Extremal dependence of random scale constructions


A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1,X2) = R(W1,W2), with non-degenerate R > 0 independent of (W1,W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1,W2), the shape of the support of (W1,W2), and dependence between (W1,W2). When R is distinctly lighter tailed than (W1,W2), the extremal dependence of (X1,X2) is typically the same as that of (W1,W2), whereas similar or heavier tails for R compared to (W1,W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1,X2) possible in such cases when (W1,W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.


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Financial support from the Swiss National Science Foundation grant 200021-166274 (Sebastian Engelke) and UK Engineering and Physical Sciences Research Council grant EP/P002838/1 (Jennifer Wadsworth) is gratefully acknowledged. Thomas Opitz was partially funded by the French national programme LEFE/INSU.

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Appendix A: Lemmas and proofs

A.1 Additional lemmas

The following Lemma 8 is widely known as Breiman’s lemma and is useful in several contexts throughout Section 7.

Lemma 8 (Breiman’s lemma, see Breiman (1965), Cline and Samorodnitsky (1994), and Pakes (2004), Lemma 2.1)

SupposeXF,YGare independentrandom variables. If\(\overline {F}\in \text {RV}_{-\alpha }^{\infty }\)withα ≥ 0 andY ≥ 0 with E(Yα+ε) < forsomeε > 0,then

$$\overline{F}_{XY}(x)\sim \mathrm{E}\left( Y^{\alpha}\right){\kern1.7pt} \overline{F}(x), \qquad x\to\infty.$$

Equivalently, if F ∈ETα and E(e(α+𝜖)Y) < , then \(\bar {F}_{X+Y}(x)=\overline {F\star G}(x)\sim \mathrm {E}\left (e^{\alpha Y}\right ){\kern 1.7pt} \overline {F}(x)\).

The following Lemma 9 provides some additional detail on the function τ, as defined in Section 2.

Lemma 9

If\(1-\tau _{1}(b_{1}-\cdot ) \in \text {RV}_{1/\gamma _{1}}^{0}\),forγ1 > 0,then\(b_{1} - \tau _{1}^{-1}(1-\cdot ) \in \text {RV}_{1/\gamma _{1}}^{0}\),andγ1 ≤ 1.Similarly, if\(1-\tau _{2}(b_{2}+\cdot ) \in \text {RV}_{1/\gamma _{2}}^{0}\),forγ2 > 0,then\(\tau _{2}^{-1}(1-\cdot )-b_{2} \in \text {RV}_{\gamma _{2}}^{0}\)andγ2 ≤ 1.


The argument is similar for both cases, so we focus on the first one. Let g(t) = 1 − τ1(b1t), which is decreasing as t → 0 and invertible, with \(g^{-1}(s)=b_{1} - \tau _{1}^{-1}(1-s)\). Since \(g \in \text {RV}^{0}_{1/\gamma _{1}}\), then \(g^{-1} \in \text {RV}^{0}_{\gamma _{1}}\) (Resnick (2007), Proposition 2.6(v), adapting to regular variation at zero). Now make the left-sided Taylor expansion

$$ \tau_{1}(b_{1}-t) = \tau_{1}(b_{1}) -t \tau_{1}^{\prime}(b_{1-}) + O(t^{2}), $$

and note that τ1(b1) = 1. Consequently, \(1-\tau _{1}(b_{1}-t) = t \tau _{1}^{\prime }(b_{1-}) + O(t^{2})\), and so finiteness of \(\tau _{1}^{\prime }(b_{1-})\) will imply that the index of regular variation of g is at least 1. Define the convex function μ : (0,) → (0,) by μ(x) = ν(1,x), so that τ(x) = 1/μ(1/x − 1). Since τ1 is increasing on (0,b1), μ is increasing on ((1 − b1)/b1,). We have

$$ \tau^{\prime}(x_{-}) = x^{-2} \mu^{\prime}((1/x-1)_{+}) / \mu(1/x-1)^{2}, $$

and so \(\tau _{1}^{\prime }(b_{1-}) = b_{1}^{-2} \mu ^{\prime }([(1-b_{1})/b_{1}]_{+}) / \mu ((1-b_{1})/b_{1})^{2}\). For h ∈ (0, 1), convexity entails

$$ \mu((1-b_{1})/b_{1} + h) \leq h \mu((1-b_{1})/b_{1}+1) +(1-h)\mu((1-b_{1})/b_{1}), $$

and so

$$ 0\leq \frac{\mu((1{\kern1.7pt}-{\kern1.7pt}b_{1})/b_{1} {\kern1.7pt}+{\kern1.7pt} h) {\kern1.7pt}-{\kern1.7pt} \mu((1{\kern1.7pt}-{\kern1.7pt}b_{1})/b_{1})}{h} \leq \mu((1-b_{1})/b_{1}+1)-\mu((1-b_{1})/b_{1}) \!<\!\infty. $$

