > 2 and assume (7) holds true for the case of n − 1. Then, for all x ∈ (1, ∞) and(θ1 , . . . , θn ) ∈ ∆+n,P(θ1 Y1 + · · · + θn Yn ≥ x) = P(Yn ≥ δ) + P(θ1 Y1 + · · · + θn Yn ≥ x; Yn < δ)
≥ P(Yn ≥ δ) + P(θ1 Y1 + · · · + θn−1 Yn−1 ≥ x − θn ; Yn < δ)
θ1 Y1 + · · · + θn−1 Yn−1 x − θn
= P(Yn ≥ δ) + P≥
P(Yn < δ)
1 − θn1 − θn
1 1 − θn1
≥ +1−
δ x − θnδ

1 1 − θn
1
1
> +
1−
= .
δ
x
δ
x

(9)

By induction, (4) holds for n independent Pareto(1) random variables.
(b) Since {(y1 , . . . , yn ) : θ1 y1 + · · · + θn yn < t} is a decreasing set, if Y1 , . . . , Yn are weakly negatively
associated, for any t > 1 and n ≥ 2,
P(θ1 Y1 + · · · + θn−1 Yn−1 ≥ t; Yn < δ) = P(Yn < δ) − P(θ1 Y1 + · · · + θn−1 Yn−1 < t; Yn < δ)Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

9≥ P(Yn < δ) − P(θ1 Y1 + · · · + θn−1 Yn−1 < t)P(Yn < δ)

= P(θ1 Y1 + · · · + θn−1 Yn−1 ≥ t)P(Yn < δ).
Therefore, if Y1 , . . . , Yn are weakly negatively associated, the equal signs in (8) and (9) will become
greater-than-or-equal signs and the proof in step (a) can still go through. Hence, (4) holds forweakly negatively associated Pareto(1) random variables.
(c) To show (4) for WNAID super-Pareto random variables, the following two lemmas will be
helpful. Their proofs are in Appendix A, and they do not assume super-Pareto distributions.
ddLemma 1. Suppose that X = f (Y ) for an increasing convex function f , X1 = X, and X1 , . . . , Xndare WNAID. There exist WNAID random variables Y1 , . . . , Yn with Y1 = Y such thatd

(X1 , . . . , Xn ) = (f (Y1 ), . . . , f (Yn )).
Lemma 2. Let Y, Y1 , . . . , Yn be any random variables, and (θ1 , . . . , θn ) ∈ ∆n . If Y ≤stPnthen f (Y ) ≤st i=1 θi f (Yi ) for any increasing convex function f .

Pn

i=1 θi Yi ,By Lemma 1 and the definition of super-Pareto distribution, for any WNAID super-Paretodrandom variables X1 , . . . , Xn with X1 = X, there exist WNAID Y1 , . . . , Yn ∼ Pareto(1) such thatd

(X1 , . . . , Xn ) = (f (Y1 ), . . . , f (Yn )). Therefore, for (4), it suffices to showf (Y ) ≤stnXθi f (Yi ),

(10)

i=1

which follows by combining Lemma 2 and the result in step (b). As it is shown in the previous
Pn
Pn
steps that Y <st i=1 θi Yi , it is clear that for t > f (1), P ( i=1 θi f (Xi ) > t) > P (f (X) > t), where
θi > 0 for at least two i ∈ [n]. Hence, the strictness statement after (4) also holds.
(d) Next we show (ii). Let A1 , . . . , An be events independent of (X1 , . . . , Xn ) and event A be
 T

Pn
T
independent of X satisfying P(A) = i=1 θi P(Ai ). For S ⊆ [n], let BS = i∈S Ai ∩ i∈S c Aci . For(θ1 , . . . , θn ) ∈ ∆n , we writenX

θi Xi 1Ai =

i=1

P

S⊆[n]

1BS

X

θi Xi .

i∈S

P

for any S ⊆ [n]. As first-order stochastic dominance is closedP
under mixture (e.g., Theorem 1.A.3 (d) of Shaked and Shanthikumar (2007)), i∈S θi Xi 1BS ≥stPi∈S θi X1BS for any S ⊆ [n]. Since BS and BR are mutually exclusive for any distinct S, R ⊆ [n],By (4),i∈S θi Xi ≥st

X

i∈S θi X

we havenX
i=1

θi Xi 1Ai =

X
S⊆[n]

1BS

X
i∈S

θi Xi ≥stXX
S⊆[n] i∈S

θi X1BS .

