Frédéric Meunier

Thomas Pradeau

F R Éd

Éric Meunier

SENSITIVITY OF EQUILIBRIUM IN ATOMIC CONGESTION GAMES WITH PLAYER-SPECIFIC COST FUNCTIONS

Keywords: congestion games, sensitivity, atomic splittable games

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Introduction

In many contexts, users share common resources to realize some tasks while being not coordinated. Examples of such resources are means of transportation, machines in a flexible manufacturing environment, or arcs in a telecommunication network. Congestion may appear on resources, leading to strategic behaviors. Game theory is a useful approach to understand and predict the behaviors of the users and the resulting congestion on the resources. The games arising in such a context, called congestion games, have been studied since the 50's, addressing many questions, such as the existence and the uniqueness of the Nash equilibrium, the way to compute it and related complexity questions, or its efficiency with respect to the social optimum (via the so-called "Price of Anarchy").

One stream of questions is about the sensitivity analysis, defined as the evaluation of the impact of the input (graph, cost functions and demands) on the equilibrium. In practice, these analyses are used for designing networks, estimating origin-destination matrices, or fixing pricing rules. Such analyses have been mainly applied to nonatomic congestion games and formulas have been designed in order to perform the sensitivity analysis, see [START_REF] Roger | Sensitivity analysis for equilibrium network flow[END_REF], [START_REF] Qiu | Sensitivity analysis for variational inequalities defined on polyhedral sets[END_REF], [START_REF] Michael | Transportation network analysis[END_REF]. Nonatomic means that there is a continuous set of users, each with a negligible impact on the congestion. Taking a more theoretical point of view, [START_REF] Hall | Properties of the equilibrium state in transportation networks[END_REF] proved that the equilibrium flows of nonatomic games played on a network is continuous with respect to the demand when all users have the same cost functions. This result have been extended for cost functions depending on the flows on all arcs of the network by [START_REF] Dafermos | Sensitivity analysis for the asymmetric network equilibrium problem[END_REF]. A more general study, concerning in particular the differentiability, has been made by [START_REF] Patriksson | Sensitivity analysis of aggregated variational inequality problems, with application to traffic equilibria[END_REF] and [START_REF] Patriksson | Sensitivity analysis of traffic equilibria[END_REF]. The latter gave a characterization for the existence of a directional derivative of the equilibrium flow with respect to the demand. [START_REF] Josefsson | Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design[END_REF] showed that while equilibrium costs are directionally differentiable, this does not hold for the flows.

A natural intuition would be that an increase of the demand gives an increase of the equilibrium cost. [START_REF] Hall | Properties of the equilibrium state in transportation networks[END_REF] proved that this result is true for two-terminal graphs when players have the same cost functions, and [START_REF] Lin | Stronger bounds on Braess's paradox and the maximum latency of selfish routing[END_REF] gave an alternate combinatorial proof of this result. However this intuition is false in general, as noted for example by [START_REF] Fisk | More paradoxes in the equilibrium assignment problem[END_REF]. [START_REF] Dafermos | Sensitivity analysis for the asymmetric network equilibrium problem[END_REF] proved that an "average" total cost will necessarily increase. More recently, [START_REF] Englert | Sensitivity of Wardrop equilibria[END_REF] proved that there are networks for which a slight increase of the total demand changes the strategies of all users. This "microscopic" instability, in the sense that the total flow on each arc does not change too much, happens in particular for the class of generalized Braess graphs introduced by [START_REF] Roughgarden | On the severity of Braess's paradox: designing networks for selfish users is hard[END_REF].

Our purpose is to make a sensitivity analysis for atomic splittable games. We consider a finite set of non-negligible players having a stock, the demand, to divide among different resources. For example, a freight company may have to choose between several means of transportation. This situation can be modeled by an atomic game on a network with parallel arcs, each arc representing a resource. This kind of games have been extensively studied, see for example [START_REF] Orda | Competitive Routing in Multi-User Communication Networks[END_REF], [START_REF] Altman | Competitive routing in networks with polynomial costs[END_REF], [START_REF] Richman | Topological Uniqueness of the Nash Equilibrium for Selfish Routing with Atomic Users[END_REF], [START_REF] Bhaskar | Equilibria of atomic flow games are not unique[END_REF][START_REF] Bhaskar | The price of collusion in series-parallel networks[END_REF]. Further results and extensions on Nash equilibria in this context can be found in [START_REF] Gairing | Routing (un-) splittable flow in games with player-specific linear latency functions[END_REF], [START_REF] Harks | The impact of collusion on the price of anarchy in nonatomic and discrete network games[END_REF], [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF]. These works consider mostly the case when every player is impacted in the same way by the congestion.

In this paper, we study the sensitivity of the equilibrium in atomic splittable games on parallelarcs graphs with player-specific cost functions. Questions like the continuity of the equilibrium and its differentiability are addressed, as well as questions about the consequence of a partial transfer of demand from a player to another one with a higher demand. The impact of making coalitions is a special case of this latter question.

Model and main results

2.1. Model. We are given a two-terminal graph with a set A of parallel arcs and K ≥ 2 players identified with the integers 1, . . . , K. Throughout the paper, the set {1, . . . , K} is denoted [K]. We consider the model of atomic splittable games: each player k has a demand d k ∈ R + and cost functions c k a : R + → R + , assumed to be increasing, differentiable, and strictly convex. A feasible strategy for the player k is an element

x k = (x k a ) a∈A of R A + such that a∈A x k a = d k . The quantity x k
a is then the flow of player k on the arc a. When each player has chosen a strategy, the total flow on arc a is the quantity x a = K k=1 x k a and the support of the strategy is the set of arcs "used" by this player supp(

x k ) = {a ∈ A : x k a > 0}. The cost supported by player k is a∈A c k a (x a ). A vector x = (x 1 , . . . , x K ) a∈A of feasible strategies is an equilibrium if for every player k a∈A x k a c k a (x a ) = min (y k a )∈R A + : a y k a =d k a∈A y k a c k a (y a ).
2.2. Properties. The following characterization of the equilibrium is standard and has been used for instance in [START_REF] Haurie | On the relationship between Nash-Cournot and Wardrop equilibria[END_REF].

Proposition 1. The vector x is an equilibrium if and only if, for all k, the vector x k is a feasible strategy for player k and

a∈A (c k a (x a ) + x k a c k a (x a ))(y k a -x k a )
≥ 0, for any other feasible strategy y k .

The following proposition is an alternate and again standard characterization of the equilibrium.

Proposition 2. The vector x is an equilibrium if and only if, for all k, the vector x k is a feasible strategy for player k and c k a (x a ) + x k a c k a (x a ) = π k for all a ∈ supp(x k ), where

π k = min a∈A c k a (x a ) + x k a c k a (x a ) . c k a (x a ) + x k a c k a (x a )
is the marginal cost of arc a for player k. Proposition 2 states thus that at equilibrium, the marginal costs of the arcs in the support of every player are all equal.

For the games considered in this paper, an equilibrium always exists, see [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF], [START_REF] Orda | Competitive Routing in Multi-User Communication Networks[END_REF]. Furthermore, the graph belongs to the class of nearly-parallel graphs and thus the equilibrium is unique [START_REF] Richman | Topological Uniqueness of the Nash Equilibrium for Selfish Routing with Atomic Users[END_REF].

2.3. Results. The uniqueness of equilibrium allows to define for each player k the map

e k : R K + -→ R A +
by defining e k (d) to be the strategy of player k at equilibrium when the demands are given by the components of d = (d 1 , . . . , d K ). Define

e : d ∈ R K + -→ (e 1 (d), . . . , e K (d)) ∈ (R A + ) K . It is a continuous map, see Proposition 3 of Section 3.
A demand transfer is a part δ ∈ [0, d i ] removed of the demand of a player i and added to the demand of a player j, the demand of the other players remaining the same. The new demand of player i is then d i -δ and the new demand of player j is then d j + δ.

Theorem 1. Suppose that |A| = 2 or K = 2, and that the cost functions are twice continuously differentiable. Let d 0 ∈ R K + be a demand vector such that for any pair of players, a sufficiently small nonzero demand transfer does not modify the supports of the players. Then e(•) is differentiable at d 0 .

Generically, all d 0 ∈ R K + satisfy this condition (it is for instance a consequence of Corollary 1 of Section 4.1). We do not know whether this theorem still holds when |A| ≥ 3 and K ≥ 3. However, we cannot expect e(•) to be differentiable everywhere, even if there is only one player, see Section 6.

Our two other main results are about the case when the players i and j involved in the transfer described above are such that d i ≤ d j .

Theorem 2. Suppose that |A| = 2 or K = 2. If a player i transfers a part of his demand to a player j with d i ≤ d j , then, on each arc, the flow of player i (resp. player j) decreases (resp. increases) or remains constant equal to zero.

The social cost is defined by

Q( x) = K k=1 a∈A x k a c k a (x a ).
It is the cost experienced by all players together.

Theorem 3. Suppose that c 1 a = • • • = c K a for all a ∈ A (we are no longer in the player-specific cost setting). Suppose moreover that at least one of the following conditions is satisfied.

• |A| = 2 • K = 2 and the cost functions are twice continuously differentiable . Then if a player i transfers a part of his demand to a player j with d i ≤ d j , the social cost at equilibrium decreases or remains constant.

A consequence of this theorem is that if the total demand k d k = d is fixed, the equilibrium with highest social cost is obtained when all players have the same demand equal to d/k. In the contrary, social cost is reduced when a monopole takes in charge the whole demand. Nevertheless, this is true under the conditions of the theorem. If these conditions are relaxed, there are situations where the conclusion of Theorem 3 does not hold, see Section 6.

Theorem 3 is a generalization of Theorem 3.23 of [START_REF] Wan | Contributions à la théorie des jeux d'évolution et de congestion[END_REF] when there are no nonatomic players. [START_REF] Wan | Contributions à la théorie des jeux d'évolution et de congestion[END_REF] proved that when there are two arcs the social cost at equilibrium cannot increase when two players merge, i.e. in our context when player i transfers all his demand to player j. The question whether our result remains valid with nonatomic players deserves future work.

Theorems 1, 2, and 3 are respectively proved in Sections 3, 4, and 5. We emphasize that the proof of Theorem 3 uses Theorems 1 and 2.

3. Regularity of equilibrium 3.1. Continuity. The following proposition holds for the model described in Section 2.1 in its full generality. It contrasts with Theorem 1, which requires the cost function to be twice continuously diffrrentiable.

Proposition 3. The map e(•) is continuous.

Proof. Let = {p ∈ R A + : a∈A p a = 1} be the (|A| -1)-dimensional simplex and F : K × R K + → (R A + ) K be defined by F k a ( p, d) = c k a K =1 p a d + p k a d k c k a K =1 p a d d k ,
for every a and k. The application F is continuous in both variables.

According to Proposition 1, e(d) is the equilibrium if and only if e k a (d) = p k a d k for all a and k, where p satisfies the variational inequality

(V I(d)) F ( p, d) • ( q -p) ≥ 0 for any q ∈ K .
By a sequential argument, using the continuity of F and the compactness of K , elementary calculations give that the solution of (V I(d)) is continuous with respect to d.