Hence μ([(1 − b1)/b1]+) < , giving \(\tau _{1}^{\prime }(b_{1-})<\infty \). Thus the index of regular variation of g is at least 1. □

The following Lemma 10 clarifies the influence of negative values in convolutions of exponential-tailed distributions. It allows us to extend certain results from the literature formulated for nonnegative random variables to the real line.

Lemma 3 (Convolutions of exponential-tailed distributions with negative values)

Fori = 1, 2 and probabilitydistributionsFi ∈ETαdefined over\(\mathbb {R}\)withα > 0,denote\(p_{i}=\bar {F}_{i}(0)\in [0,1]\)theprobability of nonnegative values. Using the convention0/0 = 0,let\(F_{i}^{+}\)with\(1-F_{i}^{+}(x)=\bar {F}_{i}(x)/p_{i}\),x ≥ 0, denote the conditionaldistribution ofFiovernonnegative values, and let\(F_{i}^{-}(x)=F_{i}(x)/(1-p_{i})\),x < 0, denote the conditionaldistribution ofFiover negativevalues. We use the notation\(M_{H}(\alpha )={\int }_{-\infty }^{\infty }\exp (\alpha y)H(\mathrm {d}y)\)fora given distribution H. Then:

  1. 1.

    If\(\bar {F_{i}^{+}}(x)/\bar {F_{1}^{+}\star F_{2}^{+}}(x)\rightarrow 0\)whenxfori = 1, 2,then

    $$ \bar{F_{1}\star F_{2}} \sim p_{1}p_{2}\bar{F_{1}^{+}\star F_{2}^{+}}. $$
  2. 2.

    If \(\bar {F_{1}^{+}\star F_{2}^{+}} \sim c_{1}\bar {F_{1}^{+}}+c_{2}\bar {F_{2}^{+}}\) with constants 0 ≤ c1,c2 < , then

    $$ \bar{F_{1}\star F_{2}}(x) \sim \bar{F}_{1}(x) \left( p_{2}c_{1}+(1-p_{2})M_{F_{2}^{-}}(\alpha)\right)+\bar{F}_{2}(x) \left( p_{1}c_{2}+(1-p_{1})M_{F_{1}^{-}}(\alpha)\right). $$

    Specifically, if \(c_{1}=M_{F_{2}^{+}}(\alpha )\) and \(c_{2}=M_{F_{1}^{+}}(\alpha )\), then

    $$ \bar{F_{1}\star F_{2}}(x)\sim M_{F_{2}}(\alpha) \bar{F}_{1}(x)+M_{F_{1}}(\alpha) \bar{F}_{2}(x), $$

    and if \(c_{1}=M_{F_{2}^{+}}(\alpha )\) and c2 = 0, then

    $$ \bar{F_{1}\star F_{2}}(x)\sim M_{F_{2}}(\alpha) \bar{F}_{1}(x)+\bar{F}_{2}(x)(1-p_{1})m_{F_{1}^{-}}(\alpha). $$


We start with the mixture representation

$$ \bar{F_{1}\star F_{2}}(x)=p_{1}p_{2}\bar{F_{1}^{+}\star F_{2}^{+}}(x)+p_{1}(1-p_{2})\bar{F_{1}^{+}\star F_{2}^{-}}(x)+(1-p_{1})p_{2}\bar{F_{1}^{-}\star F_{2}^{+}}(x), \quad x \geq 0. $$

We can then use the equation \(p_{i}\bar {F_{i}^{+}}(x)=\bar {F_{i}}(x)\), x ≥ 0, and the following inequalities for {i1,i2} = {1, 2},

$$ 0=\bar{F_{i_{1}}^{-}\star F_{i_{2}}^{-}}(x) = \bar{F_{i_{1}}^{-}}(x)\leq \bar{F_{i_{1}}^{+}\star F_{i_{2}}^{-}}(x) \leq \bar{F_{i_{1}}^{+}}(x) \leq \bar{F_{i_{1}}^{+}\star F_{i_{2}}^{+}}(x) , \quad x\geq 0. $$

