Strong approximation of the empirical distribution function for absolutely regular sequences in R^d.

Strong approximation of the empirical distribution function for absolutely regular sequences in R

.

08/03/2013

J´erˆome Dedecker^a, Florence Merlev`ede^b and Emmanuel Rio^c.

a Universit´e Paris Descartes, Laboratoire MAP5, UMR 8145 CNRS, 45 rue des Saints-P`eres, F-75270 Paris cedex 06, France. E-mail: jerome.dedecker@parisdescartes.fr

bUniversit´e Paris-Est, LAMA (UMR 8050), UPEMLV, CNRS, UPEC, F-77454 Marne-La-Vall´ee, France.

E-mail: florence.merlevede@univ-mlv.fr

bUniversit´e de Versailles, Laboratoire de math´ematiques, UMR 8100 CNRS, Bˆatiment Fermat, 45 Avenue des Etats-Unis, 78035 Versailles, FRANCE. E-mail: rio@math.uvsq.fr

Running head: Strong approximation for the empirical process.

Key words: Strong approximation, Kiefer process, empirical process, stationary sequences, absolutely regular sequences.

Mathematical Subject Classification(2010): 60F17, 60G10.

Abstract

We prove a strong approximation result with rates for the empirical process associated to an ab- solutely regular stationary sequence of random variables with values in R^d. As soon as the absolute regular coefficients of the sequence decrease more rapidly thann^1−pfor somep∈]2,3], we show that the error of approximation between the empirical process and a two-parameter Gaussian process is of order n^1/p(logn)^λ(d) for some positive λ(d) depending on d, both in L¹ and almost surely. The power of n being independent of the dimension, our results are even new in the independent setting, and improve earlier results. In addition, for absolutely regular sequences, we show that the rate of approximation is optimal up to the logarithmic term.

1 Introduction

Let (X_i)_i∈Zbe a strictly stationary sequence of random variables inR^d equipped with the usual product order, with common distribution functionF. Define the empirical process of (X_i)_i∈Z by

RX(s, t) = X

1≤k≤t

1X_k≤s−F(s)

, s∈R^d, t∈R⁺. (1.1)

In this paper we are interested in extensions of the results of Kiefer for the process R_X to absolutely regular processes. Let us start by recalling the known results in the case of independent and identically distributed (iid) random variables X_i. Kiefer (1972) obtained the first result in the case d = 1. He constructed a continuous centered Gaussian processKX with covariance function

E KX(s, t)KX(s⁰, t⁰)

= (t∧t⁰)(F(s∧s⁰)−F(s)F(s⁰)) in such a way that

sup

(s,t)∈R×[0,1]

|R_X(s,[nt])−K_X(s,[nt])|=O(a_n) almost surely, (1.2) withan=n^1/3(logn)^2/3. The two-parameter Gaussian processKXis known in the literature as the Kiefer process. Cs¨org¨o and R´ev´esz (1975a) extended Kiefer’s result to the multivariate case. For iid random variables with the uniform distribution over [0,1]^d, they obtained (1.2) withan =n(d+1)/(2d+4)(logn)². Next they extended this result to iid random variables inR^d with a density satisfying some smoothness conditions (see Cs¨org¨o and R´ev´esz (1975b)).

In the univariate case, a major advance was made by Koml´os, Major and Tusn´ady (1975): they obtained (1.2) withan= (logn)²(we refer to Castelle and Laurent-Bonvalot (1998) for a detailed proof)

via a new method of construction of the Gaussian process. Concerning the strong approximation by a sequence of Gaussian processes in the cased= 2, Tusn´ady (1977) proved that when the random variables X_iare iid with uniform distribution over [0,1]², then one can construct a sequence of centered continuous Gaussian processes (G_n)_n≥1 inR² with covariance function

Cov(G_n(s), G_n(s⁰)) =n((s₁∧s⁰₁)(s₂∧s⁰₂)−s₁s₂s⁰₁s⁰₂), withs= (s₁, s₂) ands⁰= (s⁰₁, s⁰₂), such that

sup

s∈[0,1]²

|RX(s, n)−Gn(s)|=O(log²n) almost surely. (1.3) Adapting the dyadic method of Koml´os, Major and Tusn´ady (sometimes called Hungarian construc- tion), several authors obtained new results in the multivariate case. For iid random variables inR^d with distribution with dependent components (without regularity conditions on the distribution), Borisov (1982) obtained the almost sure rate of approximation O(n(d−1)/(2d−1)logn) in the Tusn´ady strong ap- proximation. Next, starting from the result of Borisov (1982), Cs¨org¨o and Horv´ath (1988) obtained the almost sure rate O(n(2d−1)/(4d)(logn)^3/2) for the strong approximation by a Kiefer process. Up to our knowledge, this result has not yet been improved in the case of general distributions with dependent components. Ford≥3 and Tusn´ady’s type results, Rio (1994) obtained the rateO n^(d−1)/(2d)(logn)^1/2 for random variables with the uniform distribution or more generally with smooth positive density on the unit cube (see also Massart (1989) in the uniform case). Still in the uniform case, concerning the strong approximation by a Kiefer process, Massart (1989) obtained the almost sure rateO n^d/(2d+2)(logn)²

for anyd≥2, which improves the results of Cs¨org¨o and R´ev´esz (1975a). In fact the results of Massart (1989) and Rio (1994) also apply to Vapnik-Chervonenkis classes of sets with uniformly bounded perimeters, such as the class of Euclidean balls. In that case, Beck (1985) proved that the error term cannot be better than n^(d−1)/(2d). Consequently the result of Rio (1994) for Euclidean balls is optimal, up to the factor √

logn. However, there is a gap in the lower bounds between the class of Euclidean balls and the class of orthants, which corresponds to the empirical distribution function. Indeed, concerning the lower bounds in Tusn´ady’s type results, Beck (1985) showed that the rate of approximation cannot be less than cd(logn)^(d−1)/2 wherecd is a positive constant depending ond. To be precise, he proved (see his Theorem 2) that when the random variablesXiare iid with the uniform distribution over [0,1]^d, then for any sequence of Brownian bridges (Gn)n≥1in R^d,