3.2.

Proof of Theorem 1. The strategies at equilibrium are completely determined by the supports of the players. This is due to the increasing assumption of the costs. When the supports are given, the equilibrium is characterized by equalities and then the inverse function theorem makes the job. This is the core idea of the proof of Theorem 1.

Lemma 1. Suppose that the cost functions are twice continuously differentiable. Choose an S k ⊆ A for each player k. Define the application H :

K k=1 R S k + × R K + → K k=1 R S k × R K by H( y, π) = (G( y, π), D( y)), where for a ∈ S k , y k ∈ S k , and π ∈ R K , we have G k a ( y, π) = c k a (y a ) + y k a c k a (y a ) -π k and D k ( y) = a∈S k y k a .
Let z be such that z a > 0 for every a ∈ S k and every player k, and let π ∈ R K . Then H is continuously differentiable and invertible in a neighborhood of ( z, π), with a continuously differentiable inverse.

Proof. Since c a is twice continuously differentiable for every a, the map H is continuously differentiable. We will show that the Jacobian matrix J of H at ( z, π) is nonsingular. The conclusion will then follow from the inverse function theorem.

We define

g k a = c k a (z a ) + z k a c k a (z a ) for every a ∈ S k and h k a = c k a (z a )
for every a ∈ A. Note that because of the assumptions, they are positive when a ∈ S k . A direct calculation gives the entries of J. For k ∈ [K] and a ∈ S k , we have

∂G k a ∂y b ( z, π) =    g k a for a = b, k = g k a + h k a for a = b, k = 0 for a = b. ∂G k a ∂π ( z, π) = 0 for k = -1 for k = ∂D k ∂y b ( z, π) = 0 for k = 1 for k = . ∂D k ∂π ( z, π) = 0.
We prove now that J is nonsingular by showing that its kernel contains only the zero vector. We denote by K a the set of players effectively using arc a:

K a = {k ∈ [K] : a ∈ S k }. Let ( λ, µ) ∈ K k=1 R S k × R K be in the kernel of J.
The equality J( λ, µ) = 0 can be written under the form (1)

       M a λ a = µ a for a ∈ A, a∈S k λ k a = 0 for k ∈ [K].
where λ a = (λ k a ) k∈Ka and µ a = (µ k ) k∈Ka , with

M a = (m k, ) being the |K a | × |K a | matrix such that m k, = g k a for k = g k a + h k a for k = .
It can be readily checked that M a is a nonsingular matrix because the g k a and the h k a are positive. In order to consider matrices with same dimension, we define Λ ∈ K k=1 R A by Λ k a = λ k a for a ∈ K a and 0 elsewhere. The system (1) is then equivalent to the following system:

       Λ a = M a µ for a ∈ A, a∈A Λ a = 0, where M a is a K × K matrix obtained from M -1
a by setting the missing coefficients to zero. These conditions imply that

a∈A Λ a = a∈A M a µ = 0.
A Z-matrix is a square matrix whose nondiagonal coefficients are nonpositive. According to a theorem by [START_REF] Minkowski | Zur theorie der einheiten in den algebraischen zahlkörpern[END_REF], if each column of a Z-matrix has a positive sum, then it is a nonsingular matrix. For more details, see [Berman and Plemmons, 1979, Chapter 6]. It can be checked that each M a is such a Z-matrix. Their sum is thus also such a Z-matrix, and is thus nonsingular. It gives µ = 0 and then Λ a = 0 for all a ∈ A. In particular, λ = 0. Therefore, J is nonsingular.

Proof of Theorem 1. Let d 0 be as in the statement. Define S k to be supp(e k (d 0 )). Let x = e(d 0 ). There exists

π ∈ R K + such that        c k a (x a ) + x k a c k a (x a ) -π k = 0 for k ∈ [K], a ∈ S k a∈A x k a = d k 0 for k ∈ [K] x k a = 0 for k ∈ [K], a / ∈ S k c k a (x a ) + x k a c k a (x a ) -π k > 0 for k ∈ [K], a / ∈ S k .
Indeed, it is a consequence of Proposition 2, except the fact that the last inequality is strict. We prove now this latter fact. Assume for a contradiction that there is a player k such that c k a (x a ) + x k a c k a (x a ) -π k = 0, while a / ∈ S k . The continuity of e and Corollary 2 (see Section 4) would then imply that a slight transfer of demand to player k would change the support of his strategy, which is in contradiction with the assumption regarding the non-modification of the support. Corollary 2 can be used since we have assumed that |A| = 2 or K = 2.

We can apply Lemma 1: there is a neighborhood V of (0,

d 0 ) in k∈[K] R S k × R K such that H -1 exists on V and is differentiable. Define P : (z, π) ∈ k∈[K] R S k + × R K + -→ y ∈ k∈[K] R A + with y k a = z k a if a ∈ S k 0 otherwise. Let ( z, π) = H -1 (0, d)) for some (0, d) ∈ V . It is straightforward to check that (P ( z, π), π
) is a solution of the system above with d k 0 replaced by d k , and thus that P ( z, π) satisfies the condition of Proposition 2. In other word, e(d) = P (H -1 (0, d)) for (0, d) ∈ V . The map P is differentiable everywhere and H -1 is differentiable on V . It leads to the desired conclusion.

4. Equilibrium and demand transfers: proof of Theorem 2 4.1. A more general result. Theorem 2 is a direct consequence of the following proposition. We define d δ the vector of demands after player i has transferred a part δ > 0 of his demand to player j: If x i a ≤ y i a , then x i a = y i a = 0 and x a ≤ y a . If x j a ≥ y j a , then x j a = y j a = 0 and x a ≥ y a . In case |A| = 2, we have moreover (x k a -y k a )(x a -y a ) ≤ 0 for k = i, j. This proposition is proved in the next subsection. We first show two corollaries. They will be useful in the proof of Theorem 3. The second one is also used in the proof of Theorem 1.

d i δ = d i -δ, d j δ = d j + δ,
Corollary 1. Suppose that |A| = 2 or K = 2, and let x = e(d) and y = e(d δ ). Then,

• supp(y i ) ⊆ supp(x i ) and supp(x j ) ⊆ supp(y j ).

• {a ∈ A : y a < x a } ⊆ supp(x i ) and {a ∈ A : y a > x a } ⊆ supp(y j ).

Proof. Let a be an arc in supp(y i ). We have y i a = 0. Proposition 4 implies then that x i a > y i a , which shows that supp(y i ) ⊆ supp(x i ). The inclusion supp(x j ) ⊆ supp(y j ) is proved similarly. This shows the first point.

Let a be such that y a < x a . Proposition 4 implies then that x i a > y i a , which shows that a ∈ supp(x i ). The other inclusion is proved similarly. This shows the second point.

We consider now the "limit" case when the cost of an arc with no flow is equal to the marginal cost.

Corollary 2. Suppose that |A| = 2 or K = 2, and let x = e(d) and y = e(d δ ).

If b / ∈ supp(x j ) is such that c j b (x b ) = min a∈A (c j a (x a ) + x j a c j a (x a )), then b ∈ supp(y j ). If b / ∈ supp(y i ) is such that c i b (y b ) = min a∈A (c i a (y a ) + y j a c i a (y a )), then b ∈ supp(x i ).
Proof. We only prove the first point, the proof of the second one being similar. If x b < y b , we have b ∈ supp(y j ) because of Corollary 1. We can thus assume for the remaining of the proof that x b ≥ y b . Suppose for a contradiction that b / ∈ supp(y j ). Consider an arc a ∈ supp(y j ). Proposition 4 gives that x j a < y j a . We have moreover

c j a (x a ) + x j a c j a (x a ) ≥ c j b (x b ) ≥ c j b (y b ) ≥ c j a (y a ) + y j a c j a (y a )
. The first inequality is a consequence of the assumption on b, the second one is a consequence of the inequality x b ≥ y b , the third one is a consequence of Proposition 2. Since c j a and c j a are increasing, we get that x a > y a .

Thus any arc a ∈ supp(y j ) is such that x a > y a . Now, take such an arc α, i.e. such that x α > y α . Since the total demand is the same before and after the transfer, there must be an arc α = α such that x α < y α . According to Corollary 1, we have α ∈ supp(y j ). This is in contradiction with the fact that such an arc must satisfy x α > y α . 4.2. Proof of Proposition 4. We start with a technical lemma.

Lemma 2. Let d u , d v ∈ R K + and let u = e(d u ) and v = e(d v ) for k ∈ [K]. Let a and b be two arcs such that u a ≤ v a and u b ≥ v b . If k is a player such that u k a < v k a , then u k b = v k b = 0 or u k b < v k b . Proof. If u k b = 0, then u k b ≤ v k b , with equality if and only if u k b = v k b = 0. We can thus suppose that u k b > 0. Proposition 2 gives c k b (u b )+u k b c k b (u b ) ≤ c k a (u a )+u k a c k a (u a ). Since 0 ≤ u k a < v k a , Proposition 2 gives c k a (v a ) + v k a c k a (v a ) ≤ c k b (v b ) + v k b c k b (v b
). These two equations together with the facts that c k a and c k a are increasing, u a ≤ v a , and δ , there is at least one arc a such that x i a > y i a . Lemma 2 with u = y and v = x implies that y i b = x i b = 0 or y i b < x i b for all arcs b ∈ A. We get the desired conclusion for i. The proof of the conclusion for j is similar and omitted. Finally, in case |A| = 2, Lemma 2 with u = x and v = y together with the fact that d k = d k δ for k = i, j shows that x k a = y k a for both arcs a in A, leading to the desired inequality. We can thus assume for the remaining of the proof that x = y.

u k a < v k a give that c k b (u b ) + u k b c k b (u b ) < c k b (v b ) + v k b c k b (v b ).
We first deal with the case K = 2. Since x = y, there is an arc α ∈ A such that x α = y α . Without loss of generality x α < y α , and there is an arc β ∈ A such that x β > y β . We have x α < y α for a player ∈ {i, j} and x β > y β for a player ∈ {i, j} (there are only two players).

For every arc b such that x b ≥ y b , apply Lemma 2 with a = α, u = x, v = y, and k = . Summing these inequalities over all arcs b, we get that d ≤ d δ and d ≥ d δ , and we conclude that = j and = i.

We have proved that x i a = y i a = 0 or x i a > y i a for every arc a such that x a ≤ y a , and x i a ≥ y i a for every arc a such that x a ≥ y a . It remains to prove that x i a > y i a is strict for arcs a such that

x a > y a . Consider then such an arc a. Since x j a ≤ y j a and there are only two players, we have a strict inequality for player i too: x i a > y i a . A similar argument for player j leads to the desired conclusion.

Suppose that |A| = 2. Let α and β be the two arcs and suppose without loss of generality that x α < y α and x β > y β . Applying Lemma 2 with a = α, b = β, u = x, and v = y, we get that x k α ≥ y k α for all k = j (using the fact the d k ≥ d k δ for such a player k). Since x α < y α , we get that x j α < y j α . For a player k = i, j, we have d k = d k δ . Thus x k β ≤ y k β for such a player. Note that we already get the last statement of the proposition.

