ON BERNSTEIN–KANTOROVICH INVARIANCE PRINCIPLE IN H ¨ OLDER SPACES AND WEIGHTED SCAN STATISTICS

https://doi.org/10.1051/ps/2019027 www.esaim-ps.org

ON BERNSTEIN–KANTOROVICH INVARIANCE PRINCIPLE IN H ¨ OLDER SPACES AND WEIGHTED SCAN STATISTICS

Alfredas Raˇ ckauskas

and Charles Suquet

Abstract. Let ξn be the polygonal line partial sums process built on i.i.d. centered random vari- ables Xi,i≥1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|^max(2,r) and the joint weak convergence inC[0,1] ofn^−1/2ξnto a Brownian motionW with the moments convergence of Ekn^−1/2ξnk^r∞to EkWk^r∞. For 0< α <1/2 andp(α) = (1/2−α)⁻¹, we prove that the joint convergence in the separable H¨older spaceH^o_α ofn^−1/2ξn toW jointly with the one of Ekn^−1/2ξnk^rα to EkWk^rα holds if and only ifP(|X1|> t) =o(t^−p(α)) whenr < p(α) or E|X1|^r <∞ whenr≥p(α). As an application we show that for everyα <1/2, all theα-H¨olderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of therth moments of someα-H¨olderian weighted scan statistics where the natural border for αis 1/2−1/pwhen E|X1|^p<∞. In the case where theXi’s are pregularly varying, we can complete these results forα >1/2−1/pwith an appropriate normalization.

Mathematics Subject Classification.60F17, 62G30.

Received July 10, 2018. Accepted November 14, 2019.

1. Introduction

Let (Zn)_n≥1 be a sequence of random elements in some separable metric space S endowed with its Borel σ-fieldS. LetZ be a random element inS. Assume for notational simplicity thatZ and theZn’s are all defined on the same probability space (Ω,F,P). ThenZn converges in distribution to Z, denoted by

Z_n−−−−→^d

n→∞ Z,

if its distributionµn= P◦Z_n⁻¹ converges weakly toµ= P◦Z⁻¹. This means that for every continuous bounded functionf :S→R,

Z

S

fdµn= Ef(Zn)−−−−→

n→∞ Ef(Z) = Z

S

fdµ. (1.1)

∗Research supported by the Research Council of Lithuania, grant No. S-MIP-17-76.

Keywords and phrases: Fonctional central limit theorem, H¨older space, moments, quantile process, regular variation, scan statistics, Wasserstein distance.

1 Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania.

2 Laboratoire P. Painlev´e (UMR 8524 CNRS), Bˆat. M2, Cit´e Scientifique, Universit´e de Lille, 59655 Villeneuve d’Ascq Cedex, France.

*∗Corresponding author:Charles.Suquet@univ-lille.fr

c

The authors. Published by EDP Sciences, SMAI 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Relaxing the boundedness assumption on f in (1.1) leads to the classical question of convergence of moments.

WhenS is a separable Banach space with normk k, one is interested in extending the convergence in (1.1) to the case of functions satisfying for some positive constants c1,c2,r,

|f(x)| ≤c₁kxk^r+c₂, x∈S. (1.2) It is well known that this extension is valid if and only if (kZnk^r)_n≥1is uniformly integrable (see [3], Thm. 5.4) that is

a→∞lim sup

n≥1

EkZ_nk^r1_{kZnk>a}= 0. (1.3)

Let us note that if (kZnk^r)n≥1 is uniformly integrable, necessarily sup

n≥1

EkZ_nk^r<∞. (1.4)

In this paper we focus on the convergence of moments in the functional central limit theorem. Let (Xi)i≥1

be an i.i.d. sequence of real valued random variables with null expectation and variance one if they exist, S_n:=X₁+· · ·+X_n andξ_n the random polygonal line with vertices (k/n, S_k),k= 0,1, . . . , n :

ξn(t) =S_[nt]+ (nt−[nt])X_[nt]+1, t∈[0,1]. (1.5) From Bernstein theorem [2] it is known that forr >0 the joint convergence

n^−1/2S_n−−−−→^d

n→∞ G and lim

n→∞E

n^−1/2S_n

r

= E|G|^r,

where Gis a GaussianN(0,1) random variable, is equivalent to the finiteness of E|X₁|^max{2,r}. Note also that in the case where r= 2 the convergence of the corresponding moment is trivial and that for 0< r <2 the convergence of E

n^−1/2S_n

r follows immediately from EX₁²<∞by uniform integrability of n⁻¹S_n², n≥1 . Let us denote byW a standard Brownian motion viewed as a random element in the space C[0,1] of continuous functions x: [0,1]→R endowed with the uniform norm kxk_∞= sup{|x(t)|, t∈[0,1]}. The classical Donsker- Prokhorov theorem provides the equivalence :

n^−1/2ξ_n −−−−→^d

n→∞ W in C[0,1] if and only if EX₁²<∞.

Forr >0, the Bernstein–Kantorovich functional central limit theorem (see [11], Thm. 11.2.1, p. 219) provides the equivalence between E|X1|^max{2,r}<∞and the joint convergence

n^−1/2ξn

−−−−→d

n→∞ W in C[0,1] with Ekn^−1/2ξnk^r_∞−−−−→

n→∞ EkWk^r_∞. (1.6)

It turns out that the condition E|X1|^r<∞for somer >2 provides also the convergence in distribution of n^−1/2ξn to W in a stronger topology than the C[0,1]’s one. Define for 0≤α <1 the H¨older spaceH^o_α[0,1] as the set of functionsx: [0,1]→Rsuch that

ωα(x, δ) := sup

0<t−s≤δ s,t∈[0,1]

|x(t)−x(s)|

(t−s)^α −−−→

δ→0 0, (1.7)

endowed with the norm

kxkα=|x(0)|+ωα(x,1), (1.8)

which makes it a separable Banach space (isomorphic to C[0,1] in the special caseα= 0).