P

sup

s∈[0,1]^d

|RX(s, n)−Gn(s)| ≤cd(logn)^(d−1)/2

< e⁻ⁿ. Beck’s result implies in particular that, for anyn≥2,

(logn)^(1−d)/2E

sup

s∈[0,1]^d

|RX(s, n)−Gn(s)|

≥cd/2. (1.4)

The results of Beck (1985) motivated new research in the multivariate case. For the empirical distribution function and Tusn´ady type results, Rio (1996) obtained the rateO n^5/12(logn)^c(d)

for random variables with the uniform distribution, where c(d) is a positive constant depending on the dimension d, without the help of Hungarian construction. Here the power ofndoes not depend on the dimension: consequently this result is better than the previous results if d ≥7. It is worth noticing that, although this subject has been treated intensively, up to now, the best known rates for the strong approximation by a Kiefer process in the multivariate case are of the order n^1/3 ford= 2, up to some power of logn, even in the uniform case. Furthermore these rates depend on the dimension, contrary to the result of Rio (1996) for Tusn´ady type approximations.

We now come to the weakly dependent case. Contrary to the iid case, there are only few results concerning the rate of approximation. Up to our knowledge, when (X_i)_i∈Z is a geometrically strongly mixing (in the sense of Rosenblatt (1956)) strictly stationary sequence of random variables in R^d, the best known result concerning rates of convergence, is due to Doukhan and Portal (1987) stating that one can construct a sequence of centered continuous Gaussian processes (Gn)_n≥1 in R^d with common covariance function

Λ(s, s⁰) =X

k∈Z

Cov(1X₀≤s,1X_k≤s⁰),

such that the Ky-Fan distance between{n^−1/2RX(s, n), s∈R^d} and{Gn(s), s∈R^d} iso(n^−a) for any a < 1/(15d+ 12). In their paper, they also give some rates in case of polynomial decay of the mixing coefficients. Concerning the strong approximation by a Kiefer process in the stationary and strongly mixing case, Theorem 3 in Dhompongsa (1984) yields the rate O(n^1/2(logn)^−λ) for some positive λ, under the strong mixing conditionα_n=O(n^−a) for somea >2 +d, improving slightly previous results of Phillip and Pinzur (1980) (hereλdepends onaandd).

Strong mixing conditions seem to be too poor to get optimal rates of convergence. Now recall that, for irreducible, aperiodic and positively recurrent Markov chains, the coefficients of strong mixing and the coefficients of absolute regularity are of the same order (see for example Rio (2000), chap. 9). Since absolute regularity is a stronger condition, it is more convenient to consider absolute regularity, at least in the case of irreducible Markov chains. Let

β(A,B) = 1

2sup X

i∈I

X

j∈J

|P(Ai∩Bj)−P(Ai)P(Bj)| ,

the maximum being taken over all finite partitions (Ai)_i∈I and (Bi)_i∈J of Ω respectively with elements in A and B. For a strictly stationary sequence (Xk)_k∈Z, letF0 =σ(Xi, i ≤0) andGk =σ(Xi, i≥k).

The sequence (Xk)_k∈Z is said to be absolutely regular in the sense of Rozanov and Volkonskii (1959) or β-mixing if

βn=β(F0,Gn)→0, asn→ ∞.

Concerning the strong approximation by a Kiefer process in the stationary andβ-mixing case, Theorem 1 in Dhompongsa (1984) yields the rateO(n^(1/2)−λ) for some positiveλ, under the assumptionβn=O(n^−a) for some a >2 +d. Nevertheless this mixing condition is clearly too restrictive andλis not explicit.

We now come to our results. For absolutely regular sequences, the finite dimensional convergence of {n^−1/2RX(s, n) :s∈R^d}to a Gaussian process holds under the summability condition P

k≥0βk <∞, and this condition is sharp. Rio (1998) proved that this summability condition also implies the functional central limit theorem for {n^−1/2RX(s, n) :s∈R^d}in the sense of Dudley (1978) for anyd≥1. Assume now that the strongerβ-mixing condition

βn =O(n^1−p) for somep >2 (1.5)

holds true. In Section 2, we shall prove that, in the cased= 1, one can construct a stationary absolutely regular Markov chain satisfying (1.5), whose marginals are uniformly distributed over [0,1], and such that, for any construction of a sequence (G_n)_n>0 of continuous Gaussian processes on [0,1],

lim inf

n→∞(nlogn)^−1/pE

sup

s∈(0,1]

|RX(s, n)−Gn(s)|

>0.