Applying again Lemma 2 with a = α, b = β, u = x, v = y, but this time for k = j, we get that x j β = y j β = 0 or x j β < y j β . This already gives the conclusion for player j. Since x β > y β , it implies that x i β > y i β . It remains to prove that the conclusion holds for player i. Let a be an arc such that x i a ≤ y i a . We necessarily have a = α. Applying Lemma 2, this times with a = β, b = α, u = y, and v = x allows then to conclude. 5. Social cost at equilibrium when players have same cost functions: proof of Theorem 3

5.1. Main steps of the proof. The proof of Theorem 3 relies on two results: Corollary 1 and the following proposition, proved in the remaining of the section. We define d δ as in Section 4

d i δ = d i -δ, d j δ = d j + δ, and d k δ = d k for k = i, j.
Throughout the section, the inequality d i ≤ d j is assumed.

Proposition 5. Suppose that we are under the condition of Theorem 3 and that there exists δ > 0 such that supp(e k (d)) = supp(e k (d δ )) for both players k = i and k = j. Then there exists η > 0

such that h → Q( e(d h )) is nonincreasing on [0, η].
If e(•) is differentiable, e.g. the cost functions are twice continuously differentiable (Theorem 1), then we can actually show that h → Q( e(d h )) is differentiable with a nonpositive derivative at 0.

The remaining of the section is devoted to the proof of this proposition. We finish this subsection by explaining how this proposition can be used to prove Theorem 3.

Proof of Theorem 3. Corollary 1 implies that when player i transfers a part of his demand to player j, the support of player i does not increase and the support of player j does not decrease. It implies that for k = i and k = j, the set supp(e k (d h )) changes a finite number of times when h goes from 0 to d i . Hence, for almost all h ∈ [0, d i ], there exists some δ > 0 (depending on h ) such that supp(e k (d h )) = supp(e k (d h +δ )) for both k = i and k = j. Thus, Proposition 5 implies that h → Q( e(d h )) is nonincreasing almost everywhere on [0,

d i ]. Since h → Q( e(d h )) is continuous according to Proposition 3, the map h → Q( e(d h )) is nonincreasing on the whole interval [0, d i ].
5.2. Technical lemmas. Throughout this subsection, we assume that we are no longer in the player-specific cost function setting, i.e. we suppose that c 1 a = • • • = c K a for all a ∈ A. The cost function attached to arc a is denoted c a without superscript. We emphasize that the result of this subsection do not necessarily hold when we assume the costs to be player-specific.

We have divided this subsection into two parts. The first one deals with results regarding the supports of the strategies. The second one deals with the derivate of the social cost at equilibrium seen as a function of the transfer.

5.2.1.

Lemmas about the supports. The following lemma has been proved by [START_REF] Orda | Competitive Routing in Multi-User Communication Networks[END_REF]. It is stated here without proof.

Lemma 3. Let x = e(d). If d k 1 ≤ d k 2 , then x k 1 a ≤ x k 2
a for every arc a. The next lemma deals with the case when the condition of Proposition 5 is satisfied and when moreover the supports of player i and j are identical. In this case, the social cost is not only nonincreasing, it is even constant.

Lemma 4. Suppose that there exists δ ∈ (0, d i ] such that supp(e i (d)) = supp(e j (d)) = supp(e i (d δ )) = supp(e j (d δ )).

Then there exists

η > 0 such that h → Q( e(d h )) is constant on [0, η].
Proof. Let x = e(d) and denote y(h) = e(d h ). Denote by S 0 be the common support: S 0 = supp(x i ) = supp(x j ). Let

π k = min a∈A c a (x a ) + x k a c a (x a ) .
It is the marginal cost at equilibrium for player k when the demand is d. Let η = min(δ 0 , δ 1 ), where

δ 0 = min a∈S 0 x i a c a (x a ) a∈S 0 1 c a (x a )
and

δ 1 = min b / ∈S 0 c b (x b ) -π j a∈S 0 1 c a (x a )
.

η is well defined since c a > 0 for every x > 0, and is nonnegative, according to Proposition 2. Moreover, Corollary 2 shows that η > 0.

Let h ≤ η. We consider the strategy vector z defined by

z k a = x k a + d k h -d k βc a (x a )
1 {a∈S 0 } for every player k where β = a∈S 0 1 c a (xa) . The remaining of the proof consists in showing that z = y(h). Since such a z satisfies z a = x a for all a (checking straightforward), it will show that x a = y a (h) for all a, and in particular that Q(x) = Q(y(h)).

We first check that z leads to feasible strategies. For each player k and arc a, we have z k a ≥ 0 since h ≤ δ 0 . We have moreover a∈A z k a = d k h for each player k. Hence, we have feasible strategies. We check now that z is an equilibrium for d h , by checking that it satisfies the condition of Proposition 2. Consider first a player k ∈ {i, j}. For every arc a ∈ S 0 , we have

c a (z a ) + z k a c a (z a ) = c a (x a ) + x k a c a (x a ) + d k h -d k β = π k + d k h -d k β
Consider now an arc a / ∈ S 0 . By definition of z k a , we have z k a = x k a = 0 (we still work with k ∈ {i, j}) and thus

c a (z a ) = c a (x a ) ≥ π j + h β ≥ π k + d k h -d k β ,
where the first equality holds since z a = x a , the first inequality since h ≤ δ 1 , and the last inequality for player i since π j ≥ π i , according to Lemma 3. The condition of Proposition 2 is satisfied for every player k ∈ {i, j}. Consider now a player k = {i, j}. We have

d k h = d k and thus z k a = x k a for all a ∈ A. Hence, we have c a (z a ) + z k a c a (z a ) = π k for a ∈ supp(z k ) and c a (z a ) ≥ π k for a /
∈ supp(z k ). Again, the Proposition 2 is satisfied, this time for the players k / ∈ {i, j}. Therefore, z is an equilibrium for the demand d h .

By uniqueness of the equilibrium, z = y(h).

The following lemma can be seen as a complement of Corollary 1.

Lemma 5. Suppose that |A| = 2 or K = 2. Let δ ∈ (0, d i ) and denote x = e(d), y = e(d δ ). Suppose that supp(x i ) = supp(y i ) = supp(x j ) = supp(y j ). If x = y, then {a ∈ A, y a < x a } = supp(x i ) and {a ∈ A, y a > x a } = supp(x j ) \ supp(x i ).

Proof. Assume that x = y, i.e. that there exists an arc a on which x a = y a (and thus there are at least two such arcs, since the total demand remains constant).

Suppose first that |A| = 2. Corollary 1 gives that {a ∈ A : y a < x a } ⊆ supp(x i ) and {a ∈ A : y a > x a } ⊆ supp(x j ). Since x = y, then there exists an arc α with y α < x α and an arc α such that y α > x α . Since supp(x i ) ⊆ supp(x j ) (Lemma 3), | supp(x j )| ≤ 2, and supp(x i ) = supp(x j ), we get the conclusion.

Suppose now that K = 2. Again, there exists an arc α with y α < x α . Corollary 1 implies that α ∈ supp(x i ), and thus according to the assumption, we also have α ∈ supp(y i ). Lemma 3 implies that supp(x i ) ⊆ supp(x j ) and supp(y i ) ⊆ supp(y j ). Proposition 2 implies thus that 2c a (x a ) + x a c a (x a ) = 2c α (x α ) + x α c α (x α ) for all a ∈ supp(x i ), by summing the two marginal costs for the two players i and j, and similarly that 2c a (y a ) + y a c a (y a ) = 2c α (y α ) + y α c α (y α ) for all a ∈ supp(y i ).

Since y α < x α and since u → 2c a (u) + uc a (u) is increasing, we get y a < x a for all a ∈ supp(x i ). Combining this with the inclusion {a ∈ A : y a < x a } ⊆ supp(x i ) given by Corollary 1, we get that {a ∈ A : y a < x a } = supp(x i ).

Take now an arc a in supp(x j ) \ supp(x i ) = supp(y j ) \ supp(y i ). It is necessarily such that y j a = y a ≥ x a = x j a . Suppose for a contradiction that y a = x a , then Proposition 4 gives x j a = y j a = 0, which contradicts the way a has been taken. Thus y a > x a . Conversely, take an arc a such that y a > x a . It is not in supp(y i ). Thus y j a = y a > 0, which implies that a ∈ supp(y j ). Since supp(y j ) = supp(x j ), we get the conclusion. 5.2.2. Lemmas about the derivate of the social cost. Given z = R A + and an arc a ∈ A, we define the vector z -a ∈ R A\{a} + by z -a = (z b ) b =a . Then for every arc a ∈ A, we consider the function Q a : R

A\{a} + → R + defined by Q a (z -a ) = a =a z b c b (z b ) +   d - b =a z b   c a   d - b =a z b   , where d = k∈[K] d k is the total demand. Let x ∈ R A + be such that a∈A x a = d. We have then Q a (x -a ) = Q( x) and for any b = a (2) ∂Q a ∂z b (x -a ) = c b (x b ) + x b c b (x b ) -c a (x a ) + x a c a (x a ) .
Denote y(h) = e(d h ).

Lemma 6. Let h ∈ (0, d i ]. Suppose that supp(y i (h)) = supp(y j (h)). Let a ∈ supp(y i (h)) and b ∈ supp(y j (h)) \ supp(y i (h)). We have then Suppose first that |A| = 2. If we are under the condition of Lemma 4, the conclusion is immediate. We can thus assume that supp(x i ) = supp(y i (δ)) = supp(x j ) = supp(y j (δ)). Since there are only two arcs, there is an arc a ∈ supp(x i ) and an arc b ∈ supp(x j ) \ supp(x i ). We have Q( y(h)) = Q a (y b (h)) (there are only two arcs in A). A first application of Corollary 1 shows that supp(y i (h)) ⊆ supp(x i ). A second application shows that supp(y i (δ)) ⊆ supp(y i (h)). Since it is assumed that supp(x i ) = supp(y i (δ)) for all h ∈ [0, δ], we get that supp(y i (h) = supp(x i ) for all h ∈ [0, δ]. Similarly, supp(y j (h) = supp(x j ) for all h ∈ [0, δ]

Lemma 5 shows that the total flow on b increases when h increases (b is in supp(y j (h)) \ supp(y i (h))). The map h → y b (h) is thus an increasing map on [0, δ]. According to Lemma 6, we have

∂Q a ∂z b (y b (h)) < 0. Thus Q a is decreasing on [y b (0), y b (δ)]. Therefore the map h → Q( y(h)) is decreasing on [0, δ].
Suppose now that K = 2 and that the cost functions are twice continuously differentiable. If we are under the condition of Lemma 4, the conclusion is immediate. We can thus assume that supp(x i ) = supp(y i (δ)) = supp(x j ) = supp(y j (δ)). Again, supp(y i (h)) and supp(y j (h)) are constant for all h ∈ [0, δ].