Letα∈(0,1/2) andp(α) = (1/2−α)⁻¹. By the necessary and sufficient condition for Lamperti’s H¨olderian invariance principle [12,13], we know thatn^−1/2ξnconverges in distribution in the spaceH_α^o[0,1] to the standard Brownian motion if and only if P(|X1|> t) =o(t^−p(α)) when t tends to infinity. When E|X1|^p(α)<∞, this condition is satisfied. Our first result extends the Bernstein–Kantorovich functional central limit theorem to the spacesH^o_α[0,1].

Theorem 1.1. Letα∈(0,1/2) andp(α) = (1/2−α)⁻¹. Let r >0. Then the joint convergence n^−1/2ξn

−−−−→d

n→∞ W inH^o_α[0,1] with Ekn^−1/2ξnk^r_α−−−−→

n→∞ EkWk^r_α (1.9)

holds if and only if

t→∞lim t^p(α)P(|X1|> t) = 0, whenr < p(α), (1.10)

E|X1|^r<∞, whenr≥p(α). (1.11)

It is worth noticing here that (1.9) is equivalent to the convergence of the distribution of n^−1/2ξn to the one of W with respect to the Wasserstein distance of order rassociated to the norm k.kα,i.e. with the mass transportation cost functionc(x, y) =kx−yk^r_α, see Section3.2for details.

An immediate consequence of Theorem1.1is that (1.11) implies the convergence of Ef(n^−1/2ξ_n) to Ef(W) for any continuous functionalf :H_α^o[0,1]→Rsatisfying

|f(x)| ≤c₁kxk^r_α+c₂. (1.12)

Among the various functionals f(n^−1/2ξ_n) wheref satisfies a condition like (1.12) are the (powers of the) following weighted scan type statistics :

M_n,α= max

1≤j≤nj^−α max

0≤k≤n−j|S_k+j−S_k|.

We refer to [1,10] for valuable information about scan statistics and their applications. The following result is a corollary of more general results obtained in this paper (see Thms. 4.1and4.2).

Theorem 1.2. Letp >2.

(a) If0≤α≤1/2−1/p, andE|X₁|^p<∞ then

n→∞lim n⁻¹EM_n,α^p = Eω_α^p(W,1).

(b) If1/2−1/p < α <1 andX1 is regularly varying with exponent pthen for any 0≤r < p,

n→∞lim b^−r_n EM_n,α^r = EY_p^r,

where

bn = inf{t >0 : P(|X1| ≤t)≥1−1/n} (1.13) andYp has Fr´echet distribution with exponent p.

The paper is organized as follows. Section 2 is devoted to preliminaries where uniform integrability, regu- larly varying random variables are discussed and necessary tools on H¨older spaces are presented. In Section 3, one proves Theorem 1.1 and some comments concerning the upper bound for admissible H¨older index in the Bernstein–Kantorovich invariance principle are presented. Convergence of moments of weighted scan statistics is considered in Section4. The paper ends with an appendix devoted to some facts from Karamata theory.

2. Preliminaries 2.1. Uniform integrability

Lemma 2.1. Let(Z_n)_n≥1be a sequence of random elements in the Banach space(S,k k). Forr >0,(kZnk^r)_n≥1 is uniformly integrable if and only if

a→∞lim sup

n≥1

Z ∞ a

t^r−1P(kZnk> t) dt= 0. (2.1)

The proof is elementary and will be omitted.

2.2. H¨ olderian tools

Let Dj denotes the set of dyadic numbers of level j in [0,1], that is D0 := {0,1} and for j ≥1, Dj :=

(2l−1)2^−j; 1≤l≤2^j−1 . Ford∈Dj set d⁻ :=d−2^−j,d⁺:=d+ 2^−j,j≥0. Forx: [0,1]→Randd∈Dj

let us define

λ_d(x) :=

(x(d)−^x(d+)+x(d₂ ⁻⁾ ifj ≥1,

x(d) ifj = 0.

The following sequential norm defined onH^o_α[0,1] by kxk^seq_α := sup

j≥0

2^αjmax

d∈D_j|λd(x)|,

is equivalent to the natural norm kxk_α, see [5]. Let us define also Dj :={k2^−j, 0 ≤k < 2^j}, so that Dj = {0} ∪S

1≤i≤jD_i.

The H¨older norm of a polygonal line function is very easy to compute according to the following lemma for which we refer e.g.to [8] Lemma 3, where it is proved in a more general setting.

Lemma 2.2. Lett₀= 0< t₁<· · ·< t_n= 1be a partition of[0,1]andxbe a real-valued polygonal line function on [0,1]with vertices at the ti’s, i.e. xis continuous on [0,1]and its restriction to each interval[ti, ti+1]is an affine function. Then for any 0≤α <1,

sup

0≤s<t≤1

|x(t)−x(s)|

(t−s)^α = max

0≤i<j≤n

|x(tj)−x(ti)|

(t_j−t_i)^α .

2.3. Regularly varying random variables

Throughout this paper we implicitly assume that all the random variables considered are defined on the same probability space (Ω,F,P) and we use the following notion of regularly varying random variable.

Definition 2.3. The random variable X is regularly varying with index p >0 (denoted X ∈RV_p) if there exists a slowly varying functionLsuch that the distribution functionF(t) = P(X≤t) satisfies the tail balance condition

F(−x)∼bL(x)x^−p and 1−F(x)∼aL(x)x^−p, as x→ ∞, (2.2) where a, b∈(0,1) anda+b= 1.

We refer to [4] for an encyclopaedic treatment of regular variation.