Concerning the upper bound, Dedecker, Merlev`ede and Rio (2012) obtain a strong approximation by a Kiefer process under a weak dependence condition which is implied by the above condition, with a power-type rateO(n^(1/2)−δ) for some positiveδ depending onp. Nevertheless their result holds only for d = 1 and the value of δ is far from the optimal value (1/2)−(1/p). This gap motivates the present work. In Section 3, we prove that, if (X_i)_i∈Z is a strictly stationary sequence of random variables inR^d satisfying (1.5) forp∈]2,3], there exists a two-parameter continuous (with respect to the pseudo metric defined by (3.1)) Kiefer type processK_X such that

E

sup

s∈R^d,t∈[0,1]

|RX(s,[nt])−KX(s,[nt])|

=O(n^1/p(logn)^λ(d)). We also prove that, for another Kiefer processK_X,

sup

s∈Rd k≤n

|RX(s, k)−KX(s, k)|=O(n^1/p(logn)^λ(d)+ε+1/p) almost surely, for anyε >0.

More precisely, the covariance function Γ_X ofK_X is given by

Γ_X(s, s⁰, t, t⁰) = min(t, t⁰)Λ_X(s, s⁰) where Λ_X(s, s⁰) =X

k∈Z

Cov(1_X0≤s,1_Xk≤s⁰). (1.6)

Our proofs can be adapted to iid random variables with values in R^d and arbitrary distribution function, for any d≥2, yielding the error term in the strong approximation O(n^1/3(logn)^ε+(2d+4)/3) in the almost sure strong approximation by a Kiefer process. This result improves the results of Cs¨org¨o and Horv´ath (1988) for general distributions as soon asd≥2 and the result of Massart (1989) concerning the specific case of the uniform law as soon as d≥3 (recall that Massart’s rate isO(n^1/3(logn)²) ford= 2 andO(n^3/8(logn)²) ford= 3).

We now describe our methods of proofs. We shall apply the conditional version of the transport theorem of Kantorovich and Rubinstein to the trajectories to get a bound on the error. However, in the dependence setting, we do not apply the transport theorem directly. Indeed, we start by approxi- mating the initial process R_X by a Gaussian process with the same covariance structure as R_X, using the conditional Kantorovich-Rubinstein theorem applied in the space of trajectories, together with the Lindeberg method. Next, we use a martingale method to approximate the Gaussian process by the Kiefer process. This step is due to the fact that the Lindeberg method in the space of trajectories applies only to processes with the same covariance structure. In all these steps the error terms can be bounded by n^1/p up to some power of logn, which leads to the (nearly) optimal rates of convergence for absolutely regular sequences. Note that the Lindeberg method in the space of trajectories was introduced by Sakhanenko (1987) in the real case to bound up the Prokhorov distance between the partial sum process and the Brownian motion. This result was then extended to random vectors in Banach spaces by Bentkus and Lyubinskas (1987) using smoothing techniques introduced by the first author in his doctoral dissertation.

Later, Sakhanenko (2000) improved the results of Bentkus and Lyubinskas (1987) in the specific case of the L^∞-norm, yielding efficient estimates under some assumptions on the moments of order two and three of the Euclidean norms of the random vectors. Sakhanenko (1988, 2000) also gives some results for martingale differences under very restrictive assumptions on the conditional moments of order two. In our opinion, the smoothing technique used in Sakhanenko (2000) is not suitable in the dependent case.

Indeed the assumption on the conditional moments cannot be relaxed.

Our paper is organized as follows. In Section 2 we give an example of absolutely regular process for which we can derive lower bounds for the rates of approximation by any continuous Gaussian processes.

In Section 3 we formulate our main results concerning the upper bounds for the rates of approximation both in the dependent setting and in the independent one. The proofs of these results are given in Sections 4 and 5. Section 6 is devoted to the very technical proofs of key intermediate lemmas leading to our main results. Finally, in Section 7 we collect some auxiliary assertions and general facts.

2 Lower bounds for the rate of approximation

In this section, we give an example of a stationary absolutely regular Markov chain with state space [0,1]

and absolute regularity coefficients βk of the order ofk^1−p forp > 2 which has the following property:

with probability one, the error in the strong approximation by Gaussian processes is bounded from below by (nlogn)^1/p, for any construction of a sequence of continuous Gaussian processes, whereas theL¹-error is bounded from below by n^1/p.

Theorem 2.1 For any p >2, there exists a stationary Markov chain(Xi)_i∈Z of random variables with uniform distribution over[0,1]andβ-mixing coefficients(βn)n>0, such that:

(i) 0<lim infn→+∞n^p−1βn≤lim sup_n→+∞n^p−1βn<∞.

(ii) There exists a positive constantCsuch that, for any construction of a sequence(Gn)n>0of continuous Gaussian processes on [0,1]

(a) lim inf

n→∞ n^−1/pE

sup

s∈(0,1]

|RX(s, n)−Gn(s)|

≥C . Furthermore

(b) lim sup

n→∞

(nlogn)^−1/p sup

s∈(0,1]

|RX(s, n)−Gn(s)|>0 almost surely.

Before proving this result, we give a second theorem, which proves that the strong approximation of partial sums of functionals of the chain holds with the same error term.

Theorem 2.2 Let (Xi)i∈Z be the stationary Markov chain defined in Theorem 2.1 and let f be a map from [0,1]toR, with continuous and strictly positive derivativef⁰ on [0,1]. Let

S_n(f) =

n

X

k=1

f(X_k)−n Z 1

0

f(t)dt . Then the series Varf(X0) + 2P

k>0Cov(f(X0), f(Xk)) is absolutely convergent to some nonnegative σ²(f). Furthermore, for 2 < p < 3 and any positive ε, one can construct a sequence of iid Gaussian random variables (g⁰_k)_k>0 with lawN(0, σ²(f))such that

(a) Sn(f)−

n

X

k=1

g_k⁰ =o(n^1/pp

logn(log logn)^(1+ε)/p)almost surely.