Choose h 0 , h ∈ (0, δ), with h = h 0 . Let a ∈ supp(y i (h 0 )) as in Lemma 7. Lemma 5 shows that y b (h) -y b (h 0 ) and h -h 0 have opposite signs when b ∈ supp(y i (h 0 )) \ {a} and same sign when b ∈ supp(y j (h 0 )) \ supp(y i (h 0 )). Thus,

b∈supp(y

i (h 0 ))\{a} ∂Q a ∂z b y -a (h 0 ) y b (h) -y b (h 0 ) h -h 0 ≤ 0 for all h ∈ (0, δ), and b∈supp(y j (h 0 ))\supp(y i (h 0 )) ∂Q a ∂z b y -a (h 0 ) y b (h) -y b (h 0 ) h -h 0 ≤ 0
for all h ∈ (0, δ) (with the help of Lemma 6). This sum over arcs b not in supp(y j (h 0 )) is zero since y b (h) = y b (h 0 ) = 0. Besides, the cost functions being differentiable and since there are only two players, we can apply Theorem 1: h → y(h) is differentiable in a neighborhood of h 0 . Using the above inequalities, we get that the derivative of h

→ Q( y(h)) is nonpositive at h 0 : b =a ∂Q a ∂z b (y -a (h 0 ))y b (h 0 ) = lim h→h 0 b =a ∂Q a ∂z b (y -a (h 0 )) y b (h) -y b (h 0 ) h -h 0 ≤ 0.
Since this is true for any h 0 ∈ (0, δ), we get the conclusion: h → Q( y(h)) is nonincreasing on [0, δ] (using the continuity of h → y(h) ensured by Proposition 3 to get the conclusion for the boundary of [0, δ]).

Discussion

6.1. e(•) is not differentiable everywhere. The application e(•) is not differentiable everywhere as shown by the following example, inspired from [START_REF] Hall | Properties of the equilibrium state in transportation networks[END_REF]. Consider a parallel-link graph with two arcs a and b and only one player with demand d. Suppose that the cost functions are c a (x) = x + 1 and c b (x) = x.

A direct calculation gives that at equilibrium x a = 0 and x b = d when 0 ≤ d ≤ 1 2 , and x a = 2d-1 4 , x b = 2d+1 4 when d ≥ 1 2 . Hence e is not differentiable at the point d = 1 2 . 6.2. When there are three arcs and three players. Theorems 3 is not valid when there are three arcs and three players. We introduce the example of [START_REF] Huang | Collusion in atomic splittable routing games[END_REF], for which the social cost at equilibrium increases after a transfer.

Consider a parallel-link graph with three arcs a, b, and c, and three players 1, 2, and 3. Suppose that the players have the same cost functions c a (x) = 20x + 5000, c b (x) = x 2 + 500, and c c (x) = x 11 .

When the vector of demand is d = (0.1, 20.9, 200), the (rounded) flows at equilibrium are in the following table and the equilibrium cost is 1 558 627. After the transfer of δ = 0.1 from player 1 to player 2, we have demands d δ = (0, 21, 200). The (rounded) flows at equilibrium are in the following table and the social cost at equilibrium is 1 558 633. In particular, the cost has increased after the transfer. Moreover, the part regarding players that keep the same demand in Proposition 4 (more general result than Theorem 2) does not hold either, since (y 3 a -x 3 a )(y a -x a ) > 0, where x (resp. y) is the equilibrium flow before (resp. after) the transfer. However Theorem 2 still holds: on each arc the flow of player 1 decreases or remains constant equal to zero, and the flow of player 2 increases or remains constant equal to zero. 6.3. When we allow player-specific cost functions. Another question is whether Theorem 3 remains valid when we allow player-specific cost functions. The answer is 'no', as shown by the following example.

Consider a parallel-link graph with two arcs a and b, and two players 1 and 2. Let the cost on arc b for player 1 (resp. on arc a for player 2) be prohibitively high, in such a way that at equilibrium, for every repartition of the demand, player 1 (resp. 2) puts all his demand on arc a (resp. b). Let the costs be c 1 a (x) = 2x, c 2 b (x) = x. If d 1 = 3 and d 2 = 2, the social cost at equilibrium is 8, while after a transfer of 1, i.e. if d 1 = 4 and d 2 = 1, the social cost at equilibrium is 9. The result of Theorem 3 does not hold if we allow player-specific costs.

  and d k δ = d k for k = i, j. Proposition 4. Suppose that |A| = 2 or K = 2, and let x = e(d) and y = e(d δ ). Let a ∈ A.

  Finally, since u b ≥ v b and since c k b and c k b are increasing, we have u k b < v k b . Proof of Proposition 4. Let us first suppose that x = y. Since d i > d i

  It gives that x b = y b = 0 or x b < y b . In particular when applied with b = β, we get = . Since x b ≥ y b , we have x b ≥ y b . For every arc b such that x b ≤ y b , apply Lemma 2 with a = β, u = y, v = x, and k = . It gives that x b = y b = 0 and y b < x b . Since x b ≤ y b , we have x b ≤ y b .

  ∂Q a ∂z b (y -a (h)) < 0. Proof. Let x = y(h). Since a ∈ supp(x i ), we have c b (x b ) ≥ c a (x a ) + x i a c a (x a ) > c a (x a ). Consider the set I b of players k such that b ∈ supp(x k ). The set I b is nonempty and for every player k ∈ I b we have, according to Proposition 2, c b(x b ) + x k b c b (x b ) ≤ c a (x a ) + x k a c a (x a ). By summing over these players, we get|I b |c b (x b ) + x b c b (x b ) ≤ |I b |c a (x a ) a (x a ) < |I b |c a (x a ) + x a c a (x a ).Since |I b | ≥ 1 and c b (x b ) > c a (x a ), we have c b (x b ) + x b c b (x b ) < c a (x a ) + x a c a (x a ), and thus, using Equation (2), we have ∂Q a ∂z b (x -a ) < 0. Lemma 7. Suppose that K = 2. Let h ∈ (0, d i ]. There always exists an arc a ∈ supp(y i (h)) such that ∂Q a ∂z b (y -a (h)) ≥ 0 for all b ∈ supp(y i (h)) \ {a}.Proof. Let x = y(h). Pick an arc a ∈ arg max{c α (x α ) : α ∈ supp(x i )}.Let then b ∈ supp(x i ) \ {a}. Lemma 3 implies that supp(x i ) ⊆ supp(x j ). Thus a and b are in supp(x j ). Summing the marginal costs for players i and j, we get 2c a (x a ) + x a c a (x a ) = 2c b (x b ) + x b c b (x b ). According to the definition of a, we have c a (x a ) ≥ c b (x b ) and then c a (x a ) + x a c a (x a ) ≤ c b (x b ) + x b c b (x b ). Thus, using Equation (2), we have ∂Q a ∂z b (x -a ) ≥ 0. 5.3. Proof of Proposition 5. Proof of Proposition 5. Let x = e(d) and denote y(h) = e(d h ).


Clément Huneau

Hugues Chabriat

Habib Benali

Potential of dynamic models to investigate neurovascular coupling using ASL fMRI

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Stéphane Girard

Gilles Stupfler

Intriguing properties of extreme geometric quantiles

Keywords: AMS Subject Classifications: 62H05, 62G20, 62G32 Extreme quantile, geometric quantile, consistency, asymptotic normality

Central properties of geometric quantiles have been well-established in the recent statistical literature. In this study, we try to get a grasp of how extreme geometric quantiles behave. Their asymptotics are provided, both in direction and magnitude, under suitable moment conditions, when the norm of the associated index vector tends to one. Some intriguing properties are highlighted: in particular, it appears that if a random vector has a finite covariance matrix, then the magnitude of its extreme geometric quantiles grows at a fixed rate. We take profit of these results by defining a parametric estimator of extreme geometric quantiles of such a random vector. The consistency and asymptotic normality of the estimator are established, and contrasted with what can be obtained for univariate quantiles. Our results are illustrated on both simulated and real data sets. As a conclusion, we deduce from our observations some warnings which we believe should be known by practitioners who would like to use such a notion of multivariate quantile to detect outliers or analyze extremes of a random vector.

Introduction

Let X be a random vector in R d . Up to now, several definitions of multivariate quantiles of X have been proposed in the statistical literature. We refer to [START_REF] Serfling | Quantile functions for multivariate analysis: approaches and applications[END_REF] for a review of various possibilities for this notion. Here, we focus on the notion of "spatial" or "geometric" quantiles, introduced by [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF], which generalises the characterisation of a univariate quantile shown in [START_REF] Koenker | Regression quantiles[END_REF]. For a given vector u belonging to the unit open ball B d of R d , where d ≥ 2, a geometric quantile with index vector u is any solution of the optimisation problem defined by arg min

q∈R d E( X -q -X ) -u, q , (1)
where •, • is the usual scalar product on R d and • is the associated Euclidean norm. Note that q(u) ∈ R d possesses both a direction and magnitude. It can be seen that geometric quantiles are in fact special cases of M -quantiles introduced by [START_REF] Breckling | M-quantiles[END_REF] which were further analysed by [START_REF] Koltchinskii | M-estimation, convexity and quantiles[END_REF]. Besides, such quantiles have various strong properties. First, the quantile with index vector u ∈ B d is unique whenever the distribution of X is not concentrated on a single straight line in R d (see [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF], or Theorem 2.17 in [START_REF] Kemperman | The median of a finite measure on a Banach space[END_REF]. Second, although they are not fully affine equivariant, they are equivariant under any orthogonal transformation [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF]. Third, geometric quantiles characterise the associated distribution. Namely, if two random variables X and Y yield the same quantile function q, then X and Y have the same distribution [START_REF] Koltchinskii | M-estimation, convexity and quantiles[END_REF]. Finally, for u = 0, the well-known L 2 -geometric median is obtained, which is the simplest example of a "central" quantile (see [START_REF] Small | A survey of multidimensional medians[END_REF]. We point out that one may compute an estimation of the geometric median in an efficient way, see [START_REF] Cardot | Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm[END_REF].

These properties make geometric quantiles reasonable candidates when trying to define multivariate quantiles, which is why their estimation was studied in several papers. We refer for instance to [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF], who established a Bahadur expansion for the estimator of geometric quantiles obtained by solving the sample counterpart of problem (1). [START_REF] Chakraborty | On affine equivariant multivariate quantiles[END_REF] then introduced a transformationretransformation procedure to obtain affine equivariant estimates of multivariate quantiles. This notion was extended to a multiresponse linear model by [START_REF] Chakraborty | On multivariate quantile regression[END_REF]. Recently, [START_REF] Dhar | Comparison of multivariate distributions using quantile-quantile plots and related tests[END_REF] defined a multivariate quantile-quantile plot using geometric quantiles. Conditional geometric quantiles can also be defined by substituting a conditional expectation to the expectation in (1). We refer to [START_REF] Cadre | Asymptotic normality of consistent estimate of the conditional L 1 -median[END_REF] for the estimation of the conditional geometric median and to [START_REF] Cheng | On the u-th geometric conditional quantile[END_REF] for the estimation of an arbitrary conditional geometric quantile. The estimation of a conditional median when there is an infinite-dimensional covariate is considered in [START_REF] Chaouch | Nonparametric multivariate L 1 -median regression estimation with functional covariates[END_REF].