WritingLporL^o_p,∞for the sets of random variablesXverifying respectively E|X|^p<∞or lim_t→∞t^pP(|X|>

t) = 0, we note that

RVp⊂ Lr, for 0≤r < p, RVp∩ Lp6=∅, RVp∩ L^o_p,∞6=∅, RV_p∩ L_p⁰ =∅ for p⁰> p.

The next lemma plays a key role in our results on the scan statistics built on RV_prandom variables. Its proof is detailed in the Appendix.

Lemma 2.4. LetX ∈RVp andbn be its(1−1/n)quantile defined as in (1.13).

i) For any 0< s < p,

E|X|^s1_{|X|>ybn}≤ 2p p−sb^s_n1

ny^s−p, (2.3)

fornlarge enough, uniformly iny∈[1,∞).

ii) For any s > p,

E|X|^s1_{|X|≤ybn}≤ 2p s−pb^s_n1

ny^s−p, (2.4)

fornlarge enough, uniformly iny∈[1,∞).

iii) LetXe =X1_{|X|≤ybn},X⁰=Xe−EX. Then for anye 0< s6=p, E|X⁰|^s≤K(s, p)b^s_n1

ny^s−p, (2.5)

fornlarge enough, uniformly iny∈[1,∞), where

K(s, p) = max(1,2^s−1) 4p

|p−s|.

3. Bernstein–Kantorovich theorem in H

[0, 1]

In this section first we prove Theorem1.1and then discuss some aspects of Bernstein–Kantorovich theorem in H¨older framevork.

3.1. Proof of Theorem 1.1

The necessity of the X₁’s integrability conditions for the joint convergence (1.9) is easily seen. Indeed when r < p(α), (1.10) follows from the first convergence in (1.9), see [12]. When r≥p(α), we note that |X₁| =

|ξ_n(1/n)−ξ_n(0)| ≤ kξ_nk_αn^−α.From the second convergence in (1.9), Ekn^−1/2ξ_nk^r_αis finite, at least fornlarge enough, whence E|X₁|^r≤n^(1/2−α)rEkn^−1/2ξ_nk^r_α<∞, giving the necessity of (1.11).

Now let us prove that the integrability conditions (1.10) or (1.11) are sufficient for the joint convergence (1.9).

By the H¨olderian invariance principle [12], (1.10) ora fortiori (1.11) implies thatn^−1/2ξn converges in distri- bution to W in H^o_α[0,1] and by continuous mapping that kn^−1/2ξnkα converges in distribution to kWkα. It remains to check the uniform integrability of the sequence kn^−1/2ξnk^r_α

n≥1. It is enough to consider the case where r≤p(α) only. Indeed, if r > p(α), then we choose β = 1/2−1/r (so that r=p(β)) and notice that uniform integrability of the sequence kn^−1/2ξ_nk^p(β)_β

n≥1 yields that of the sequence kn^−1/2ξ_nk^p(β)α

n≥1 since β > α.

Now to prove the uniform integrability of the sequence (kn^−1/2ξnk^r_α)_n≥1 we can obviously replace k kα by an equivalent norm. The choice of k k^seq_α seems more convenient here. So we have to prove that forr≤p(α),

a→∞lim sup

n≥1

Z ∞ a

rt^r−1P

kn^−1/2ξnk^seq_α > t

dt= 0. (3.1)

The following proof of (3.1) is essentially common to the cases r < p(α) and r=p(α) except for some nuance in the exploitation of the integrability ofX₁. From now on, we writepforp(α).

The first task in establishing (3.1) is to obtain a good estimate for P(n, t) := P

kn^−1/2ξnk^seq_α > t

. (3.2)

Write for simplicity tk,j =k2^−j, k = 0,1, . . . ,2^j, j= 1,2, . . . and tk =tk,j whenever the context dispels any doubt on the value of j. It is easily seen that for anyx∈ Hαsuch thatx(0) = 0,

kxk^seq_α ≤sup

j≥0

2^jα max

0≤k<2^j|x(tk+1,j)−x(t_k,j)|.

From this we deduce that

P(n, t)≤P1(n, t) +P2(n, t), (3.3)

with

P₁(n, t) = P

max

0≤j≤logn2^jα max

0≤k<2^j|ξ_n(t_k+1,j)−ξ_n(t_k,j)|> n^1/2t

, (3.4)

P2(n, t) = P sup

j>logn

2^jα max

0≤k<2^j|ξn(tk+1,j)−ξn(tk,j)|> n^1/2t

!

, (3.5)

where log denotes the logarithm with basis 2 (log 2 = 1).

Estimation ofP₂(n, t). Ifj >logn, thent_k+1−t_k= 2^−j<1/nand therefore witht_k ∈[i/n,(i+ 1)/n), either t_k+1 is in (i/n,(i+ 1)/n] or belongs to (i+ 1)/n,(i+ 2)/n

, where 1≤i≤n−2 depends onkandj.

In the first case we have

|ξn(tk+1)−ξn(tk)|=|Xi+1|2^−jn≤2^−jn max

1≤i≤n|Xi|.

Iftk andtk+1are in consecutive intervals, noticing that the slope of each of the two involved segments of the polygonal line is bounded in absolute value by nmax_1≤i≤n|Xi|, we get

|ξn(t_k+1)−ξ_n(t_k)| ≤ |ξn(t_k)−ξ_n((i+ 1)/n)|+|ξn((i+ 1)/n)−ξ_n(t_k+1)|

≤2^−jn max

1≤i≤n|Xi|.

With both cases taken into account we obtain P₂(n, t)≤P sup

j>logn

2^jαn^−1/22^−jn max

1≤i≤n|X_i|> t

! .