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10

Finally, we need to showPS⊆[n] (Pi∈S θi )X1BS ≥st X1A . For this, we prove the following state-ment. For mutually exclusive events B1 , . . . , Bn independent of X and (c1 , . . . , cn ) ∈ [0, 1]n , we have
X1B ≤stnX

ci X1Bi ,

(11)

i=1

where B is an event independent of X satisfying P(B) =Pn

i=1 ci P(Bi ). To show this, first notethat the statement is clearly true if c1 = · · · = cn = 0. If any components of (c1 , . . . , cn ) are zero, theproblem simply reduces its dimension. Hence, we assume that (c1 , . . . , cn ) ∈ (0, 1]n for the rest of
the proof. Let the survival function of X be P(X > x) = 1/g(x) for x ≥ zX and P(X > x) = 1 forx < zX . As X is regular, zX > 0, the concavity of g, and g(x) ≤ x/zX for x ≥ zX together implyg(t/c) ≤ g(t)/c for t ≥ 0 and c ∈ (0, 1]. For t ≥ zX , as B1 , . . . , Bn are mutually exclusive and ci ∈ (0, 1]
for all i ∈ [n],
P

n
X

!
ci X1Bi ≤ t = 1 −nXP(Bi )i=1

i=1

g(t/ci )

n

≤1−

1 X
P(B)
ci P(Bi ) = 1 −= P(X1B ≤ t).
g(t) i=1g(t)For t ∈ [0, zX ),
P

n
X

!
ci X1Bi ≤ t ≤ P

i=1

n
X

!
ci X1Bi ≤ zX

≤1−

i=1

P(B)
= P(X1B ≤ t),g(zX )

where we used g(zX ) = 1, implied by 1 ≤ g(zX ) ≤ zX /zX . This yields (11). As

Pi∈S θi ∈ [0, 1] for

any S ⊆ [n], andX
S⊆[n]

P(BS )

X

θi =

n
X
j=1

i∈S

θj

X

P(BS ) =S⊆[n],j∈SnX

θi P(Aj ) = P(A),

j=1

P

P
S⊆[n] (i∈S θi )X1BS ≥st X1A follows from (11) and the desired result (5) holds.

2.4. A few remarks
The next few remarks discuss the relation of Theorem 1 to the literature and some technical issues.
Remark 1 (EVT). In the literature of EVT, it has been observed that, for iid extremely heavytailed Pareto losses X1 , . . . , Xn ,
P

!
n
1X
Xi > t ≥ P (X > t)n i=1

holds true asymptotically as t → ∞; see, e.g., Kaas et al. (2004), Albrecher et al. (2006), andEmbrechts et al. (2009). Theorem 1 implies that the same inequality holds for any t ∈ R regardlessof whether t is large enough. This gives rise to implications for decision makers whose preferences
are not determined purely by the tail behaviour of risks.

Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

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Remark 2 (Stable distributions). As an important class of heavy-tailed distributions, stable
distributions have frequently appeared in the analysis of portfolio diversification (e.g., Ibragimov(2005), Ibragimov and Walden (2007, 2008)). Using majorization order, Ibragimov (2005) showedthat diversification increases the risk of a portfolio which consists of iid stable random variables
without finite mean. In particular, if the stable random variables are one-sided on the positive axis,
diversification will increase the total loss in first-order stochastic dominance; Ibragimov and Walden(2010) applied this result to study the problem of optimal bundling strategies for extremely heavytailed valuations. On the other hand, if the stable random variables are not one-sided, diversification
will make the total loss “more spread out”, hence different from first-order stochastic dominance.
These results were extended to the case when losses are convex transformations of iid infinite-mean
stable random variables in Ibragimov and Walden (2008). For iid symmetric infinite-mean stablerandom variables truncated by a sufficiently large number, diversification still makes the total loss
“more spread out”, as shown by Ibragimov and Walden (2007).
Remark 3 (Notions of negative dependence). Among the following notions of negative
dependence, weak negative association is weaker than (a) and (b) below, and stronger than (c).
(a) A random vector X = (X1 , . . . , Xn ) is negatively associated if for every pair of disjoint sets A,
B of [n] and any functions f and g both increasing or decreasing coordinate-wise, provided thecovariance below exists,cov(f (XA ), g(XB )) ≤ 0,