Let us note though that the previous papers focus on central properties of geometric quantiles and of their sample versions. While some of them label geometric quantiles as "extreme" when u is close to 1 (Chaudhuri, 1996, Cheng and[START_REF] Cheng | On the u-th geometric conditional quantile[END_REF] and use it in real applications (see e.g. [START_REF] Chaouch | Design-based estimation for geometric quantiles with application to outlier detection[END_REF] for an application to outlier detection), the specific properties of these extreme geometric quantiles have not been investigated yet. In this study, we provide the asymptotics of the direction and magnitude of the extreme geometric quantile q(u) when u → 1, under suitable moment conditions.

There are well-known analogue results for univariate extreme quantiles in the right tail of a distribution, see e.g. de [START_REF] De Haan | Extreme value theory: an introduction[END_REF]. A particular corollary of our results is that the magnitude of the extreme geometric quantiles of a random vector X having a finite covariance matrix grows at a fixed rate.

Moreover, in this case, the magnitude of the extreme geometric quantiles is asymptotically characterised by the covariance matrix of X. This is an intriguing property, which opens the door to a parametric estimation of extreme quantiles whose asymptotic properties are studied in this work.

The outline of the paper is as follows. Asymptotic properties of geometric quantiles are stated in Section 2.

An illustrative application to the estimation of extreme geometric quantiles is given in Section 3. Some examples and numerical illustrations of our results, including a study of a real data set, are presented in Section 4. Section 5 offers a couple of concluding remarks, in which some warnings are given to practitioners who would like to use such a geometric quantiles to detect outliers or analyze extremes of a random vector. Proofs are deferred to Section 6.

Asymptotic behaviour of extreme geometric quantiles

From now on, we assume that the distribution of X is not concentrated on a single straight line in R d and non-atomic. [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF] proved that, in this context, the solution q(u) of ( 1), namely the geometric quantile with index vector u, exists and is unique for every u ∈ B d . Let ψ : R d × R d → R be defined as

ψ(u, q) = E( X -q -X ) -u, q and assume further that t/ t = 0 if t = 0. If u ∈ R d is such that
there is a solution q(u) ∈ R d to problem (1), then the gradient of q → ψ(u, q) must be zero at q(u), that is

u + E X -q(u) X -q(u) = 0. ( 2
)
This condition immediately entails that if u ∈ R d is such that problem (1) has a solution q(u), then u ≤ 1. In fact, we can prove a stronger result:

Proposition 1. The optimisation problem (1) has a solution if and only if u ∈ B d .

Moreover, remarking that the function ψ(u, •) is strictly convex, [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF] proved the following characterisation of a geometric quantile: for every u ∈ B d , q(u) is the solution of problem (1) if and only if it satisfies equation (2). In particular, this entails that the function G : R d → B d defined by

∀q ∈ R d , G(q) = -E X -q X -q
is a continuous bijection. Proposition 2.6(iii) in [START_REF] Koltchinskii | M-estimation, convexity and quantiles[END_REF] shows that the inverse of the function G, i.e. the geometric quantile function u → q(u), is also continuous on B d .

In most cases however, computing explicitly the function G is a hopeless task, which makes it impossible to obtain a closed-form expression for the geometric quantile function. It is thus of interest to prove general results about the geometric quantile q(u), especially regarding its direction and magnitude. Our first main result focuses on the special case of spherically symmetric distributions.

Proposition 2. If X has a spherically symmetric distribution then:

(i) The map u → q(u) commutes with every linear isometry of R d . Especially, the norm of a geometric quantile q(u) only depends on the norm of u.

(ii) For all u ∈ B d , the geometric quantile q(u) has direction u if u = 0 and q(0) = 0 otherwise.

(iii) The function u → q(u) is a continuous strictly increasing function on [0, 1).

(iv) It holds that q(u) → ∞ as u → 1.

Although Proposition 2(i,iii) cannot be expected to hold true for a random variable which is not spherically symmetric, one may wonder if (ii,iv), namely that a geometric quantile shares the direction of its index vector and that the norm of the geometric quantile function tends to infinity on the unit sphere, can be extended to the general case. The next result, which examines the behaviour of the geometric quantile function near the boundary of the open ball B d , provides an answer to this question.

Theorem 1. Let S d-1 be the unit sphere of R d .

(i) It holds that q(v) → ∞ as v → 1.

(ii) Moreover, if v → u with u ∈ S d-1 and v ∈ B d then q(v)/ q(v) → u.
Theorem 1 shows two properties of geometric quantiles: first, the norm of the geometric quantile q(v)

with index vector v diverges to infinity as v ↑ 1. In other words, Proposition 2(iv) still holds for any distribution. This is a rather intriguing property of geometric quantiles, since it holds even if the distribution of X has a compact support (for instance, when X is uniformly distributed on a square).

A related point is the fact that sample geometric quantiles do not necessarily lie within the convex hull of the sample, see [START_REF] Breckling | A note on multivariate M -quantiles[END_REF] for a counter-example. Second, if v → u ∈ S d-1 , then the geometric quantile q(v) has asymptotic direction u. Proposition 2(ii) thus remains true asymptotically for any distribution.

It is possible to specify the convergences obtained in Theorem 1 under moment assumptions. Theorem 2 provides a first-order expansion of both the direction and the magnitude of an extreme geometric quantile q(αu) in the direction u, where u is a unit vector and α tends to 1.

Theorem 2. Let u ∈ S d-1 . (i) If E X < ∞ then q(αu) -{ q(αu) u + E(X -X, u u)} → 0 as α ↑ 1. (ii) If E X 2 < ∞ and Σ denotes the covariance matrix of X then q(αu) 2 (1 -α) → 1 2 (tr Σ -u ′ Σu) > 0 as α ↑ 1.
Let us note that the integrability conditions of Theorem 2 exclude any random vector X whose distribution possesses a right tail which is too heavy. For instance, condition E X < ∞ in (i) excludes for instance the multivariate Student distribution with less than one degree of freedom, while condition E X 2 < ∞ in (ii) excludes the multivariate Student distribution with less than two degrees of freedom.

Consequence 1. It appears that, if X has a finite covariance matrix Σ, then the magnitude of an extreme geometric quantile is determined (in the asymptotic sense) by Σ. In other words, since the asymptotic direction of an extreme geometric quantile in the direction u is exactly u by Theorem 1, it follows that the extreme geometric quantiles of two probability distributions which admit the same finite covariance matrix are asymptotically equivalent. This phenomenon is illustrated on simulated data in Section 4 below. This is surprising from the extreme value perspective: one could expect the behaviour of extreme geometric quantiles not to be driven by a central parameter such as the covariance matrix, as happens in the univariate context where the value of an extreme quantile depends on the tail heaviness of the probability density function of X.

Consequence 2. The map λ → q((1λ -1 )u) is a regularly varying function with index 1/2 (see [START_REF] Bingham | Regular Variation[END_REF] and therefore:

q(βu) q(αu) = 1 -α 1 -β 1/2 (1 + o(1))
when α → 1 and β → 1. In other words, given an arbitrary extreme geometric quantile, one can deduce the asymptotic behaviour of every other extreme geometric quantile sharing its direction, independently of the distribution. Again, this is fundamentally different from the univariate case when deducing the value of an extreme quantile from another one then requires the knowledge (or an estimate) of the extreme-value index of the distribution, see de [START_REF] De Haan | Extreme value theory: an introduction[END_REF], Chapter 4. A further, perhaps unexpected, consequence is that our results can actually be used to define a consistent and asymptotically Gaussian estimator of extreme geometric quantiles by using the standard empirical estimator of the covariance matrix of X, see Section 3 below.

Consequence 3. Finally, Theorem 2 provides some information on the shape of an extreme quantile contour. It is readily seen that the global maximum of the function h 1 (u) := tr Σu ′ Σu on S d-1 is reached at a unit eigenvector u min of Σ associated with its smallest eigenvalue λ min > 0. Thus, the norm of an extreme geometric quantile is asymptotically the largest in the direction where the variance is the smallest. Similarly, the global minimum of h 1 is reached at a unit eigenvector u max of Σ associated with its largest eigenvalue λ max > 0. In particular, if f is the probability density function associated with an elliptically contoured distribution [START_REF] Cambanis | On the theory of elliptically contoured distributions[END_REF], the level sets of f coincide with the level sets of the function h 2 (u) := u ′ Σu. The global maximum of h 2 is reached at the eigenvector u max while the global minimum is reached at u min . The extreme geometric quantile is therefore furthest from the origin in the direction where the density level set is closest to the origin, see Section 4 for an illustration on real data. In such a case, the extreme geometric quantile contour plot and the density level plots are in some sense orthogonal (even though they agree when the distribution of X is spherically symmetric). Of course, one should not expect a direct geometric match between quantile contours and density contours, but this phenomenon should be kept in mind when designing outlier detection procedures. In our view, this can be seen as a consequence of the lack of affine-equivariance of geometric quantiles. To tackle this issue, one may apply a transformation-retransformation procedure, see [START_REF] Serfling | Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardization[END_REF]. Such procedures admit sample analogues, see for instance [START_REF] Chakraborty | Operating transformation retransformation on spatial median and angle test[END_REF] and [START_REF] Chakraborty | On affine equivariant multivariate quantiles[END_REF], at the possible loss of geometric interpretation, see Serfling (2004).

An estimator of extreme geometric quantiles

In this paragraph, our focus is to illustrate Consequence 2 of Theorem 2 at the sample level. Let X 1 , . . . , X n be independent random copies of a random vector X having a finite covariance matrix Σ. It follows from

Theorem 2 that any extreme geometric quantile q(αu) of X, with α ↑ 1 and u ∈ S d-1 can be approximated by:

q eq (αu) := (1 -α) -1/2 1 2 (tr Σ -u ′ Σu) 1/2 u. (3)
This can be used to define an estimator of the extreme geometric quantiles of X: let X n = n -1 n k=1 X k be the sample mean and

Σ n = 1 n n k=1 (X k -X n )(X k -X n ) ′
be the empirical estimator of the covariance matrix Σ of X. Let further (α n ) be an increasing sequence of positive real numbers tending to 1. Our estimator q n (α n u) of q(α n u) is then

q n (α n u) = (1 -α n ) -1/2 1 2 tr Σ n -u ′ Σ n u 1/2 u.
The consistency of q n (α n u) is examined in the next result.

Theorem 3. Let u ∈ S d-1 and assume that α n ↑ 1. If E X 2 < ∞ then √ 1 -α n ( q n (α n u) -q(α n u)) → 0 almost surely as n → ∞.
This result actually means that the extreme geometric quantile estimator is relatively consistent in the sense that

q n (α n u) -q(α n u) q(α n u) → 0 almost surely as n → ∞, since q(α n u) -1 = O( √ 1 -α n ), see Theorem 2(ii)
. This normalisation could be expected since the quantity to be estimated diverges in magnitude. Under the additional assumption that X has a finite fourth moment, an asymptotic normality result can be established for this estimator:

Theorem 4. Let u ∈ S d-1 and assume that α n ↑ 1 is such that n(1 -α n ) → 0. If E X 4 < ∞ then n(1 -α n ) ( q n (α n u) -q(α n u)) d -→ Z as n → ∞
where Z is a Gaussian centred random vector.

covariance matrix M of the Gaussian limit of √ n( Σ n -Σ), see the proof in Section 6. Although M has a complicated expression (see e.g. [START_REF] Neudecker | The asymptotic variance matrix of the sample correlation matrix[END_REF], it can be estimated when E X 4 < ∞, which makes it possible to construct asymptotic confidence regions for extreme geometric quantiles.