In addition, for any p > 2 and any stationary and Gaussian centered sequence (gk)_k∈Z with convergent series of covariances,

(b) lim sup

n→∞

(nlogn)^−1/p

Sn(f)−

n

X

k=1

gk

>0 almost surely.

Note that Part (a) of this theorem was proved in Merlev`ede and Rio (2012). Part (b) proves that the result in Merlev`ede and Rio (2012) is optimal up to the factor (logn)(1/2)−(1/p)(log logn)^(1+ε)/p. It is worth noticing that the power of the logarithm in the loss tends to 0 as ptends to 2.

Proof of Theorem 2.1. The sequence (X_i)_i∈Zis defined from a strictly stationary Markov chain (ξ_i)_i∈Z on [0,1] as in Rio (2000), Section 9.7. Letλbe the Lebesgue measure,a=p−1 andν = (1 +a)x^a1_[0,1]λ.

The conditional distribution Π(x, .) ofξn+1, given (ξn=x), is defined by Π(x, .) = Π(δx, .) = (1−x)δx+xν ,

where δx is the Dirac measure at point x. Then the β-mixing coefficients (βn)n>0 of the stationary chain (ξi)_i∈Z with transition probability Π(x, .) satisfy (i) of Theorem 2.1 (see Rio (2000), Section 9.7).

Moreover, the stationary distribution π has distribution function F(x) =x^a, and consequently setting Xi=ξ_i^awe obtain a stationary Markov chain (Xi)i∈Zof random variables with uniform distribution over [0,1] and adequate rate ofβ-mixing. Define then the empirical measurePn by

P_n=n⁻¹

n

X

i=1

δ_Xi.

The regeneration times (T_k)_k of the Markov chain (ξ_i)_i∈Z are defined by induction as follows: T₀ = inf{n >0 :ξ_n 6=ξ_n−1} and T_k = inf{n > Tk−1 :ξ_n 6=ξ_n−1}. Letτ_k =T_k+1−T_k. It follows that the empirical measure at time T_k−1 satisfies the equality

(Tk−1)PT_k−1= (T0−1)δX₀+

k−1

X

j=0

τjδX_Tj. (2.1)

Consequently, for n≥Tk−1 the maximal jump ofRX(s, n) is greater than

∆k = max

j∈[0,k−1]τj. Next, from the continuity of Gn, forn≥Tk−1,

Dn:= sup

s∈(0,1]

|RX(s, n)−Gn(s)| ≥∆k/2. (2.2) Now the sequence (∆k)k is a nondecreasing sequence of positive integers. Notice that the random vari- ables (ξT_k, τk) are independent and identically distributed. Moreover ξT_k has the distributionν and the conditional distribution ofτk given (ξT_k =x) is the geometric distributionG(x). Hence,

P(∆k≤m) = (P(τ0≤m))^k,

and

P(τ0> m) = (1 +a)m^−1−a Z m

0

(1−y/m)^my^ady∼(1 +a)Γ(1 +a)m^−1−a asm↑ ∞. (2.3) From the above facts, it follows that

E(∆_k) = X

m≥0

P(∆_k > m)≥c_pk^1/p. (2.4) In the same way, one can prove that

k∆kk(2+a)/2≤C_pk^1/p. (2.5)

Here c_p andC_p are positive constants depending only onp.

Now, by the strong law of large numbers T_k/k converges to E(τ₀) almost surely, and therefore in probability. Consequently, for k=k_n= [n/(2E(τ₀))],

n→∞lim P(n < T_kn−1) = 0. Now

2E(D_n)≥E(∆_kn1_n≥Tkn−1)≥E(∆_kn)− k∆_knk2+a

2 (P(n < T_kn−1))^2+aa .

From (2.5), (2.4) and the above inequality, we get that, there exists some positive constantC(depending onp) such that, fornlarge enough, E(D_n)≥Cn^1/p, which completes the proof of (a) of Theorem 2.1.

To prove (b), we note that, by (2.3) ,

P(τ_k >(klnk)^1/p)∼c_p/(klogk). (2.6) Since the regeneration times τk are independent, by the converse Borel-Cantelli lemma, it follows that

P(τk >(klogk)^1/p infinitely often ) = 1. (2.7) Hence, by (2.2),

lim sup

n

(nlnn)^−1/pDT_n−1≥(1/2) almost surely.

Both this inequality and the strong law of large numbers forTn then imply (b) of Theorem 2.1.

Proof of Theorem 2.2. Letbbe a real in ]0,1] such thatf(b)<R1

0 f(t)dt(note that such a positiveb exists). With the same notations as in the proof of the previous theorem, the random variables (XT_k, τk) are independent and identically distributed, and

P(τ0> m, XT₀< b) = (1 +a)m^−1−a

Z mb^1/a 0

(1−y/m)^my^ady∼(1 +a)Γ(1 +a)m^−1−a as m↑ ∞.

Consequently, by the converse Borel-Cantelli lemma,

P(τk>(klogk)^1/p andXT_k< b infinitely often ) = 1. (2.8) Since Tn/n converges toE(τ0) almost surely, it follows that, for some positive constant c depending on E(τ0),

lim sup

n

n+[c(nlogn)^1/p]

X

i=n+1

(f(b)−f(Xi))≥0 almost surely. (2.9) Consider now a stationary and Gaussian centered sequence (gk)_k∈Zwith convergent series of covariances.