Moreover, Lemma 8 can be applied for mixed terms, yielding \(\overline {F_{i_{1}}^{+}\star F_{i_{2}}^{-}}\sim M_{F_{i_{2}}^{-}}(\alpha )\overline {F_{i_{1}}^{+}}\), and then Eqs. 38 and 39 follow from straightforward calculations. To determine the behavior for special cases of c1 and c2, observe that \((1-p_{i})M_{F_{i}^{-}}(\alpha )+p_{i}M_{F_{i}^{+}}(\alpha )=M_{F_{i}}(\alpha )\). □

A.2 Proof of lemmas in Section 7

Proof of Lemma 2

  1. 1.

    If b1 = b2 = 1 then P(W = 1) = 0 and \(\bar {F}_{W}(1-s)=\bar {F}_{Z}(\tau ^{-1}(1-s))\). By assumption, \(\bar {F}_{Z}(\tau ^{-1}(1-s)) = \ell _{Z}(1-\tau ^{-1}(1-s))(1-\tau ^{-1}(1-s))^{\alpha _{Z}}\), where \(\ell _{Z} \in {\text {RV}_{0}^{0}}\). Since \(1-\tau ^{-1}(1-s) \in \text {RV}_{\gamma }^{0}\) (Lemma 9) with limit zero, results on composition of regularly varying functions (Resnick (2007), Proposition 2.6 (iv)) implies the result, with αW = αZγ.

  2. 2.a)

    If b1 < 1 then

    $$ \bar{F}_{W}(1-s) - \P(W=1) = \P(Z\in[\tau_{1}^{-1}(1-s),b_{1}))+\P(Z\in (b_{2},\tau_{2}^{-1}(1-s)]), $$

    and since Z has a Lebesgue density,

    $$ \begin{array}{@{}rcl@{}} &&\P(Z\in[\tau_{1}^{-1}(1-s),b_{1}))+\P(Z\in (b_{2},\tau_{2}^{-1}(1-s)])\\ &\sim& f_{Z}(b_{1})(b_{1} - \tau_{1}^{-1}(1-s)) + f_{Z}(b_{2})(\tau_{2}^{-1}(1-s) - b_{2}) \in \text{RV}_{\gamma}^{0}, \end{array} $$

    again using Lemma 9.

  3. 2.b)

    A left-sided Taylor expansion of \(\tau _{1}^{-1}\) about 1 gives

    $$ \tau_{1}^{-1}(1-s) = b_{1} - (\tau_{1}^{-1})'(1_{-}) s + O(s^{2}), $$

    where O(s2)/s uniformly tends to 0 as s → 0, and similarly we can make a right-sided expansion for \(\tau _{2}^{-1}(1-s)\). Hence, using Eqs. 40 and 41, \( \bar {F}_{W}(1-s) - \P (W=1) = s \ell (s)\) with \(\lim _{s\to 0}\ell (s) = f_{Z}(b_{1}) (\tau _{1}^{-1})'(1_{-}) - f_{Z}(b_{2})(\tau _{2}^{-1})'(1_{-})\). Noting the link \(1/\tau _{1}^{\prime }(b_{1-}) = (\tau _{1}^{-1})'(1_{-})\), similarly for \(1/\tau _{2}^{\prime }(b_{2+})\), gives Eq. 24.

  4. 2.

    For the final part, we have

    $$ \begin{array}{@{}rcl@{}}\bar{F}_{W_{\wedge}}(\zeta(1-s)) &=& \bar{F}_{Z}(\tau_{1}^{-1}(\zeta(1-s))) - \bar{F}_{Z}(1-\tau_{1}^{-1}(\zeta(1-s)))\\ &\sim& f_{Z}(1/2)\left\{ 1 - 2\tau_{1}^{-1}(\zeta(1-s))\right\}. \end{array} $$

    Again by left-sided Taylor expansion of \(\tau _{1}^{-1}\) about ζ = τ(1/2), we have

    $$\tau_{1}^{-1}(\zeta(1-s)) = 1/2 - (\tau_{1}^{-1})'(\zeta_{-}) \zeta s + O(s^{2}),$$

    and so we obtain \(\bar {F}_{Z}(\tau _{1}^{-1}(\zeta (1-s))) - \bar {F}_{Z}(1-\tau _{1}^{-1}(\zeta (1-s))) = s \ell _{\wedge }(s)\) with \(\lim _{s\to 0}\ell _{\wedge }(s) = 2 f_{Z}(1/2)\zeta (\tau _{1}^{-1})'(\zeta _{-})\). Noting again that \(1/\tau _{1}^{\prime }(1/2_{-}) = (\tau _{1}^{-1})'(\zeta _{-})\), we arrive at Eq. 25.