Extreme geometric quantiles can thus be consistently estimated by q n (α n u), whatever the "order" α n , and an asymptotic normality result is obtained when α n ↑ 1 quickly enough. The proposed estimator is therefore able to extrapolate arbitrarily far from the original sample. This is very different from the univariate case, where the empirical quantile

q n (α n ) = inf{t ∈ R | F n (t) ≥ α n }, deduced from the empirical
cumulative distribution function F n , estimates the true quantile q(α n ) consistently only if α n converges to 1 slowly enough. The extrapolation with faster rates α n is then handled assuming that the underlying distribution function is heavy-tailed and by using adapted estimators, see e.g. [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF] and the monograph by de [START_REF] De Haan | Extreme value theory: an introduction[END_REF].

4 Numerical illustrations

Simulation study

In this section, our main results are illustrated, particularly Theorems 2, 3 and 4 in the bivariate case d = 2 to make the display easier. In this framework, u ∈ S 1 can be represented by an angle: u =

u θ = (cos θ, sin θ), θ ∈ [0, 2π). The iso-quantile curves Cq(α) = {q(αu θ ), θ ∈ [0, 2π)} and their estimates C q n (α) = { q n (αu θ ), θ ∈ [0, 2π
)} can then be considered in order to get a grasp of the behaviour of extreme quantiles in every direction. The following two distributions are considered for the random vector X:

• the centred Gaussian multivariate distribution N (0, v X , v Y , v XY ), with probability density function:

∀x, y ∈ R, f (x, y) = 1 2π √ det Σ exp   - 1 2   x y   ′ Σ -1   x y     with Σ =   v X v XY v XY v Y   .
• a double exponential distribution E(λ -, µ -, λ + , µ + ), with λ -, µ -, λ + , µ + > 0, whose probability density function is:

∀x, y ∈ R, f (x, y) = 1 4          λ + µ + e -λ+|x|-µ+|y| if xy > 0, λ -µ -e -λ-|x|-µ-|y| if xy ≤ 0.
In this case, X is centred and has covariance matrix

Σ =      1 λ 2 - + 1 λ 2 + 1 2 1 λ + µ + - 1 λ -µ - 1 2 1 λ + µ + - 1 λ -µ - 1 µ 2 - + 1 µ 2 +      .
Three different sets of parameters were used for each distribution, in order that the related covariance matrices coincide:

• N (0, 1/2, 1/2, 0) and E(2, 2, 2, 2) with spherical covariance matrices;

• N (0, 1/8, 3/4, 0) and E(4, 2 2/3, 4, 2 2/3) with diagonal covariance matrices;

• N (0, 1/2, 1/2, 1/6) and E(2 √ 3, 2 √ 3, 2 3/5, 2 3/5) with full covariance matrices.

In each case, we carry out the following computations:

• for each α ∈ {0.99, 0.995, 0.999}, the true quantile curves Cq(α) obtained by solving problem (1) numerically, as well as their analogues Cq eq (α) using approximation (3) are computed. The normalised squared approximation error

e(α) = (1 -α) 2π 0 q eq (αu θ ) -q(αu θ ) 2 dθ
is then recorded.

• for each value of α, we draw N = 1000 replications of an n-sample (X 1 , . . . , X n ) of independent copies of X, with n ∈ {100, 200, 500}. The estimated quantile curves C q (j) n (α) corresponding to the j-th replication and the associated normalised squared error

E (j) n (α) = (1 -α) 2π 0 q (j) n (αu θ ) -q(αu θ ) 2 dθ
are computed as well as the mean squared error

E n (α) = N -1 N j=1 E (j) n (α).
The true quantile curves, as well as the approximated and the estimated ones are displayed on Figures 123456in the case n = 200 and α = 0.995. The true quantile curves look very similar in Figures 1 and4, in Figures 2 and5 and Figures 3 and6 (in which the words "best", "median" and "worst" are to be understood with respect to the L 2 error). This is in accordance with Theorem 2: eventually, extreme geometric quantiles only depend on the covariance matrix of the underlying distribution. Moreover, the approximated quantiles curves are close to the true ones in all cases, and the estimated quantile curves are satisfying in all situations with a moderate variability. Similar results were observed for n = 100, 500 and α = 0.99, 0.999. We do not report the graphs here for the sake of brevity; we do however display the approximation and estimation errors in Table 1. Unsurprisingly, the estimation error E n (α) decreases as the sample size n increases. Both approximation and estimation errors e(α) and E n (α) have a stable behaviour with respect to α.

Real data illustration

The finite sample behaviour of extreme geometric quantiles is illustrated on a two-dimensional dataset extracted from the Pima Indians Diabetes Database. This data set, which is available at the webpage ftp.ics.uci.edu/pub/machine-learning-databases/pima-indians-diabetes, was already considered by [START_REF] Cheng | On the u-th geometric conditional quantile[END_REF] and [START_REF] Chaouch | Design-based estimation for geometric quantiles with application to outlier detection[END_REF], among others. In the latter study, geometric iso-quantile curves with a high α are used to detect outliers in the data set. Using extreme quantiles for outlier detection was advocated in e.g. [START_REF] Burridge | Additive outlier detection via extreme-value theory[END_REF]Robert Taylor (2006), Huberta et al. (2012) in the univariate case and He and Einmahl (2014) using depth-based quantile regions in the multivariate case; see also the monograph by [START_REF] Aggarwal | Outlier analysis[END_REF].

After working on the data set so as to eliminate missing values, the data set consists of n = 392 pairs

(X i , Y i )
, where X i is the body mass index (BMI) of the ith individual and Y i is its diastolic blood pressure.

The centered data cloud is represented in Figure 7 with blue crosses, along with the geometric iso-quantile curve with α = 0.95. While geometric quantiles with a moderate α tend to give a fair idea of the shape of the data cloud (see e.g. [START_REF] Chaouch | Design-based estimation for geometric quantiles with application to outlier detection[END_REF], the same cannot be said for extreme geometric quantiles on this example. This is an illustration of the phenomenon described in Consequence 3 in Section 2: the norm of an extreme geometric quantile is the largest in the direction where the variance is the smallest. We are thus led to think that here, outlier detection would be dangerous without a preliminary transformation-retransformation procedure [START_REF] Chakraborty | On affine equivariant multivariate quantiles[END_REF].

Concluding remarks

In this paper, we established the asymptotics of extreme geometric quantiles. A particular consequence of our results is that, if the underlying distribution possesses a finite covariance matrix Σ, then an extreme geometric quantile may be estimated accurately, no matter how extreme it is, with the help of the standard empirical estimator of Σ. This result is supported by our numerical study. The situation is very different from the univariate case, in which the asymptotic decay of a survival function can be linked to the asymptotic behaviour of an extreme quantile.

An additional issue, illustrated on a real data set, is that although central geometric quantile contours may roughly match the shape of the data cloud, this does not necessarily stay true for extreme iso-quantile curves. This is why we would advise practitioners to be cautious when using such a notion of multivariate quantile to detect outliers or analyze the extremes of a random vector. We believe that one can tackle this problem by applying a transformation-retransformation procedure, see [START_REF] Serfling | Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardization[END_REF] at the population level, and [START_REF] Chakraborty | Operating transformation retransformation on spatial median and angle test[END_REF] and [START_REF] Chakraborty | On affine equivariant multivariate quantiles[END_REF] at the sample level. Future work on extreme geometric quantiles thus includes building and studying their analogues for transformed-retransformed data.

Finally, let us underline again that this work was carried out under moment conditions such as the existence of finite first and second-order moments for X . It would definitely be interesting to see if our conclusions carry over, to some extent, to the case when these assumptions are violated.

Proofs

Some preliminary results are collected in Paragraph 6.1, their proofs are postponed to Paragraph 6.3. The proofs of the main results are provided in Paragraph 6.2.

Preliminary results

The first lemma provides some technical tools necessary to show Theorem 2(ii).

Lemma 1. Let ϕ : R d × R + × S d-1 → R be the function defined by

ϕ(x, r, v) = r 2 1 + x -rv, v x -rv . Then, for all v ∈ S d-1 , ϕ(•, •, v) is nonnegative and ∀x ∈ R d , ∀r ≤ x , ϕ(x, r, v) ≤ 2r 2 and ∀r > x , ϕ(x, r, v) ≤ x 2 .
In particular, ϕ(x, r, v) ≤ 2 x 2 for every

(x, r, v) ∈ R d × R + × S d-1 .
The next lemma is the first step to prove Theorem 2(i).

Lemma 2. Let u ∈ S d-1 . If E X < ∞ then, for all v ∈ R d , q(αu) αu - q(αu) q(αu) , v → -E X -X, u u, v as α ↑ 1.
Lemma 3 below is a result which is similar to Lemma 2.

Lemma 3. Let u ∈ S d-1 . If E X 2 < ∞ then q(αu) 2 αu - q(αu) q(αu) , q(αu) q(αu) → - 1 2 E X -X, u u 2 as α ↑ 1.
Lemma 4 is the first step to prove Theorem 4. It is essentially a refinement of Lemma 2.

Lemma 4. Let u ∈ S d-1 . If E X 2 < ∞ then, for all v ∈ R d , q(αu) q(αu) αu - q(αu) q(αu) , v + E X -X, u u, v → u, v Var X, u - 1 2 u, v E X -X, u u 2 + u, v E(X -X, u u) 2 -Cov( X, u , X, v ) as α ↑ 1.
Lemma 5 below is a refinement of Lemma 3. It is the second step to prove Theorem 4.

Lemma 5. Let u ∈ S d-1 . If E X 3 < ∞ then q(αu) q(αu) 2 αu - q(αu) q(αu) , q(αu) q(αu) + 1 2 E X -X, u u 2 → E X, u X, E(X -X, u u) -X -X, u u 2 as α ↑ 1.

Proofs of the main results

Proof of Proposition 1. From [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF], it is known that if u ∈ B d then problem (1) has a unique solution q(u) ∈ R d . To prove the converse part of this result, use equation ( 2) to get

E X -q(u) X -q(u) = u .
Introduce the coordinate representations X = (X 1 , . . . , X d ) and q(u) = (q 1 (u), . . . , q d (u)). The Cauchy-Schwarz inequality yields

u 2 = E X -q(u) X -q(u) 2 = d i=1 E X i -q i (u) X -q(u) 2 ≤ d i=1 E (X i -q i (u)) 2 X -q(u) 2 = 1.
Furthermore, equality holds if and only if for all i ∈ {1, . . . , d}, there exists µ i ∈ R such that

X i -q i (u) X -q(u) = µ i
almost surely. In particular, if w = (µ 1 , . . . , µ d ), this entails X ∈ D = q(u) + Rw almost surely, which cannot hold since the distribution of X is not concentrated in a single straight line in R d . It follows that necessarily u 2 < 1, which is the result.