Noting that forj >logn, 2^j(−1+α)n^1/2< n^−1/2+α=n^−1/p, this leads to P2(n, t)≤nP

|X1|> tn^1/p

. (3.6)

To control the contribution of P₂(n, t) when estimating the integral in (3.1), we note that for everyn≥1, Z ∞

a

rt^r−1nP

|X1|> tn^1/p

dt=n^1−r/p Z ∞

an^1/p

rs^r−1P(|X1|> s) ds. (3.7) In the case where r=p, as E|X₁|^p=R∞

0 ps^p−1P(|X₁|> s) dsis supposed finite, we can bound the right hand side of (3.7) by R∞

a ps^p−1P(|X1|> s) ds uniformly in n≥1. In the case where r < p, the hypothese (1.10) implies that

P(|X1|> t)≤Kt^−p, t >0, (3.8)

for some constantK depending only on the distribution ofX₁. Hence n^1−r/p

Z ∞ an^1/p

rs^r−1P(|X1|> s) ds≤ Kr

p−ra^r−p, n≥1. (3.9)

Gathering both cases we obtain that for r≤p,

a→∞lim sup

n≥1

Z ∞ a

rt^r−1nP

|X1|> tn^1/p

dt= 0 (3.10)

and consequently,

a→∞lim sup

n≥1

Z ∞ a

rt^r−1P₂(n, t) dt= 0. (3.11)

Estimation of P1(n, t). Let uk = [ntk]. Then uk ≤ntk ≤1 +uk and 1 +uk ≤uk+1 ≤ntk+1 ≤1 +uk+1. Therefore

|ξn(tk+1)−ξn(tk)| ≤ |ξn(tk+1)−Su_k+1|+|Su_k+1−Su_k|+|Su_k−ξn(tk)|.

Since|Su_k−ξn(tk)| ≤ |X1+u_k| and|ξn(tk+1)−Su_k+1| ≤ |X1+u_k+1| we obtain

P1(n, t)≤P1,1(n, t) + 2P1,2(n, t), (3.12) where

P1,1(n, t) := P sup

0≤j≤logn

2^jαn^−1/2 max

0≤k<2^j|Su_k+1−Su_k|> t 2

!

P1,2(n, t) := P

0≤j≤logmax n2^jαn^−1/2 max

1≤i≤n|Xi|> t 4

.

InP_1,2(n, t), max_0≤j≤logn2^jα≤n^α, so

P_1,2(n, t)≤P

n^−1/2+α max

1≤i≤n|Xi|> t 4

≤nP

|X1|> tn^1/p 4

.

Using (3.10) we obtain that forr≤p,

a→∞lim sup

n≥1

Z ∞ a

rt^r−1P1,2(n, t) dt= 0. (3.13)

To estimateP1,1(n, t), we use a truncation method. Define fort >0 and 0< δ≤1,

Xe_i:=X_i1_{|Xi|≤δtn1/p}, X_i⁰:=Xe_i−EXe_i.

Let Seu_k and S_u⁰_k be the random variables obtained by replacingXi with respectivelyXei in Su_k or withX_i⁰ in S_uk. We introduce also

Pe1,1(n, t, δ) := P sup

0≤j≤logn

2^jαn^−1/2 max

0≤k<2^j|Seu_k+1−Seu_k|> t 2

!

, (3.14)

P_1,1⁰ (n, t, δ) := P sup

0≤j≤logn

2^jαn^−1/2 max

0≤k<2^j|S_u⁰_k+1−S_u⁰_k|> t 4

!

. (3.15)

First, since on the event{max_1≤i≤n|Xi| ≤δtn^1/p},Seu_k =Su_k for everyk, we note that P_1,1(n, t)≤Pe_1,1(n, t, δ) + P

1≤i≤nmax |X_i|> δtn^1/p

≤Pe1,1(n, t, δ) +nP |X1|> δtn^1/p

. (3.16)

Invoking again (3.10) reduces the control of the contribution of P1,1(n, t) when bounding the integral in (3.1) to the one of Pe_1,1(n, t, δ).

Now to work with centered random variables, we prove that Pe1,1(n, t, δ)≤P_1,1⁰ (n, t, δ) fort large enough, uniformly inn≥1. Comparing (3.14) and (3.15), we see thatPe1,1(n, t, δ)≤P_1,1⁰ (n, t, δ) will be satisfied provided

that

max

0≤j≤logn2^jαn^−1/2 max

0≤k<2^j u_k+1

X

i=1+u_k

|EXe_i|< t

4. (3.17)

As j ≤ logn, 1 ≤ uk+1−uk ≤ 2n2^−j, (3.17) reduces to 2n^1/2|EXe1| < t/4. Since EX1 = 0, EXe1 =

−EX11_{|X1|>δtn1/p}, so we just have to check that

E|X1|1_{|X1|>δtn1/p}< t

8n^−1/2, (3.18)

fort large enough, uniformly inn≥1. By a Fubini argument, E|X₁|1_{|X1|>δtn1/p}=

Z ∞ 0

P

|X₁|>max(δtn^1/p, s) ds

=

Z δtn^1/p 0

P(|X1|> δtn^1/p) ds+ Z ∞

δtn^1/p

P(|X1|> s) ds

=δtn^1/pP(|X₁|> δtn^1/p) + Z ∞

δtn^1/p

P(|X₁|> s) ds.