If follows from both the Borel-Cantelli lemma and the usual tail inequality for Gaussian random variables that, for any positiveθ,

lim inf

n

n+[c(nlogn)^1/p]

X

i=n+1

(gi+θ)≥0 almost surely.

Taking θ = (R1

0 f(t)dt−f(b))/2 in the above inequality, we then infer from the two above inequalities that

lim sup

n→∞

1 [c(nlogn)^1/p]

n+[c(nlogn)^1/p]

X

i=n+1

gi+

Z 1 0

f(t)dt−f(Xi)

≥θ almost surely, which implies Theorem 2.2.

3 Upper bounds for the rate of approximation

In this section, we state the main result of this paper, which is a Kiefer type approximation theorem for absolutely regular sequences. In all this section, we assume that the underlying probability space (Ω,A,P), is rich enough to contain a sequence (U_i)_i∈Z = (η_i, δ_i, υ_i, _i)_i∈Z of iid random variables with uniform distribution over [0,1]⁴, independent of (Xi)_i∈Z.

Theorem 3.1 Let (Xi)_i∈Z be a strictly stationary sequence of random variables in R^d. Let Fj be the distribution function of thej-th marginal ofX0. Assume that the absolutely regular coefficients of(Xi)_i∈Z are such thatβn=O(n^1−p)for somep∈]2,3]. Then

1. for all(s, s⁰)inR^2d, the seriesΛX(s, s⁰)defined by (1.6) converges absolutely.

2. For any(s, s⁰)∈R^2d and(t, t⁰)in R⁺×R⁺, letΓX(s, s⁰, t, t⁰) = min(t, t⁰)ΛX(s, s⁰). There exists a centered Gaussian processKX with covariance function ΓX, whose sample paths are almost surely uniformly continuous with respect to the pseudo metric

d((s, t),(s⁰, t⁰)) =|t−t⁰|+

d

X

j=1

|F_j(s_j)−F_j(s⁰_j)|, (3.1) and such that

(a) E

sup

s∈R^d, k≤n

|R_X(s, k)−K_X(s, k)|

=O(n^1/p(logn)^λ(d)).

Furthermore, one can construct another centered Gaussian process KX with the above covariance function in such a way that

(b) sup

s∈R^d, k≤n

|RX(s, k)−KX(s, k)|=O(n^1/p(logn)^λ(d)+ε+1/p) almost surely, for any ε >0 In both itemsλ(d) = ^3d₂ + 2−^2+d_2p

1_p∈]2,3[+ 2 +^4d₃ 1p=3.

From the above theorem, in the independent setting, the error in theL¹ approximation is bounded up by n^1/3(logn)^2+4d/3, whereas the almost sure error is bounded up byn^1/3(logn)^ε+(9+4d)/3, for any ε >0. However, in that case, the powers of logncan be improved as follows.

Theorem 3.2 Let (Xi)_i∈Z be a sequence of independent and identically distributed random variables in R^d. Then one can construct a centered Gaussian process KX with covariance function

ΓX(s, s⁰, t, t⁰) = min(t, t⁰)(F(s∧s⁰)−F(s)F(s⁰))wheres∧s⁰= (min(s1, s⁰₁), . . . ,min(sd, s⁰_d)), whose sample paths are almost surely uniformly continuous with respect to the pseudo metricddefined in (3.1), and such that

(a) E

sup

s∈R^d, k≤n

|RX(s, k)−KX(s, k)|

=O(n^1/3(logn)^(2d+3)/3).

Furthermore, one can construct another centered Gaussian processKXwith the above covariance function in such a way that

(b) sup

s∈R^d, k≤n

|RX(s, k)−KX(s, k)|=O(n^1/3(logn)^+(2d+4)/3) almost surely, for anyε >0.

Recently, Merlev`ede and Rio (2012) obtained efficient strong approximation results for partial sums of real-valued random variables. In the bounded case, under the mixing condition βn =O(n^1−p), they obtain in their Theorem 2.1 (see Item 1(b)) the rate of almost sure approximationO(n^1/p(logn)^(1/2)+ε).

According to the results of Section 2, the power ofncannot be improved, contrary to the previous papers on the same subject. Starting from Theorem 3.1, we can derive an extension of this result to partial sums of random vectors inR^d, in the same way as Borisov (1983) derives strong approximation of partial sums from the strong Kiefer approximation of Koml´os, Major and Tusn´ady.

Corollary 3.1 Let (Xi)i∈Z be a strictly stationary and absolutely regular sequence of bounded random vectors in R^d. Assume that its absolutely regular coefficients are such that βn = O(n^1−p) for some p ∈]2,3]. Then the series of covariance matrices P

k∈ZCov(X₀, X_k) is convergent to a non-negative definite symmetric matrixΓ. Furthermore, there exists a sequence(Z_i)_i≥1of iid random vectors with law N(0,Γ)such that, setting ∆k =Pk

i=1(Xi−E(Xi)−Zi),

(a) E sup

k≤n

k∆_kk

=O(n^1/p(logn)^λ(d)).

In addition, there exists another sequence (Z_i)_i≥1 of iid random vectors with law N(0,Γ) such that, for any positive ε,

(b) sup

k≤n

k∆kk=o(n^1/p(logn)^λ(d)+ε+1/p)almost surely.

In both items,λ(d) is defined in Theorem 3.1.