Proof of Proposition 2. (i) Note that (2) implies that, for any linear isometry h of R d and every u ∈ B d ,

h(u) + E h(X) -h • q(u) X -q(u) = 0.
Since h is a linear isometry, the random vectors X and h(X) have the same distribution and the equality Xq(u) = h(X)h • q(u) holds almost surely. It follows that

h(u) + E X -h • q(u) X -h • q(u) = 0.
Since h(u) ∈ B d , it follows that h • q(u) = q • h(u), which completes the proof of the first statement.

(ii) To prove the second part of Proposition 2, start by noting that since X and -X have the same distribution, it holds that E (X/ X ) = 0. The case u = 0 is then obtained via (2). If u = 0, up to using the first part of the result with a suitable linear isometry, we shall assume without loss of generality that u = (u 1 , 0, . . . , 0) for some constant u 1 ∈ (0, 1). It is then enough to prove that there exists some constant q 1 (u) > 0 such that q(u) = (q 1 (u), 0, . . . , 0). To this end, let us remark that, on the one hand, if v 1 ∈ R and w = (1, 0, . . . , 0) then

∀j ∈ {2, . . . , d}, E X j X -v 1 w = 0, (4)
since, for every j ∈ {2, . . . , d}, the random vectors X and (X 1 , . . . , X j-1 , -X j , X j+1 , . . . , X d ) have the same distribution. On the other hand, the dominated convergence theorem entails that the function

v 1 → E X 1 -v 1 X -v 1 w
is continuous, converges to 1 at -∞, is equal to 0 at 0 and converges to -1 at +∞. Thus, the intermediate value theorem yields that there exists some constant q 1 (u) > 0 such that

u 1 + E X 1 -q 1 (u) X -q 1 (u)w = 0. (5)
Consequently, collecting (4) and ( 5) yields

u + E X -q 1 (u)w X -q 1 (u)w = 0
and it only remains to apply (2) to finish the proof of the second statement.

(iii) To show the third statement, use the first result to obtain that the function g : u → q(u) is indeed well-defined; since the geometric quantile function is continuous, so is g. Assume that g is not strictly increasing: namely, there exist u 1 , u 2 ∈ B d such that u 1 < u 2 and q(u 1 ) ≥ q(u 2 ) . Since q(0) = 0, it is a consequence of the intermediate value theorem that one may find u, v ∈ B d such that u < v and q(u) = q(v) . Let h be an isometry such that h(u/ u ) = h(v/ v ); then

q(h(u)) = q(u) = q(v) = q(h(v)) and q(h(u)) q(h(u)) = h(u) h(u) = h(v) h(v) = q(h(v)) q(h(v)) .
In other words, q(h(u)) and q(h(v)) have the same direction and magnitude, so that they are necessarily equal, which entails that h(u) = h(v) because the geometric quantile function is one-to-one. This is a

contradiction because h(u) = u < v = h(v)
, and the third statement is proven.

(iv) Assume that q(u) does not tend to infinity as u → 1; since g is increasing, it tends to a finite positive limit r. In other words, q(u) ≤ r for every u ∈ B d , which is a contradiction since the geometric quantile function maps B d onto R d , and the proof is complete.

Proof of Theorem 1. (i) If the first statement were false, then one could find a sequence (v n ) contained in B d such that v n → 1 and such that ( q(v n ) ) does not tend to infinity. Up to extracting a subsequence, one can assume that ( q(v n ) ) is bounded. Again, up to extraction, one can assume that (v n ) converges to some v ∞ ∈ S d-1 and that (q(v n )) converges to some q ∞ ∈ R d . Moreover, it is straightforward to show that for every

u 1 , u 2 , q 1 , q 2 ∈ R d |ψ(u 1 , q 1 ) -ψ(u 2 , q 2 )| ≤ {1 + u 2 } q 2 -q 1 + q 1 u 2 -u 1 so that the function ψ is continuous on R d × R d . Recall then that the definition of q(v n ) implies that for every q ∈ R d , ψ(v n , q(v n )) ≤ ψ(v n , q
) and let n tend to infinity to obtain

q ∞ = arg min q∈R d ψ(v ∞ , q).
Because v ∈ S d-1 , this contradicts Proposition 1, and the proof of the first statement is complete:

q(v) → ∞ as v → 1.
(ii) Pick a sequence (v n ) of elements of B d converging to u and remark that from (2),

v n + E X -q(v n ) X -q(v n ) = 0
for every integer n. Hence, for n large enough, the following equality holds:

v n + E X q(v n ) - q(v n ) q(v n ) -1 X q(v n ) - q(v n ) q(v n ) = 0. ( 6
)
Since the sequence (q(v n )/ q(v n ) ) is bounded it is enough to show that its only accumulation point is u. Let then u * be an accumulation point of this sequence. Since q(v n ) → ∞, we may let n → ∞ in ( 6)

and use the dominated convergence theorem to obtain uu * = 0, which completes the proof.

Proof of Theorem 2. (i) Let (u, w 1 , . . . , w d-1 ) be an orthonormal basis of R d and consider the following expansion:

q(αu) q(αu) = b(α)u + d-1 k=1 β k (α)w k (7)
where b(α), β 1 (α), . . . , β d-1 (α) are real numbers. It immediately follows that

q(αu) q(αu) -u - 1 q(αu) {E(X) -E(X), u u} = (b(α) -1)u + d-1 k=1 q(αu) β k (α) -E X, w k q(αu) w k . (8) Lemma 2 implies that q(αu) αu - q(αu) q(αu) , w k = -q(αu) β k (α) → -E X, w k as α ↑ 1 (9)
for all k ∈ {1, . . . , d -1}. Besides, let us note that q(αu)/ q(αu)

∈ S d-1 entails b 2 (α) + d-1 k=1 β 2 k (α) = 1. ( 10
)
Theorem 1 shows that b(α) → 1 as α ↑ 1 and thus (9) yields:

q(αu) (1 -b(α)) = 1 2 q(αu) (1 -b 2 (α))(1 + o(1)) = 1 2 q(αu) d-1 k=1 β 2 k (α)(1 + o(1)) → 0 as α ↑ 1. (11)
Collecting ( 8), ( 9) and ( 11), we obtain

q(αu) q(αu) -u - 1 q(αu) {E(X) -E(X), u u} = o 1 q(αu) as α ↑ 1
which is the first result.

(ii) Recall ( 7) and use Lemma 2 to obtain q(αu) αu -q(αu) q(αu) , w k → -E X, w k as α ↑ 1, for all k ∈ {1, . . . , d -1}, leading to

q(αu) 2 β 2 k (α) → [E X, w k ] 2 as α ↑ 1 (12)
for all k ∈ {1, . . . , d -1}. Recall (10) and use Lemma 3 to get

q(αu) 2 [αb(α) -1] → - 1 2 E X -X, u u 2 as α ↑ 1. (13)
Since (u, w 1 , . . . , w d-1 ) is an orthonormal basis of R d , one has the identity

X -X, u u 2 = d-1 k=1 X, w k 2 . (14)
Collecting ( 12), ( 13) and ( 14) leads to

q(αu) 2 1 -αb(α) - 1 2 d-1 k=1 β 2 k (α) → 1 2 d-1 k=1
Var X, w k as α ↑ 1.

Therefore,

q(αu) 2 1 -αb(α) - 1 2 1 -b 2 (α) → 1 2 d-1 k=1 Var X, w k as α ↑ 1, (15)
and easy calculations show that

1 -αb(α) - 1 2 1 -b 2 (α) = 1 2 (1 -α)(1 + α) + (α -b(α)) 2 . ( 16
)
Finally, in view of Lemma 2, q(αu) αu -q(αu) q(αu) , u → 0 as α ↑ 1 which is equivalent to

q(αu) 2 (α -b(α)) 2 → 0 as α ↑ 1. (17)
Collecting ( 15), ( 16) and ( 17), we obtain

q(αu) 2 (1 -α) → 1 2 d-1 k=1
Var X, w k as α ↑ 1.

Remarking that, for every orthonormal basis (e 1 , . . . , e d ) of R d ,

d k=1 Var X, e k = d k=1 e ′ k Σe k = tr Σ (18) proves that q(αu) 2 (1 -α) → 1 2 (tr Σ -u ′ Σu) ≥ 0 as α ↑ 1.
Finally, note that if we had tr Σu ′ Σu = 0 then by ( 18) we would have that Var X, w k = 0 for all k ∈ {1, . . . , d -1}. Thus the projection of X onto the orthogonal complement of Ru would be almost surely constant and X would be contained in a single straight line in R d , which is a contradiction. This completes the proof of Theorem 2.

Proof of Theorem 3. Note that

√ 1 -α n q n (α n u) → 1 2 (tr Σ -u ′ Σu) 1/2 u (19)
almost surely as n → ∞. Moreover, by Theorems 1 and 2

√ 1 -α n q(α n u) = √ 1 -α n q(α n u) q(α n u) q(α n u) → 1 2 (tr Σ -u ′ Σu) 1/2 u (20)
almost surely as n → ∞. Combining ( 19) and ( 20) completes the proof.

Proof of Theorem 4. Consider the following representation:

n(1 -α n ) ( q n (α n u) -q(α n u)) = T 1,n + T 2,n + T 3,n with T 1,n = √ n 1 2 {tr Σ n -u ′ Σ n u} 1/2 - 1 2 {tr Σ -u ′ Σu} 1/2 q(α n u) q(α n u) , T 2,n = √ n 1 2 {tr Σ -u ′ Σu} 1/2 - √ 1 -α n q(α n u) q(α n u) q(α n u)
and

T 3,n = -n(1 -α n ) q n (α n u) q(α n u) q(α n u) -u .
We start by examining the convergence of T 1,n . Observe first that

T 1,n = √ n 1 √ 2 {tr Σ n -u ′ Σ n u} -{tr Σ -u ′ Σu} {tr Σ n -u ′ Σ n u} 1/2 + {tr Σ -u ′ Σu} 1/2 q(α n u) q(α n u) = √ n {tr Σ n -u ′ Σ n u} -{tr Σ -u ′ Σu} 2 √ 2{tr Σ -u ′ Σu} 1/2 u(1 + o P (1)) as n → ∞
in view of Theorem 1(i) and from the consistency of Σ n . Denote by M the Gaussian centred limit of √ n( Σ n -Σ) (see e.g. [START_REF] Neudecker | The asymptotic variance matrix of the sample correlation matrix[END_REF]. Since the map

A → tr A -u ′ Au is linear, it follows that √ n {tr Σ n -u ′ Σ n u} -{tr Σ -u ′ Σu} 2 √ 2{tr Σ -u ′ Σu} 1/2 d -→ Y as n → ∞
where Y is a centred Gaussian random variable. Now, clearly Z := Y u is a Gaussian centred random vector and we have

T 1,n d -→ Z as n → ∞. (21)
The sequence T 2,n is controlled in the following way: using Lemmas 4 and 5 and following the steps of the proof of Theorem 2(ii), we obtain

q(α n u) 2 (1 -α n ) = 1 2 (tr Σ -u ′ Σu) + O( q(α n u) -1 ) = 1 2 (tr Σ -u ′ Σu) + O( √ 1 -α n ) as n → ∞.
As a consequence

T 2,n = O n(1 -α n ) = o(1) as n → ∞. ( 22
)
We conclude by controlling T 3,n . Theorem 2 entails

T 3,n = O P n(1 -α n ) q n (α n u) q(α n u) = O P   n(1 -α n ) tr Σ n -u ′ Σ n u tr Σ -u ′ Σu 1/2   = O P n(1 -α n ) = o P (1) as n → ∞ (23)
by the consistency of Σ n . Combining ( 21), ( 22) and ( 23) completes the proof.

