Extreme value techniques for stress scenario selection under elliptical symmetry and beyond

A.1 Proof of Lemma 3.4

From (2.5), X = R ⁢ Q ⊤ ⁢ S , where Q ⊤ ⁢ Q = Σ , and thus Y = c ⊤ ⁢ X = R ⁢ ( Q ⁢ c ) ⊤ ⁢ S . Let

In spherical coordinate, we have

Let I ⁢ ( Θ ) = ( Q ⁢ c ) ⊤ ⁢ S and D I = { θ ∈ D : I ⁢ ( θ ) > 0 } . For any x > 0 ,

The above limit indicates P ⁢ ( Y ≥ t ⁢ x ) / P ⁢ ( Y ≥ t ) → x − 1 / γ as t → ∞ , which means that 𝑌 is regularly varying with index 1 / γ . Let

From (A.1), we have

A.2 Proof of Lemma 3.5

We first provide a result from [17], which will be used several times in the later proofs.

We prove Lemma 3.5 assuming L = g ⁢ ( X ) = w ⊤ ⁢ X for some weight vector 𝐰. The proof is similar for the situation where ( X , L ) is jointly elliptically distributed with positive definite scale matrix. Let e j be the vector where the 𝑗-th component is 1 and all other components are 0 ( j ∈ { 1 , … , d } ). Using spherical coordinates, we can write

Let D I 2 = { θ ∈ D : I 2 ⁢ ( θ ) > 0 } . From Lemma A.1, for any y > 0 and q ∈ [ 0 , 1 / γ ) ,

Thus

where r ⁢ ( γ , q ) ∈ R for q ∈ [ 0 , 1 / γ ) . Letting y = 1 , we obtain

Furthermore,

since a ⁢ ( t ) ⁢ A ̃ R ⁢ ( t ) → 0 and ρ < 0 . Combining (A.3) and (A.4), we prove

A.3 Proof of Lemma 3.6

We prove Lemma 3.6 under the situation that L = g ⁢ ( X ) = w ⊤ ⁢ X for some weight vector 𝐰. The proof is similar when ( X , L ) is jointly elliptically distributed with positive definite scale matrix. From [17, Appendix A.1], we have

where Z 1 ⁢ = d ⁢ ∥ Q ⁢ w ∥ ⁢ e 1 ⊤ ⁢ R ⁢ S . From Lemma A.1, we have

which indicates that

Furthermore, according to Lemma 3.4, there exists a function A ̃ Z 1 ⁢ ( t ) ∝ A ̃ R ⁢ ( t ) such that

Combining (A.5) and (A.6), we have, as p → 0 ,

The integral in the above asymptotic equality is finite since ρ < 0 and γ > 0 . Note that, from (A.1), we have lim t → ∞ P ⁢ ( L ≥ t ) / P ⁢ ( R ≥ t ) = c ∈ ( 0 , ∞ ) . Hence

from Condition 3 (b) and the fact that k / ( n ⁢ p ) → ∞ . Thus

As a consequence, we obtain k ⁢ ( μ ̃ j ⁢ ( p ) m ̃ j ⁢ ( p ) − 1 1 − γ ) → 0 .

A.4 Proof of Lemma 3.7

Let 𝑋 be a regularly varying random variable satisfying the second-order condition with the auxiliary function A X as in (3.10) and A ̃ X as in (3.11). Recall that the two functions are connected via A ̃ X ⁢ ( x ) = A X ⁢ ( 1 / P ⁢ ( X ≥ x ) ) .

Let d n = ( k n ⁢ p ) γ . We can write

Following arguments similar to (A.7), we can prove k ⁢ A ̃ R ⁢ ( U L ⁢ ( n / k ) ) → 0 as n → ∞ . Combined with Lemma 3.5 and the fact that k / ( n ⁢ p ) → ∞ , the last two terms converge to 0.

For the first term, Lemma 3.4 indicates there exists a function A L ⁢ ( t ) ∝ A R ⁢ ( t ) such that

Based on [12, Theorems 2.3.6 and 2.3.9], (A.8) implies that, for any ϵ , δ > 0 , there exists a t 0 = t 0 ⁢ ( ϵ , δ ) > 1 such that, for all t , t ⁢ x ≥ t 0 ,

where

Letting t = n / k → and x = k / ( n ⁢ p ) → , we have

Let v n = U L ⁢ ( 1 / p ) / U L ⁢ ( n / k ) . As k / ( n ⁢ p ) → ∞ , we have v n → ∞ . From (A.9), we have

Note that d n / v n → 1 and k ⁢ A 0 ⁢ ( n / k ) = O ⁢ ( k ⁢ A L ⁢ ( n / k ) ) → 0 since A L ⁢ ( ⋅ ) ∝ A R ⁢ ( ⋅ ) . Choosing δ < − ρ , we prove convergence of the first term to 0.

A.5 Proof of Theorem 3.8

For simplicity, let u n = U L ⁢ ( n / k ) and define L ̃ n = L n − k , n / u n . From [12, Theorem 2.2.1] and Karamata’s theorem, we have, as n → ∞ , L ̃ n ⁢ → P ⁢ 1 and

At first, we establish two lemmas, which will be used to prove Theorem 3.8.