Proof of Corollary 3.1. Adding some constant vector to the initial random vectors if necessary, we may assume that the components of the random vectors X_i are non-positive. For each integeri, we set X_i= (X_i⁽¹⁾, X_i⁽²⁾, . . . , X_i^(d)). From our assumptions, there exists some positive constantM such that, for any integeriand anyX_i belongs to [−M,0]^d. Then, for anyj in [1, d],

X_i^(j)=− Z 0

−M

1_X(j) i ≤tdt .

Let then K_X^(j)(t, k) = KX((1, . . . ,1, t,1, . . . ,1), k), where t is the j-th component. Define the random vectorsZk for any positive integerk by

Z_k^(j)=− Z 0

−M

K_X^(j)(t, k)−K_X^(j)(t, k−1)

dt for any j∈[1, d].

Then the so defined sequence (Zk)k>0is a Gaussian sequence (this means that, for any positive integern, (Z1, . . . , Zn) is a Gaussian vector) and, from the definition of the covariance ofKX, the random vectors Zk are not correlated. It follows that (Zk)k>0 is a sequence of independent Gaussian random vectors.

Now, from the definition ofZ_k,

Cov(Z_k^(j), Z_k⁽ⁱ⁾) =X

m∈Z

Z 0

−M

Z 0

−M

Cov(1_X(i)

0 ≤t,1_X(j)

m≤s)dtds . Hence, interverting the summation and the integral in the above formula, we get that

Cov(Z_k^(j), Z_k⁽ⁱ⁾) = X

m∈Z

Cov(X₀⁽ⁱ⁾, X_m^(j)) = Γij, which implies thatZ_k has the prescribed covariance. Next

X_k^(j)−E(X_k^(j))−Z_k^(j)= Z 0

−M

K_X^(j)(t, k)−K_X^(j)(t, k−1) +P(X_k^(j)≤t)−1_X(j) k ≤t

dt .

Let then ∆^(j)_k denote the j-th component of ∆k andR^(j)_X (t, k) =RX((1, . . . ,1, t,1, . . . ,1), k), wheret is thej-th component. From the above identity,

∆^(j)_k = Z 0

−M

(K_X^(j)(t, k)−R^(j)_X(t, k))dt . It follows that, for any integer j in [1, d],

sup

k≤n

|∆^(j)_k | ≤M sup

s∈Rd k≤n

|RX(s, k)−KX(s, k)|.

Part (a) (resp. Part (b)) of Corollary 3.1 follows then from both these inequalities and Part (a) (resp.

Part (b)) of Theorem 3.1.

4 Proof of Theorem 3.1

In this section we shall sometimes use the notation a_n b_n to mean that there exists a numerical constant C not depending on n such that a_n ≤ Cb_n, for all positive integers n. We shall also use the notationsF_k=σ(X_j, j≤k) andF_∞=W

k∈ZF_k.

For any (s, s⁰)∈R^2d, by using Lemma 7.4 withU =1X₀≤s, V =1X_k≤s⁰, r= 1 ands=∞, we get that

|Cov(1X0≤s,1_Xk≤s0)| ≤2β_k. SinceP

k≥0β_k<∞, Item 1 of Theorem 3.1 follows.

To prove Item 2, we first transform the random variablesX_i. With this aim, for anykinZand anyj in {1, . . . , d}, we denote byX_k^(j)thej-th marginal ofX_k. By Lemma 7.4 applied withp= 1 andq=∞, there exists some non-negative random variable b(x, k) with values in [0,1] such that, for any functions f :R^d →Rand g:R^d →[−1,1],

Cov(f(X0), g(Xk))≤2E(b(X0, k)|f(X0)|) andE(b(X0, k))≤βk. (4.1) Let then

bj(X0, k) =E(b(X0, k)|X₀^(j)).

We now introduce another probability on Ω. LetP^∗0,j be the probability on Ω whose density with respect to Pis

C_j⁻¹ 1 + 4

∞

X

k=1

bj(X0, k)

withCj= 1 + 4

∞

X

k=1

E(bj(X0, k)). (4.2) LetP_jbe the law ofX₀^(j). Notice then that the image measureP_j^∗ofP^∗0,j byX₀^(j)is absolutely continuous with respect to P_j with density

C_j⁻¹ 1 + 4

∞

X

k=1

bj(x, k)

. (4.3)

LetFP_j^∗be the distribution function ofP_j^∗, and letFP_j^∗(x−0) = sup_z<xFP_j^∗(z). Let (ηi)_i∈Zbe a sequence of iid random variables with uniform distribution over [0,1], independent of the initial sequence (Xi)i∈Z. Define then

Y_i^(j)=FP_j^∗(X_i^(j)−0) +ηi(FP_j^∗(X_i^(j))−FP_j^∗(X_i^(j)−0)) and Yi= Y_i⁽¹⁾, . . . , Y_i^(d)t

. (4.4) Note that (Yi)_i∈Z forms a strictly stationary sequence of random variables with values in [0,1]^d whose β-mixing coefficients are also of orderβn =O(n^1−p). In addition, it follows from Lemma F.1. in Rio (2000) thatX_i^(j)=F_P⁻¹∗

j (Y_i^(j)) almost surely, whereF_P⁻¹∗

j is the generalized inverse of the cadlag function F_P^∗

j. Hence

RX(·,·) =RY (FP₁^∗(·), . . . , FP_d^∗(·)),·

almost surely , where

RY(s, t) = X

1≤k≤t

1Y_k≤s−E(1Y_k≤s)

, s∈[0,1]^d, t∈R⁺. Furthermore

P(Y₀^(j)∈[a, b])≤CjP^∗0,j(Y₀^(j)∈[a, b]) =Cj(b−a) (4.5) where the last inequality comes from the fact that the random variables Y₀^(j) are uniformly distributed on [0,1] underP^∗0,j (see Item 1 of Lemma 5.1 in Dedecker, Merlev`ede and Rio (2012)). HenceY₀^(j) has a density with respect to the Lebesgue measure uniformly bounded byCj.