Proofs of the preliminary results

Proof of Lemma 1. The fact that ϕ is nonnegative and the inequality ∀r ≤ x , ϕ(x, r, v) ≤ 2r 2 (24) are straightforward consequences of the Cauchy-Schwarz inequality. Furthermore, ϕ can be rewritten as

ϕ(x, r, v) = r 2 x -x, v v 2 x -rv [ x -rv -x -rv, v ]
.

Let us now remark that, if x < r, then, by the Cauchy-Schwarz inequality, xrv, v = x, vr < 0 which makes it clear that

ϕ(x, r, v)1l { x <r} ≤ r 2 x -x, v v 2 x -rv 2 1l { x <r} =: ψ(x, r, v)1l { x <r} . (25)
Since xrv 2 = x 2 -2r x, v + r 2 , the function ψ(x, •, v) is differentiable on ( x , +∞) and some easy computations yield

∂ψ ∂r (x, r, v) = 2r x 2 -r x, v x -x, v v 2 x -rv 4 . If x, v ≤ 0 then ψ(x, •, v) is increasing on ( x , +∞) and thus ∀r > x , ψ(x, r, v) ≤ lim r→+∞ ψ(x, r, v) = x -x, v v 2 ≤ x 2 . (26)
Otherwise, if x, v > 0 then ψ(x, •, v) reaches its global maximum over [ x , +∞) at x 2 / x, v and therefore,

∀r > x , ψ(x, r, v) ≤ ψ x, x 2 x, v , v = x 2 . ( 27
)
Collecting ( 25), ( 26) and ( 27) yields

ϕ(x, r, v)1l { x <r} ≤ x 2 1l { x <r} . (28)
Combining ( 24) and ( 28) shows that ϕ(x, r, v) ≤ 2 x 2 for every r > 0 and every v ∈ S d-1 and completes the proof of the result.

Proof of Lemma 2. Let v ∈ R d and W α (•, v) : R d → R be the function defined by

W α (x, v) = x q(αu) - q(αu) q(αu) -1 -1 x q(αu) - q(αu) q(αu) , v .
For α close enough to 1, (2) entails

αu - q(αu) q(αu) , v + E (W α (X, v)) + 1 q(αu) E X, v = 0. ( 29
)
It is therefore enough to show that

q(αu) E (W α (X, v)) → -u, v E X, u as α ↑ 1. ( 30
)
Since, for every x ∈ R d , x q(αu) -q(αu) q(αu)

2 = 1 - 2 q(αu)
x, q(αu) q(αu)

+ x 2 q(αu) 2 , (31)
it follows from a Taylor expansion and Theorem 1 that q(αu) W α (X, v) →u, v X, u almost surely as α ↑ 1.

(32)

Besides,

x q(αu) -q(αu) q(αu)

-1

-1 = x q(αu) -q(αu) q(αu)

-1

1 + x q(αu) -q(αu) q(αu)

-1 2 q(αu)

x, q(αu) q(αu) -x 2 q(αu) 2 , and the Cauchy-Schwarz inequality yields x q(αu) -q(αu) q(αu)

-1

x q(αu) -q(αu) q(αu) , v ≤ v .

Thus, using the triangular inequality and the Cauchy-Schwarz inequality, it follows that

|W α (x, v)| ≤ v 1 + x q(αu)
q(αu) q(αu)

-1

x q(αu) 2 + x q(αu) .

Consequently, one has

q(αu) |W α (x, v)| 1l { x ≤ q(αu) } ≤ 3 v x 1l { x ≤ q(αu) } .
Furthermore, the reverse triangle inequality entails, for x ∈ R d such that x > q(αu) :

1 + x q(αu) -q(αu) q(αu)

-1 ≤ q(αu) x ,
and therefore,

q(αu) |W α (x, v)| 1l { x > q(αu) } ≤ 3 v x 1l { x > q(αu) } .
Finally,

q(αu) |W α (X, v)| ≤ 3 v X
so that the integrand in ( 30) is bounded from above by an integrable random variable. One can now recall (32) and apply the dominated convergence theorem to obtain (30). The proof is complete.

Proof of Lemma 3. Let Z α : R d → R be the function defined by

Z α (x) = 1 + x -q(αu)
xq(αu) , q(αu) q(αu) .

For α close enough to 1, (2) yields αu -q(αu) q(αu) , q(αu) q(αu)

+ E (Z α (X)) = 0 (33)
and it thus remains to prove that

q(αu) 2 E (Z α (X)) → 1 2 E X -X, u u 2 as α ↑ 1.
To this end, rewrite Z α as

Z α (x) = 1 - x q(αu) - q(αu) q(αu) -1 1 - 1 q(αu)
x, q(αu) q(αu) .

It thus follows from equation ( 31), Theorem 1 and a Taylor expansion that Z α (x) = 1 2 q(αu) 2 xx, q(αu) q(αu) q(αu) q(αu) , x (1 + o(1))

for all x ∈ R d . Using Theorem 1 again, we then get

q(αu) 2 Z α (X) → X 2 -X, u 2 = X -X, u u 2 almost surely as α ↑ 1. ( 35
)
To conclude the proof, let ϕ :

R d × R + × S d-1 → R be the function defined by ϕ(x, r, v) = r 2 1 + x -rv, v x -rv .
Note that q(αu) 2 Z α (x) = ϕ(x, q(αu) , q(αu)/ q(αu) ). By Lemma 1:

q(αu) 2 Z α (X) = ϕ(X, q(αu) , q(αu)/ q(αu) ) ≤ 2 X 2
and the right-hand side is an integrable random variable. Use then (35) and the dominated convergence theorem to complete the proof.

Proof of Lemma 4. Let v ∈ R d and recall the notation

W α (x, v) = x q(αu) - q(αu) q(αu) -1 -1 x q(αu) - q(αu) q(αu) , v
from the proof of Lemma 2. From (29) there, it is enough to show that

q(αu) E ( q(αu) W α (X, v) + u, v X, u ) → 1 2 u, v E X -X, u u 2 -u, v Var X, u + Cov( X, u , X, v ) -u, v E(X -X, u u) 2 (36)
as α ↑ 1. Use now (31) in the proof of Lemma 2, Theorem 2(i) and a Taylor expansion to obtain after some cumbersome computations that

q(αu) ( q(αu) W α (X, v) + u, v X, u ) = 1 2 u, v X -X, u u 2 -u, v X, u ( X, u -E X, u ) + X, u ( X, v -E X, v ) -u, v X, E(X -X, u u) + 2 j=0 X j ε j (α, X, q(αu))
with probability 1, where for all j ∈ {0, 1, 2}, ε j (α, y, z) → 0 as max(1α, y / z ) ↓ 0. In particular

q(αu) ( q(αu) W α (X, v) + u, v X, u ) → 1 2 u, v X -X, u u 2 -u, v X, u ( X, u -E X, u ) -u, v X, E(X -X, u u) + X, u ( X, v -E X, v ) almost surely as α ↑ 1. ( 37
)
The proof shall be complete provided we can apply the dominated convergence theorem to the left-hand side of (37). To this end, let δ ∈ (0, 1) be such that α ∈ (1δ, 1) and X q(αu) < δ ⇒ max 0≤j≤2 |ε j (α, X, q(αu))| ≤ 1.

Equality (37) thus entails for α close enough to 1:

q(αu) q(αu) W α (X, v) + u, v X, u 1l { X <δ q(αu) } ≤ P 1 ( X )1l { X <δ q(αu) }
where P 1 is a real polynomial of degree 2. Besides, it is a consequence of the definition of W α (X, v) and the Cauchy-Schwarz inequality that

q(αu) q(αu) W α (X, v) + u, v X, u 1l { X ≥δ q(αu) } ≤ 2(1 + δ) v δ 2 X 2 1l { X ≥δ q(αu) } .
One can conclude that there exists a real polynomial P 2 of degree 2 such that

q(αu) q(αu) W α (X, v) + u, v X, u ≤ P 2 ( X )
so that the integrand in ( 36) is bounded by an integrable random variable. Recall (37) and apply the dominated convergence theorem to complete the proof.

Proof of Lemma 5. The proof is similar to that of Lemma 4. Recall from the proof of Lemma 3 the notation Z α (x) = 1 + xq(αu)

xq(αu) , q(αu) q(αu) .

From (33) there, it is enough to show that

q(αu) E q(αu) 2 Z α (X) - 1 2 E X -X, u u 2 → E X, u X -X, u u 2 -X, E(X -X, u u) (38)
as α ↑ 1. We first use (31) in the proof of Lemma 2, equation (34) in the proof of Lemma 3, Theorem 2(i) and a Taylor expansion to obtain after some burdensome computations that

q(αu) q(αu) 2 Z α (X) - 1 2 X -X, u u 2 = X, u X -X, u u 2 -X, E(X -X, u u) + 3 j=0 X j ε j (α, X, q(αu)) (39)

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 1234567 Figure1: Spherical Gaussian distribution N (0, 1/2, 1/2, 0) for α = 0.995. Top left: comparison between a numerical method and the use of the equivalent (3) for the computation of the iso-quantile curve, full line: numerical method, dashed line: asymptotic equivalent. Top right, bottom left and bottom right: best, median and worst estimates of the iso-quantile curve for n = 200, full line: numerical method, dashed-dotted line: estimator q n .

with probability 1, where for j ∈ {0, 1, 2, 3}, ε j (α, y, z) → 0 as max(1α, y / z ) ↓ 0. Especially q(αu) q(αu) 2 Z α (X) -1 2 X -X, u u 2 → X, u X -X, u u 2 -X, E(X -X, u u) (40)

as α ↑ 1. Our aim is now to apply the dominated convergence theorem to the left-hand side of (38). To this end, pick δ ∈ (0, 1) such that

Equality (39) thus entails for α close enough to 1:

where P 1 is a real polynomial of degree 3. Moreover, the Cauchy-Schwarz inequality yields

Consequently, there exists a real polynomial P 2 of degree 3 such that

We conclude that the integrand in ( 38) is bounded by an integrable random variable. Recall (40) and apply the dominated convergence theorem to complete the proof. Centred Gaussian N (0, 1/8, 3/4, 0) 0.990 6.05 • 10 -4 1.79 • 10 -3 1.17 Double exponential E(2 √ 3, 2 √ 3, 2 3/5, 2 3/5) 0.990 6.17 • 10 -4 4.37 • 10 -3 2.71 • 10 -3 1.42