(12)

where XA = (Xk )k∈A and XB = (Xk )k∈B (Alam and Saxena (1981) and Joag-Dev and Proschan(1983)). It is known that random vectors following multivariate normal distributions with
non-positive correlations are negatively associated, and so are those obtained from increasingtransforms of such normal random vectors (Joag-Dev and Proschan (1983)). Choosing A = {i},B = [n] \ A, f (y) = 1{y≤x} and g(y) = 1{y∈S} in (12) yields (2), and hence weak negative
association is implied.
(b) A random vector X is negative regression dependent if for every i ∈ [n], the random variable
E[g(X−i )|Xi ] is a decreasing function of Xi for any coordinate-wise increasing function g suchthat the conditional expectation exists; see Lehmann (1966), who only formulated the casen = 2. This notion for general n > 2 is called negative dependence through stochastic orderingby Block et al. (1985). To check that this notion is stronger than weak negative association, it
suffices to take the function g(x) = −1{x∈S} for a decreasing set S ⊆ Rn−1 .
(c) A random vector X = (X1 , . . . , Xn ) is negatively orthant dependent if for all x = (x1 , . . . , xn ) ∈Qn
Qn
Rn , P(X ≤ x) ≤ i=1 P(Xi ≤ xi ) and P(X > x) ≤ i=1 P(Xi > xi ). The fact that (2) implies
negative orthant dependence follows from (2) and (3). Negative orthant dependence is notChen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

12

sufficient for the proof of Theorem 1, because in the proof (see Section 2.3, step (b)) we need
the inequality P(θ1 Y1 + · · · + θn−1 Yn−1 < t, Yn ≤ δ) ≤ P(θ1 Y1 + · · · + θn−1 Yn−1 < t)P(Yn ≤ δ),
which holds under weak negative association of (Y1 , . . . , Yn ) but not under negative orthantdependence.
Remark 4 (Majorization). Let X1 , . . . , Xn be iid super-Pareto random variables. Inspired by
Theorem 1, a question is whethernX

ηi Xi ≤st

n
X

i=1

θi Xi

(13)

i=1

holds for two vectors (θ1 , . . . , θn ) ∈ ∆n and (η1 , . . . , ηn ) ∈ ∆n increasing in majorization order; thatPn
Pn
is, i=1 φ(θi ) ≤ i=1 φ(ηi ) for all continuous and convex functions φ (see Marshall et al. (2011)).
Theorem 1 corresponds to the case (η1 , . . . , ηn ) = (1, 0, . . . , 0). This question seems to be beyond
the current techniques. For results similar to (13) on some distributions, see Proschan (1965) and
Ibragimov (2005).

3. Conclusion
Our main result (Theorem 1) establishes that a weighted average of WNAID super-Pareto randomvariables, possibly triggered by different events, is larger than one such loss in the sense of firstorder stochastic dominance. This result implies that the diversification of many super-Pareto losses
without finite mean increases the risk assessment of a portfolio, according to the superadditivityof VaR. Some technical questions remain open and are discussed in Remark 4.

A. Proofs of all other results
Proof of Proposition 1

We first show the equivalence statement. For the “⇐” direction, letP(X ≤ x) = 1 − 1/g(x) for x ∈ [zX , ∞), where g is strictly increasing and concave on [zX , ∞). Let
f (y) = g −1 (y) for y > g(zX ) and f (y) = zX for 1 ≤ y ≤ g(zX ). It is straightforward to see that fordany Pareto(1) random variable Y , f (Y ) = X. Next, we show the “⇒” direction. For x < ∞, theright-continuous generalized inverse of f is f −1+ (x) = inf {t : f (t) > x}. For x ≥ f (1), P(f (Y ) ≤ x) =P(Y ≤ f −1+ (x)) = 1 − 1/f −1+ (x). As f is increasing, convex, and non-constant, g := f −1+ is strictlyincreasing and concave. Hence, g is concave and strictly increasing on [zX , ∞).
The statement on regularity follows by observing two facts. First, f (1) > 0 implies that zX =f (1) > 0. Moreover, since f is convex and f (0) = 0, we have f (y) ≥ yf (1) for all y > 1, which gives
g(x) ≤ x/zX for x ≥ zX via g = f −1+ .
Proof of Proposition 2