Proof of Lemma 5

The result for nonnegative S and V is found in Theorem 4(v) of Cline (1986). The extension to negative values then follows from Lemma 10(1). □


of Lemma 6 The result is given in Theorem 6(ii,iii) of Cline (1986) for \(\bar {F}(0)=1\) and \(\bar {G}_{1}(0)=1\). For point 1, the extension to negative values in F and G1 follows from observing that Theorem 6(iii) of Cline (1986) implies

$$ \bar{F\star G_{1}}\sim \bar{F^{+}\star G_{1}^{+}}, \qquad x\rightarrow\infty, $$

where F+ is obtained from F by setting F+(0) = F(0) and \(\bar {F^{+}}(0)=1\), and the same construction is taken for \(G_{1}^{+}\); i.e., F+ and \(G_{1}^{+}\) arise from projecting negative values to 0. For point 2, the extension to negative values can be shown using Lemma 10(2). Indeed, the same limit \(M_{G_{1}}(\alpha )\) arises for \(\bar {F\star G_{1}}(x)/\bar {F}(x)\) if we project negative values in F and G1 to 0 or not. □

Appendix B: Tail classes and examples

Definitions of tail classes are given in Section 1.1. The following lemma summarizes important relationships between such tail classes. In this section, we refer to the class of heavy-tailed distributions by HT, and to superheavy-tailed distributions by SHT.

Lemma 4 (Relationships between tail classes)

The following relationships between distribution classes hold:

  1. 1.

    \(\text {RV}_{\alpha }^{\infty }\subset \text {CE}_{0}\)forα > 0,

  2. 2.

    \(\text {ET}_{0}\subsetneq \text {HT}\) .

  3. 3.

    For ETαwithα > 0,we have:

    • \(F(\exp (\cdot ))\in \text {ET}_{\alpha } \Leftrightarrow F \in \text {RV}_{\alpha }^{\infty }\),

    • CEα ⊂ETα,

    • ETα,β>− 1 ∩CE = .

  4. 4.

    For WTβ,we have:

    • \(\text {WT}_{1}\subset \bigcup _{\alpha >0}\text {ET}_{\alpha }\),

    • WTβ ⊂CE0forβ < 1,

    • LWTβ ⊂SHT forβ < 1.

  5. 5.

    By denotingF1F2if\(\overline {F}_{1}(x)/\overline {F}_{2}(x)\rightarrow 0\)forx,we have:

    • If\(\tilde {\alpha }<\alpha \),then\(\text {WT}_{\beta >1}\prec \text {ET}_{\alpha } \prec \text {ET}_{\tilde {\alpha }} \prec \text {WT}_{\beta <1}\prec \text {LWT}_{\beta >1}\prec \text {RV}_{\alpha >0}^{\infty }\prec \text {RV}_{\tilde {\alpha }}^{\infty }\prec \text {SHT}\).

    • \(\text {CE}_{\alpha >0}\prec \text {ET}_{\tilde {\alpha },\beta }\)for\(\tilde {\alpha }\leq \alpha \)andanyβ > 0.

We recall the membership in tail classes for well-known parametric distribution families in Table 1, see Johnson et al. (1994, 1995) for reference about parameters. Here we abstract away from the usual parameter symbols of these distributions to avoid conflicting notations with general tail parameters. We refer parameters as scl and loc if scl × X + loc has scale scl and location loc, where X has scale 1 and location 0. Another parameter shp may be related to shape for some distributions.

Table 4 Membership in tail classes (columns) for distribution families (rows)

Appendix C: Additional illustrations

Figure 4 illustrates further examples of norms ν and related functions τ(z) and τ(1 − z), as defined in Section 2.

Fig. 4

Further illustration of different norms ν and their related functions τ(z) (solid line) and τ(1 − z) (dashed line)

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Engelke, S., Opitz, T. & Wadsworth, J. Extremal dependence of random scale constructions. Extremes 22, 623–666 (2019). https://doi.org/10.1007/s10687-019-00353-3

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  • Copula
  • Extreme value theory
  • Residual tail dependence
  • Tail dependence

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