Now, in the case where r < p, using (3.8) we obtain E|X₁|1_{|X1|>δtn1/p}≤δ^1−pK

t^p

1 + 1 p−1

tn^1/p−1=δ^1−p pK

(p−1)t^ptn^−1/2−α. Therefore (3.18) is satisfied for everyt > t0 not depending onn, since we can choose

t0=δ^1/p−1

8pK (p−1)

^1/p

. (3.19)

The same holds in the case where r=p, replacing (3.8) by Markov’s inequality andK by E|X1|^p. Now it only remains to deal with sup_n≥1R∞

a rt^r−1P_1,1⁰ (n, t, δ) dt. For anyq > p, we have P_1,1⁰ (n, t, δ) = P

max

0≤j≤logn2^jαn^−1/2 max

0≤k<2^j|S_u⁰_k+1−S_u⁰

k|> t 4

≤

logn

X

j=0

X

0≤k<2^j

P

|S⁰_uk+1−S_u⁰_k|> tn^1/2 2^jα+2

≤ 4^q t^qn^q/2

logn

X

j=0

2^jαq X

0≤k<2^j

E|S_u⁰_k+1−S_u⁰_k|^q. (3.20)

Next we bound up E|S⁰_uk+1−S_u⁰_k|^q by using the Rosenthal inequality:

E|S_u⁰_k+1−S⁰_uk|^q ≤Cq Var(S⁰_uk+1−S_u⁰_k)^q/2+

uk+1

X

i=1+uk

E|X_i⁰|^q

! ,

were Cq is a universal constant, i.e. not depending on the distribution of the Xi’s. As the X_i⁰s are i.i.d. and uk+1−uk ≤2n2^−j, this gives

E|S_u⁰

k+1−S_u⁰

k|^q ≤C_q

(2n2^−j)^q/2(EX₁⁰²)^q/2+ 2n2^−jE|X₁⁰|^q .

Going back to (3.20) with this bound, we obtain

P_1,1⁰ (n, t, δ)≤ 4^qCq

t^qn^q/2



(2n)^q/2(EX₁⁰²)^q/2

logn

X

j=0

2j(αq−q/2+1)+ 2nE|X₁⁰|^q

logn

X

j=0

2^jαq





Asq > p= (1/2−α)⁻¹, 1−q/2 +qαis negative, whence for everyn,

logn

X

j=0

2j(αq−q/2+1)< 1

1−2^1−q/2+qα = 1 1−2^1−q/p. Next, for everyn≥1,

logn

X

j=0

2^jαq< 2^qα 2^qα−1n^qα. Moreover

EX₁⁰²= VarXe₁≤EXf₁²≤EX₁² and

E|X₁⁰|^q= E|Xf1−EXf1|^q ≤2^qE|Xe1|^q.

With all these partial estimates, the upper bound obtained forP_1,1⁰ (n, t, δ) becomes

P_1,1⁰ (n, t, δ)≤ C1(p, q)(EX₁²)^q/2

t^q +C2(p, q)n^1−q/pE|Xe1|^q

t^q ,

with

C₁(p, q) = 2^5q/2C_q

1−2^1−q/p and C₂(p, q) = 2^3q+qα+1C_q 2^qα−1 . Now we have for r≤p,

Z ∞ a

rt^r−1P_1,1⁰ (n, t, δ) dt≤C₁(p, q)(EX₁²)^q/2r

q−r a^r−q+C₂(p, q)I_r,q(a, n), (3.21) where

Ir,q(a, n) =n^1−q/p Z ∞

a

rt^r−q−1E|Xe1|^qdt.

It is worth recalling here that E|Xe₁|^q depends onn,δandt. As the first term in the upper bound (3.21) neither depends onnor onδand goes to 0 asatends to infinity, it remains only to investigate the asymptotic behavior of sup_n≥1Ir,q(a, n) whenatends to infinity. To transformIr,q(a, n), we use the fact that ifY is a positive random variable andf a C¹ non decreasing function on [0,∞) with f(0) = 0, then by the Fubini-Tonelli theorem, for any positive constantc,

E f(Y)1_{Y≤c}

= Z c

0

f⁰(s) P(s < Y ≤c) ds≤ Z c

0

f⁰(s) P(Y > s) ds.

Applying this to E|Xe₁|^q= E|X₁|^q1_{|X1|≤δtn1/p}, we obtain

Ir,q(a, n)≤n^1−q/p Z ∞

a

rt^r−q−1

Z δtn^1/p 0

qs^q−1P(|X1|> s) dsdt=:Jr,q(a, n).

Exchanging the order of integrations inJr,q(a, n) gives J_r,q(a, n) =n^1−q/p

Z ∞ 0

(Z ∞

max(a,sδ⁻¹n^−1/p)

rt^r−q−1dt )

qs^q−1P(|X₁|> s) ds

= qr

q−rn^1−q/p Z ∞

0

max(a, sδ⁻¹n^−1/p)^r−qs^q−1P(|X1|> s) ds

= qr q−r

n^1−q/pa^r−qJ⁰+n^1−r/pδ^q−rJ⁰⁰

, (3.22)

where

J⁰ =

Z δan^1/p 0

s^q−1P(|X₁|> s) ds, J⁰⁰= Z ∞

δan^1/p

s^r−1P(|X₁|> s) ds.

We boundJ⁰ forr≤p, using (3.8), agreeing for simplicity thatK= E|X1|^p whenr=p. This gives

J⁰ ≤K

Z δan^1/p 0

s^q−p−1ds= K

q−pa^q−pδ^q−pn^q/p−1. (3.23)

For J⁰⁰, the same method would lead to a divergent integral in the special case r=p, so we restrict the use of (3.8) to the case wherer < p. This gives

J⁰⁰≤ ( _K

p−ra^r−pδ^r−pn^r/p−1 whenr < p, R∞

δan^1/ps^p−1P(|X1|> s) ds whenr=p. (3.24) Going back to I_r,q(a, n) and accounting (3.22)–(3.24) we obtain (recalling thatδ≤1)

Ir,q(a, n)≤ Kqr

(q−p)(p−r)a^r−pδ^r−p≤ Kqr

(q−p)(p−r)a^r−p, (r < p). (3.25) In the caser=p, bounding the integral in (3.24) by ¹_pE|X1|^p=K/p, we obtain

Ip,q(a, n)≤ q²

(q−p)²Kδ^q−p. (3.26)

Recapitulating all the estimate proposed throughout the proof, we see that for every a > t0(δ) defined by (3.19) and everyn≥1,

Z ∞ a

rt^r−1P(n, t) dt≤ Z ∞

a

rt^r−1P2(n, t) dt+ 2 Z ∞

a

rt^r−1P1,2(n, t) dt +

Z ∞ a

rt^r−1nP |X1|> δtn^1/p dt +pC1(p, q)(EX₁²)^q/2

q−p a^p−q+C2(p, q)Ir,q(a, n). (3.27) In the case where r < p, recalling (3.11), (3.13), (3.10) and (3.25), it follows

a→∞lim sup

n≥1

Z ∞ a

rt^r−1P(n, t) dt= 0.