Note that (1) implies that ESp (X) ≤ ESp (where ESp is defined as
1
ESp (X) =1−p

Z 1
VaRu (X)du.
p

Pn

i=1 θi Xi ) for all p ∈ (0, 1),

Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

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Since ESp is convex and X1 , . . . , Xn are identically distributed, we have
!
n
n
X
X
ESp (Xi ) = ESp (X), p ∈ (0, 1).
ESp (X) ≤ ESp
θi Xi ≤ θi
i=1

i=1

Using positive homogeneity of ESp , it follows that the equality

Pn

i=1 ESp (θi Xi ) = ESp (Pn

i=1 θi Xi )holds for each p ∈ (0, 1). By Theorem 5 of Wang and Zitikis (2021), this implies that(θ1 X1 , . . . , θn Xn ) is p-concentrated for each p; this result requires X1 , . . . , Xn to have finite mean.
Using Theorem 3 of Wang and Zitikis (2021), the above condition implies that (X1 , . . . , Xn ) iscomonotonic. For definitions of comonotonicity and p-concentration, see Wang and Zitikis (2021).
Since X1 , . . . , Xn are identically distributed, comonotonicity further implies that X1 = · · · = Xnalmost surely.
Proof of Lemma 1

If f is constant, then there is nothing to show. If f is not constant, thenthere exists z ∈ R such that f (x) is strictly increasing for x > z. Denote by q = P(X = zX ). Fori ∈ [n], let Wi be a uniform transform of Xi , that is, Wi is a standard uniform random variable
such that FX−1 (Wi ) = Xi a.s. A uniform transform always exists in an atomless probability space;see Lemma A.32 of Föllmer and Schied (2016). For all i ∈ [n], let Ai = {Xi ≤ zX } andUi = qVi 1Ai + Wi 1Aci ,where V1 , . . . , Vn are iid standard uniform random variables independent of (X1 , . . . , Xn ). Notethat for each i ∈ [n], Ui is also a uniform transform of Xi , and the distribution of (U1 , . . . , Un )is one possible copula of (X1 , . . . , Xn ). Let Yi = FY−1 (Ui ) for i ∈ [n]. Note that FX−1 = f ◦ FY−1
and f is strictly increasing for x > z. Hence Yi is a strictly increasing function of Xi given Vidfor each i ∈ [n]. Moreover, (X1 , . . . , Xn ) = (f (Y1 ), . . . , f (Yn )) as they have the same copula andmarginal distributions. It remains to show that Y1 , . . . , Yn are weakly negatively associated. For
any decreasing set A ⊆ Rn−1 and x ∈ R with FY (x) > 0, denote by β = 1/FY (x), and we haveP((Y1 , . . . , Yn−1 ) ∈ A|Yn ≤ x) = P((Y1 , . . . , Yn−1 ) ∈ A, Yn ≤ x)β= E[P((Y1 , . . . , Yn−1 ) ∈ A, Yn ≤ x|V1 , . . . , Vn )]β≤ E[P((Y1 , . . . , Yn−1 ) ∈ A|V1 , . . . , Vn )P(Yn ≤ x|V1 , . . . , Vn )]β= E[P((Y1 , . . . , Yn−1 ) ∈ A|V1 , . . . , Vn−1 )P(Yn ≤ x|Vn )]β= E[P((Y1 , . . . , Yn−1 ) ∈ A|V1 , . . . , Vn−1 )]E[P(Yn ≤ x|Vn )]β= P((Y1 , . . . , Yn−1 ) ∈ A)P(Yn ≤ x)β = P((Y1 , . . . , Yn−1 ) ∈ A),
where the inequality holds because for each i ∈ [n], conditional on Vi , Yi is a strictly increasing
function of Xi , and X1 , . . . , Xn are weakly negatively associated. Therefore, Y1 , . . . , Yn are also
weakly negatively associated.

Chen, Embrechts, and Wang: An unexpected stochastic dominanceWorking paper version (March 2024)

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Proof of Lemma 2Pn
Pn
As f is convex and increasing, we have f (Y ) ≤st f ( i=1 θi Yi ) ≤ i=1 θi f (Yi ),where the first inequality holds as (4) holds for Y1 , . . . , Yn and the second inequality is due to theconvexity of f .

Acknowledgments
The authors thank an Area Editor, an Associate Editor, and three anonymous referees for various relevant
comments on the paper. In particular, the inclusion of negative dependence was inspired by a comment froman Associate Editor, and the open question in Remark 4 was inspired by comments from the referees. The
authors thank Yuming Wang and Wenhao Zhu for ideas that led to a preliminary version of Theorem 1. We
also thank Hansjörg Albrecher, Jan Dhaene, John Ery, Taizhong Hu, Massimo Marinacci, Alexander Schied,and Qihe Tang for helpful comments on a previous version of the paper. Ruodu Wang is supported by the
Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823 and CRC-2022-00141).

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