For a strictly stationary sequence (Zi)_i∈Z of random variables with values in R^d, let GZ be a two parameters Gaussian process with covariance function ΓZ defined as follows: for any (s, s⁰)∈ R^2d and (t, t⁰)∈(R⁺)²

ΓZ(s, s⁰, t, t⁰) = min(t, t⁰)ΛZ(s, s⁰) where ΛZ(s, s⁰) =X

k∈Z

Cov(1Z₀≤s,1Z_k≤s⁰), (4.6)

provided that ΛZ is well defined.

Let us now give an upper bound on the variance ofGY. Below, we prove that, for anyu= (u1, . . . , ud) andv= (v₁, . . . , v_d) in [0,1]^d and any positive integern,

Var GY(u, n)−GY(v, n)

≤nC(β)

d

X

i=1

|ui−vi|, whereC(β) = 1 + 4X

k>0

βk. (4.7)

Note that, if (u1, . . . , ud) = (FP₁^∗(s1), . . . , FP_d^∗(sd)) and (v1, . . . , vd) = (FP₁^∗(s⁰₁), . . . , FP_d^∗(s⁰_d)), then the following equalities hold in distribution: G_Y(u, n) =G_X(s, n) andG_Y(v, n) =G_X(s⁰, n). Hence

Var G_Y(u, n)−G_Y(v, n)

= Var G_X(s, n)−G_X(s⁰, n) . Now, by definition of the covariance of G_X,

n⁻¹Var GX(s, n)−GX(s⁰, n)

= lim

N→∞N⁻¹Var RX(s, N)−RX(s⁰, N) . Hence, by (4.1) and Corollary 1.4 in Rio (2000),

n⁻¹Var GX(s, n)−GX(s⁰, n)

≤E |1X₀≤s⁰−1X₀≤s|(1 + 4X

k>0

b(X0, k) . Now

|1X0≤s⁰−1X0≤s| ≤

d

X

j=1

|1_X(j)

0 ≤s⁰_j −1_X(j) 0 ≤sj|. Taking into account the definition ofP^∗0,j, it follows that

Var G_X(s, n)−G_X(s⁰, n)

≤n

d

X

j=1

C_jP^∗0,j |1_X(j)

0 ≤s⁰_j −1_X(j) 0 ≤sj|

.

Now, by definition of P^∗j, P^∗0,j |1_X(j)

0 ≤s⁰_j −1_X(j) 0 ≤sj|

=P^∗j(] min(s_j, s⁰_j),max(s_j, s⁰_j)]) =|ui−v_i|, whence

Var GX(s, n)−GX(s⁰, n)

≤n

d

X

j=1

Cj|uj−vj| ≤nC(β)

d

X

i=1

|ui−vi|, which completes the proof of (4.7) .

We shall prove in what follows that the conclusion of Theorem 3.1 holds for the stationary sequence (Yi)i∈Zand the associated continuous Gaussian processKY with covariance function ΓY defined by (4.6).

This will imply Theorem 3.1 by taking fors= (s1, . . . , sd), K_X(s, t) =K_Y (F_P^∗

1(s₁), . . . , F_P^∗

d(s_d)), t , since for any (s, s⁰) = (s₁, . . . , s_d),(s⁰₁, . . . , s⁰_d)

∈R^2d,

ΓX(s, s⁰, t, t⁰) = ΓY (FP₁^∗(s1), . . . , FP_d^∗(sd)),(FP₁^∗(s⁰₁), . . . , FP_d^∗(s⁰_d)), t, t⁰ . We start by a reduction to a grid and a discretization.

4.1 Reduction to a grid

In this section, we consider a strictly stationary sequence of random variables Zi in R^d with marginal distributions with support included in [0,1] and bounded densities. Our aim is to compare the maximal deviation over the unit cube with the maximal deviation over a grid. LetAn denote the set ofxin [0,1]^d

such that nxis a multivariate integer. The main result of the section is that, if the marginal densities are each bounded by M, then, for any integerk≤n,

sup

s∈[0,1]^d

|RZ(s, k)−GZ(s, k)| ≤ sup

s∈An

|RZ(s, k)−GZ(s, k)|+dM+ sup

ks−s⁰k∞≤1/n

|GZ(s, k)−GZ(s⁰, k)|, (4.8) where we recall that GZ is a two parameters Gaussian process with covariance function ΓZ defined by (4.6).

We now prove the above inequality. For each s= (s1, . . . , sd), we set π−(s) = n⁻¹([ns1], . . . ,[nsd]) and π+(s) =π−(s) +n⁻¹(1, . . . ,1). From the monotonicity of the multivariate distribution functionF and the empirical distribution functionF_k,

kFk(π₋(s))−kF(π+(s))≤RZ(s, k)≤kFk(π+(s))−kF(π₋(s)).

Next letFi denote the distribution function of the i-th coordinate ofZ0. From our assumption F_i(t)−F_i(s)≤M|t−s|.