We note in passing that in this case we did not really need the freedom to tune the value ofδ, the simple choice δ= 1 would have done the job as well.

In the special case where r=p, we have only to modify the treatment of the last term in the bound (3.27).

Asq > p, for anyε >0, we can fix aδ >0 such thatC2(p, q)(1−p/q)⁻²Kδ^q−p< ε. Accounting (3.11), (3.13), (3.10) and (3.26), there is somea1depending onε,p,qand on the distribution ofX1, such that for everya≥a1

and everyn≥1,

Z ∞ a

pt^p−1P(n, t) dt <6ε.

Asεwas arbitrary, the uniform convergence (3.1) is established and the proof is complete.

3.2. Comments

If we fix p > 2 and consider X1 ∈ Lp, then the best possible H¨olderian index corresponding to the p’th moments convergence is α=α(p) := 1/2−1/pas shows the following result.

Theorem 3.1. Letp >2. If for anyX1∈ Lp, sup

n≥1

Ekn^−1/2ξnk^p_β<∞ (3.28)

then β≤α(p) = 1/2−1/p.

Proof. Put r= (1/2−β)⁻¹. By looking at the increments of n^−1/2ξ_n betweenk/nand (k+ 1)/n, 0≤k < n, we see that

kn^−1/2ξ_nk^p_β≥

n^−1/2+β max

1≤i≤n|X_i| ^p

,

so (3.28) implies

sup

n≥1

Z ∞ 0

t^p−1P

1≤i≤nmax |Xi|> n^1/rt

dt <∞,

which can be recast as

sup

n≥1

n^−p/r Z ∞

0

s^p−1P

1≤i≤nmax |Xi|> s

ds <∞. (3.29)

It is well known that when theX_i’s are i.i.d., nP(|X1|> s)≤1 ⇒ P

1≤i≤nmax |Xi|> s

≥(1−e⁻¹)nP(|X1|> s). (3.30)

Now choose for |X1|the distribution given by

P(|X₁|> s) = c

s^p(lns)²1_[2,∞)(s).

Then P(|X1|> n^1/p) =cp²n⁻¹(lnn)⁻²≤n⁻¹ forn≥n0, so we deduce from (3.29) and (3.30) that sup

n≥n0

n^1−p/r Z ∞

n^1/p

1

s(lns)²ds= sup

n≥n0

pn^1−p/r lnn <∞, which holds if and only if p/r≥1 or equivalentlyβ ≤α(p).

Let us turn now on some probability distances considerations. Recall the Kantorovich functionals which are involved in the Monge-Kantorovich minimal cost of mass transportation problem :

A_c(P₁, P₂) = inf Z

S×S

c(x, y)P(dx,dy), P ∈ P^(P1,P2)

, (3.31)

where (S, d) is a separable metric space, P^(P1,P2) denotes the set of all probabilities on the Borel σ-field of S×Swith given marginalsP1,P2andc(x, y) =H(d(x, y)) whereH(0) = 0,H is non decreasing on [0,∞) and satisfies the Orlicz condition sup_t>0H(2t)/H(t)<∞. It is known, see Theorem 11.1.1 in [11] that if for some a∈S,R

Sc(x, a)P_n( dx)<∞for everyn≥1, then

n→∞lim Ac(Pn, P0) = 0 if and only if P_n−−−−→^weakly

n→∞ P₀ and lim

n→∞

Z

S

c(x, b)(P_n−P₀)(dx) = 0,

(3.32)

for some (and therefore for any)b∈S.

Let us denote by A^α_r the Kantorovich functional obtained by choosing H(t) =t^r and S=H^o_α[0,1] with d(x, y) =kx−yk_α (recall that H^o₀[0,1] is isomorphic to C[0,1]). We observe that (A^α_r)^1/r is the Wasserstein distance W_r associated to the spaceH^o_α[0,1]. WriteP_n for the distribution ofn^−1/2ξ_n and P₀ for the Wiener measure. Then (3.32) can be rewritten as

n→∞lim A^α_r(P_n, P₀) = 0 if and only if n^−1/2ξ_n−−−−→^d

n→∞ W inH^o_α[0,1] and Ekn^−1/2ξnk^r_α−−−−→

n→∞ EkWk^r_α.

(3.33)

From this point of view, Theorem 1.1 means that the convergence of A^α_r(Pn, P0) to 0 is equivalent to the moment condition (1.10) or (1.11) according tor < p(α) orr≥p(α). Similarly, from the Bernstein Kantorovich

invariance principle in C[0,1], the convergence ofA⁰_r(Pn, P0) to 0 is equivalent to E|X1|^max(r,2)<∞. As already hinted in the introduction, we see that starting from the classical Donsker-Prokhorov invariance principle in C[0,1] (A⁰₂(Pn, P0) →0 iff EX₁² < ∞) and looking for a stronger convergence in the framework of C[0,1]

(A⁰_p(Pn, P0)→0) at the price of a stronger moment assumption E|X1|^p <∞ (p > 2), we obtain a similar convergence (A^α_p(Pn, P0)→0) with a stronger topological path’s space.

3.3. An application to uniform quantile processes

As a corollary of Theorem1.1, we look now at the convergence of moments for the uniform quantile process.

