Extremes for multivariate expectiles

Proof.

We give some details on the proof for the first item, the second one may be obtained in the same way.

Under Assumption 1, for all i∈{1,…,d}, F¯Xi∈RV-θ⁢(+∞). There exists, for all i, a positive measurable function Li∈RV0⁢(+∞) such that

Then, for all (i,j)∈{1,…,d}2 and all t,s>0,

and under Assumption 1,

Using Karamata’s representation for slowly varying functions (Theorem 2.2), there exist a constant c>0, a positive measurable function c⁢(⋅) with limx↑+∞⁡c⁢(x)=c>0 such that for all ϵ>0, there exists t0 such that for all t>t0,

Taking 0<ϵ<θ-1, we conclude

Proof.

We start by proving

Using Assumption 1, it is sufficient to show

Assume that

We shall prove that, in this case, (3.3) cannot be satisfied. Taking, if necessary, a subsequence (αn→1), we may assume that

We have (recall (3.1))

Furthermore, for all i∈{2,…,d},

On one side,

so that Lemma 3.2 implies

Let i∈J01. Taking, if necessary, a subsequence, we may assume that xix1→0, and therefore,

Now,

Thus

Consider the second term of (A.1)

Karamata’s theorem (Theorem 2.3) gives

which leads to

Finally, we get

We have shown

so the first equation of optimality system (3.3) implies

This is absurd since the xk’s are non-negative, and consequently

Now, we prove that the components of the extreme multivariate expectile also satisfy

Using Assumption 1, it is sufficient to show

Let us assume that

We shall see that, in this case, (3.3) cannot be satisfied. Taking, if necessary, a convergent subsequence, we may assume

In this case,

On another side, let i∈J∞1. Taking, if necessary, a subsequence, we may assume x1=o⁢(xi). Lemma 3.2 and Proposition 3.1  (ii) give

Moreover,

We deduce

Going through the limit (α→1) in the first equation of the optimality system (3.3), divided by x1⁢F¯X1⁢(x1), leads to

which is absurd because

We can finally conclude that

Proof.

Since the random vector 𝐗 is comonotonic, its survival copula is

We deduce the expression of the functions λUi⁢j,

So

Under Assumption 1 and by Proposition 3.3, let (η,β2,…,βd) be a solution of the equation system

for all k∈{1,…,d}. So η=1θ-1 and βk=ck1θ is the only solution to this system. ∎

Proof.

Taking, if necessary, a convergent subsequence (αn)n∈ℕ, αn→1, we consider that the limits

exist.

Using the notation JC={i∈{2,…,d}∣0<βi<+∞}, for all i∈JC,

because

On another hand,

and

Then, using

and Potter’s bounds (2.4) associated to F¯X1, we deduce that, for all ϵ1>0 and 0<ϵ2<1, there exists x10⁢(ϵ1,ϵ2) such that

Application of the dominated convergence theorem leads to

We denote by J∞ the set J∞={i∈{2,…,d}∣βi=+∞}. So, for all i∈J∞, x1=o⁢(xi) and

In the same way as in the previous case and using Potter’s bounds, we show that

from which we deduce

Let J0 be the set J0={i∈{2,…,d}∣βi=0}. For all i∈J0, we have xi=o⁢(x1). Then

because

In addition, for all ϵ>0, we have

because the dominated convergence theorem is applicable using Potter’s bounds and

since ci=0.

Let κ>0. For all ϵ>0, there exists α0 such that, for all α>α0,

Then

We deduce

so

We have therefore shown that

Proof.

We suppose that

Taking if necessary a convergent subsequence (αn)n∈ℕ with αn→1, we consider that the limits limα→1⁡xix1=βi exist and that

We use the notations

The first equation of the optimality system (1.1) divided by x1⁢F¯X1⁢(x1) can be written as

By (3.1),

and by Lemma 4.2,

so going through the limit (α→1) in the previous equation leads to

Nevertheless,