Now, for any s= (s1, . . . sd) andt= (t1, . . . , td) withs≤t, 0≤F(t)−F(s)≤

d

X

i=1

(Fi(ti)−Fi(si)), which, together with the above inequality, ensures that

0≤kF(π₊(s))−kF(π₋(s))≤k(M d/n)≤M d sincek≤n. Hence

R_Z(π₋(s), k)−dM ≤R_Z(s, k)≤R_Z(π₊(s), k) +dM . Let then

D_Z(k, n) = sup

ks−s⁰k∞≤1/n

|G_Z(s, k)−G_Z(s⁰, k)|. Clearly

−GZ(π₋(s), k)−D_Z(k, n)≤ −GZ(s, k)≤ −GZ(π₊(s), k) +D_Z(k, n). Let ∆_Z=R_Z−G_Z. Adding the two above inequalities, we now get that

∆Z(π−(s), k)−dM−DZ(k, n)≤∆Z(s, k)≤∆Z(π+(s), k) +dM+DZ(k, n), which implies immediately (4.8).

4.2 Discretization

We now apply the inequality (4.8) to our problem. LetN∈N^∗ and letk∈]1,2^N+1]. We first notice that for any construction of a Kiefer processGY with covariance function ΓY defined by (4.6),

sup

1≤k≤2^N+1

sup

s∈[0,1]^d

RY(s, k)−GY(s, k)

≤ sup

s∈[0,1]^d

RY(s,1)−GY(s,1) +

N

X

L=0

DL(GY). (4.9) where

D_L(G_Y) := sup

2^L<`≤2^L+1

sup

s∈[0,1]^d

(R_Y(s, `)−R<sub>Y</sub>(s,2<sup>L</sup>))−(G<sub>Y</sub>(s, `)−G_Y(s,2^L))

. (4.10) Let then

D⁰_L(GY) = sup

2^L<`≤2^L+1

sup

s∈A₂L

|RY(s, `)−RY(s,2<sup>L</sup>)−(GY(s, `)−GY(s,2^L))|, (4.11) where we recall that A₂L is the set of x in [0,1]^d such that 2^Lx is a multivariate integer. Applying Inequality (4.8) withn= 2^L to the variablesZi=Y_i+2L and taking into account (4.5), we get that

DL(GY)≤D⁰_L(GY) +dC(β) + sup

2L <`≤2L+1 ks−s0 k∞≤2−L

|(GY(s, `)−GY(s,2<sup>L</sup>))−(GY(s<sup>0</sup>, `)−GY(s⁰,2^L))|. (4.12)

4.3 Construction of the Kiefer process

We shall construct in this section a Kiefer process K_Y with covariance function Γ_Y defined by (4.6) in such a way that forGY =KY, the terms involved in (4.12) can be suitably handled.

We start with some notations and definitions.

Definition 4.1 For two positive integersm andn, let Mm,n(R)be the set of real matrices withmlines and n columns. The Kronecker product (or Tensor product) of A= [ai,j] ∈ Mm,n(R) and B = [bi,j] ∈ Mp,q(R)is denoted byA⊗B and is defined to be the block matrix

A⊗B =







a_1,1B · · · a_1,nB

... ...

a_m,1B · · · a_m,nB





∈ Mmp,nq(R).

For any positive integer k, the k-th Kronecker power A^⊗k is defined inductively by: A^⊗1 = A and A^⊗k=ANA^⊗(k−1), andNk

i=1A_i=A₁N Nk i=2A_i

. We denote by A^tthe transposed matrix of A.

LetL∈N. For anyk∈Zand any`∈ {1, . . . , d}, letZ~<sub>k,`,L</sub>be the column vector of R²

L defined by Z~k,`,L=

1_Y

k+2L∈[0,1]^{`−1</sup>×[0,j2<sup>−L</sup>]×[0,1]<sup>d−`}

j=1,...,2^L

^t

. (4.13)

Let nowU~_k,L and U~_k,L⁽⁰⁾ be the column vectors ofR²

dL defined by

U~_k,L=

d

O

`=1

Z~_k,`,L and U~_k,L⁽⁰⁾ =U~_k,L−E(U~_k,L). (4.14)

For anyk∈ {1, . . . ,2^L}, let~ek,L be the column vector ofR²

L defined by

~e_k,L= 1_k≤m

m=1,...,2^L

^t

, (4.15)

and letS~L,d the column vector ofR²

(d+1)L

defined by S~L,d=

2^L

X

k=1

~

ek,L⊗U~_k,L⁽⁰⁾ :=

2^L

X

k=1

V~k,L. (4.16)

LetCL,dbe the covariance matrix ofS~L,d. It is then the matrix of M₂(d+1)L,2^(d+1)L(R) defined by C_L,d=E S~_L,dS~_L,d^t

. (4.17)

Let us now continue with some other definitions.

Definition 4.2 Let m be a positive integer. Let P₁ and P₂ be two probabilities on (R^m,B(R^m)). Let c be a distance on R^m associated to a norm. The Wasserstein distance of order1 betweenP₁ andP₂ with respect to the distancec is defined by

Wc(P1, P2) = inf{E(c(X, Y)),(X, Y)such thatX ∼P1, Y ∼P2}= sup

f∈Lip(c)

(P1(f)−P2(f)), where Lip(c)is the set of functions from R^minto Rthat are Lipschitz with respect toc; namely for any xandy of R^m,|f(x)−f(y)| ≤c(x, y).

Definition 4.3 Let m be a positive integer. For x = x⁽¹⁾, . . . , x^(2m)t

and y = y⁽¹⁾, . . . , y^(2m)t two vectors of R²

m, we define the following distance c_m(x, y) = sup

j∈{1,...,2^m}

|x^(j)−y^(j)|.