For the weak-H¨olderian convergence of the uniform quantile process we refer to [7].

LetU1, . . . , Un be a sample of i.i.d. random variables uniformly distributed on [0,1]. We denote byUn:i the order statistics of the sample:

0 =Un:0 ≤Un:1≤ · · · ≤Un:n ≤Un:n+1= 1, which are distinct with probability one. For notational convenience, put

un:i = EUn:i= i

n+ 1, i= 0,1, . . . , n+ 1.

The polygonal uniform quantile process χ^pg_n is the random polygonal line on [0,1] which is affine on each [u_n:i−1, un:i],i= 1, . . . , n+ 1 and satisfies

χ^pg_n (un:i) =√

n(Un:i−un:i), i= 0,1, . . . , n+ 1. (3.34) As a corollary of Theorem 10 in [7], for any 0< α < 1/2,χ^pg_n converges weakly in H_α^o[0,1] to the Brownian bridgeB. Theorem 1.1 enables us to complete this convergence by the following convergence of moments.

Corollary 3.2. Let χ^pg_n be the polygonal uniform quantile process defined above. Then for every 0≤α <1/2, and every r >0,

n→∞lim Ekχ^pg_n k^r_α= EkBk^r_α, (3.35) where B is the Brownian bridge on [0,1].

Proof. We recall the distributional equality (see e.g.[15]) (Un:1, . . . , Un:n)=^d S₁

Sn+1

, . . . , S_n Sn+1

, (3.36)

where Sk =X1+· · ·+Xk and the Xk’s are i.i.d 1-exponential random variables. Following [7], introduce the polygonal process ζn which is affine on each interval [u_n:i−1, un:i],i= 1, . . . , n+ 1 and such that

ζ_n(u_n:i) =√ n S_i

Sn+1

−u_n:i

, i= 0,1, . . . , n+ 1. (3.37) It easily follows from the distributional equalities (3.36) thatχ^pg_n andζn have the same distribution as random elements in anyH^o_α[0,1] (0≤α <1). Hence it is enough to prove the convergence of moments (3.35) withχ^pg_n replaced byζn.

PuttingX_i⁰=Xi−EXi (note that EX₁⁰²= 1) andS⁰_k=Sk−ESk, we consider also the normalized partial sums polygonal process Ξn built on the S_k⁰’s, i.e. the random polygonal line with vertices (k/n, n^−1/2S_k⁰), k= 0,1, . . . , n. As shown in the proof of Theorem 10 in [7],

ζn(t) =

pn(n+ 1) Sn+1

Ξn+1(t)−tΞn+1(1)

, t∈[0,1]. (3.38)

Since we already know thatζn, as well asχ^pg_n , converges weakly toBin anyH^o_α[0,1] for 0≤α <1/2, it remains just to check the uniform integrability of the sequence (kζnk^r_α)n≥1, for which it suffices to prove that for some s > r,

sup

n≥1

Ekζnk^s_α<∞. (3.39)

Define for each x∈C[0,1], the function x^br: [0,1]→R, t7→x(t)−tx(1). It is easily seen that the bridge linear operatorx7→x^br mapsH^o_α[0,1] into itself and that for everyx∈ H_α^o[0,1],kx^brkα≤2kxkα. Now

Ekζnk^s_α≤E

n+ 1 Sn+1

kΞ^br_n+1kα

s

≤2^sE^1/2

n+ 1 Sn+1

2s

E^1/2kΞn+1k^2s_α.

To obtain (3.39) with anys > r, we just note the following facts. First, by elementary computation, E

n+ 1 Sn+1

2s

= (n+ 1)^2s

n! Γ(n+ 1−2s) =(n+ 1)^2s(n−2s)!

n! −−−−→

n→∞ 1.

Next, since X₁⁰ has finite moments of every order, EkΞn+1k^2s_α converges to EkWk^2s_α by Theorem1.1.

4. Weighted scan statistics

In this section we consider several weighted scan type statistics. Forα≥0, define Mn,α= max

1≤`≤n

1

`^α max

0≤k≤n−`|Sk+`−Sk|,

and

Tn,α= max

1≤`≤n

1

[`(1−`/n)]^α max

0≤k≤n−`|Sk+`−Sk−(`/n)Sn|.

Theorem 4.1. Letp >2. If E|X1|^p<∞, then for any0≤α≤1/2−1/pand any 0≤r≤p,

n→∞lim n^{−r(1/2−α)}EM_n,α^r = Eω_α^r(W,1); (4.1) and

n→∞lim n^{−r(1/2−α)}ET_n,α^r = ET_α^r(B), (4.2) where

Tα(B) = sup

0<h<1

1

[h(1−h)]^α sup

0≤t<1−h

|B(t+h)−B(t)|

andB = (Bt=Wt−tW1,0≤t≤1) denotes the Brownian bridge on[0,1].

Proof. By Lemma2.2we have

n^−1/2+αM_n,α=ω_α(n^−1/2ξ_n,1) and the convergence (4.1) follows from Theorem1.1.

To prove (4.2), we use the representation

n^−1/2+αTn,α=gn(n^−1/2ξn) =g(n^−1/2ξn) +oP(1), (4.3) which is explained in details in [14]. Here the functionalg is defined by

g:H^o_α[0,1]→R⁺, x7→ sup

0<h<1

1

[h(1−h)]^α sup

0≤t<1−h

|x(t+h)−x(t)−hx(1)|

and (gn)n≥1 is an equicontinuous sequence of non negative, subadditive, homogeneous (gn(cx) = |c|gn(x)) functionalsH^o_α[0,1]→R+converging pointwise tog onH^o_α and verifying for some positive constantC

0≤g_n(x)≤g(x)≤Ckxk_α. (4.4)

By the H¨olderian invariance principle [12], continuous mapping and Slutsky lemma, (4.3) provides the con- vergence in distribution of gn(n^−1/2ξn) to g(W). Then in view of (4.4), Theorem 1.1 gives (4.2) since g(W) =Tα(B).

Theorem 4.2. Letp >0. Assume thatX1∈RVpandEX1= 0whenp >1. Then forα >max{0,1/2−1/p}, and any 0≤r < pit holds

n→∞lim b^−r_n EM_n,α^r = EY_p^r (4.5)

and

n→∞lim b^−r_n ET_n,α^r = EY_p^r, (4.6)

where bn is defined by (1.13)andYp has the Fr´echet distribution with exponentp.

Proof. From [9] we know, that

b⁻¹_n M_n,α−−−−→^d

n→∞ Y_p.

Hence, in order to prove convergence of moments we need to check uniform integrability of (b^−r_n M_n,α^r ) for each 0< r < p. Actually it is enough to prove that for each 0< r < p,

sup

n≥1

E[b⁻¹_n M_n,α]^r<∞. (4.7)

And to establish (4.7) it is clearly sufficient to prove that for some positive constant c and some integer n0

(possibly depending onr),

sup

n≥n0

Z ∞ c

y^r−1P(Mn,α> bny) dy <∞. (4.8)

To this aim we shall use the following lemma (see Lem. 3.3 in [9]).

Lemma 4.3. Assume that(Y_i, i≥1) are i.i.d. random variables. Then for ally >0 andn≥1,

P max

1≤`≤n`^−α max

1≤k≤n

k+`

X

i=k+1

Y_i

> y

!

≤2

logn

X

j=1

2^jP max

1≤k≤n2^−j

k

X

i=1

Y_i

> y(n2^−j)^α

! .

We shall estimate for y >0 the probability

P(y) = P(b⁻¹_n M_n,α> y).

To this aim, we introduce the truncated random variables

Xe_k=X_k1_{|Xk|≤ybn}, X_k⁰ =Xe_k−EXe_k, k= 1, . . . , n.

Then

P(y)≤P

max

1≤k≤n|Xk|> yb_n

+ P(b⁻¹_n Mf_n,α> y),

where Mfn,α is defined as Mn,α substituting Xk by Xek. Let M_n,α⁰ be defined as Mn,α substituting Xk byX_k⁰. We have

b⁻¹_n Mfn,α=b⁻¹_n max

1≤`≤n`^−α max

0≤k≤n−`|Sek+`−Sek|

≤b⁻¹_n M_n,α⁰ +b⁻¹_n max

1≤`≤n`^−α max

0≤k≤n−`

k+`

X

i=k+1

EXek

=b⁻¹_n M_n,α⁰ +b⁻¹_n n^1−α|EXe1|.

SinceX₁is regularly varying with index p, Lemma 2.4gives the estimates

E|X1|1_{|X1|≤ybn}≤cpn⁻¹bny^1−p ifp <1, (4.9) E|X1|1_{|X1|>ybn}≤cpn⁻¹bny^1−p ifp >1. (4.10) When 0< p <1, (4.9) gives

b⁻¹_n n^1−α|EXe1| ≤cpn^−αy^1−p≤ 1

2y, forn≥c^1/α_p andy≥2^1/p. (4.11) In the case where p >1,X1 is integrable and EX1= 0, hence EX11_{|X1|≤ybn}=−EX11_{|X1|>ybn}, so (4.10) leads also to (4.11). It follows that for n≥c^1/αp andy≥2^1/p,

P(b⁻¹_n Mfn,α> y)≤P(b⁻¹_n M_n,α⁰ > y/2). (4.12) Consequently,

P(y)≤P

1≤k≤nmax |Xk|> ybn

+ P(b⁻¹_n M_n,α⁰ > y/2).

By Lemma4.3, Markov and Doob inequalities with q > p,

P(b⁻¹_n M_n,α⁰ > y/2)≤2

logn

X

j=1

2^jP max

1≤k≤n2^−j

k

X

i=1

X_i⁰

>(n2^−j)^αybn/2

!

≤2 2q

q−1 ^{q logn}

X

j=1

2^j(n2^−j)^−αqy^−qb^−q_n E

n2^−j

X

i=1

X_i⁰

q

. (4.13)

By Rosenthal’s inequality,

E

n2^−j

X

i=1

X_i⁰

q

≤C_q (n2^−j)^q/2(EX₁⁰²)^q/2+n2^−jE|X₁⁰|^q .

By iii) in Lemma 2.4, assumingq >1 EX₁⁰²≤ 8p

|p−2|b²_n1

ny^2−p, E|X₁⁰|^q ≤p2^q+1 q−pb^q_n1

ny^q−p, whence

E

n2^−j

X

i=1

X_i⁰

q

≤C_q

(8p)^q/2

|p−2|^q/22^−jq/2b^q_ny^q−pq/2+p2^q+1

q−p2^−jb^q_ny^q−p

.

It is worth noticing here that fors >0,

logn

X

j=1

2^js≤ 1 1−2^−sn^s,

recalling that logndenotes the dyadic logarithm: 2^logn=n. Moreover we can always chooseqsuch that q >max(2, p) and 1 + (α−1/2)q >0.

Indeed if α≥1/2, the second inequality above is automatically satisfied for anyq >0 which leaves us free to choose q >max(2, p). If 1/2> α >max(1/2−1/p,0), one can findq so that

α >1/2−1/q >max(1/2−1/p,0).

The first inequality above gives 1 + (α−1/2)q >0 while the second givesq >max(2, p).

Now, going back to (4.13), we obtain for largen, uniformly iny∈[2^1/p,∞), P b⁻¹_n M_n,α⁰ > y/2

≤Cy^−p,

with

C:= 2 2q

q−1

^q 8p

|p−2|

^q/2

+p2^q+1 q−p

! 2^qα 2^qα−2^−1+q/2.