An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes

Let: Y =(Yi), where Yi = (Yi,1, ..., Yi,d), i = 1, 2, . . . , be a ddimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf F , and Fn (x) := 1 n ∑n i=1 I (Yi,1 ≤ x1, . . . , Yi,d ≤ xd) denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process Bn = √ n (Fn − F ) under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.


Introduction.
Since the publication of the celebrated papers by Brosamler [2], Lacey and Philipp [19] and Schatte [26]- [27], much attention has been drawn by the almost sure versions of distributional limit theorems, commonly referred to as the almost sure central limit theorems (AS-CLTs).The following property is investigated in the research devoted to the ASCLTs.Namely, suppose that: X 1 , X 2 , . . ., X k , . . .are some r.v.'s, f 1 , f 2 , . . ., f k , . . .denote some real-valued measurable functions, defined on R, R 2 , . . ., R k , . . ., respectively; we seek conditions under which the following almost sure (a.s.) convergence holds for some nondegenerate cdf G (1.1) lim w n I (f n (X 1 , . . ., X n ) ≤ x) = G (x) a.s., for all x ∈ C G , where: (w n ) is some sequence of weights, W N = N n=1 w n , I stands for the indicator function, and C G denotes the set of continuity points of G.
The principal purpose of our paper is to prove an empirical ASCLT, which comprises the case where some normalized empirical process is considered.We shall introduce the following notations.We denote by Y = (Y i ) = ((Y i,1 , . . ., Y i,d )), i = 1, 2, . . ., a d-dimensional, identically distributed, stationary, centered process with uniform marginal cdfs and a joint cdf F and by F n , the corresponding empirical cdf, i.e., the function such that, for any Furthermore, we define an empirical process B n by We also assume that a process (Y i ) satisfies the η-weak dependence assumption according to the condition stated in the work by Doukhan and Louhichi [6].Before we cite this condition, we shall introduce some additional notations.Namely, we define the Lipschitz modulus of a real function where In addition, we denote by Λ (1) the set of functions that are bounded by 1 and have finite Lipschitz modulus.Finally, we shall refer to the sequences We are now in a position to cite the weak dependence condition, originally stemming -as has been mentioned -from Doukhan and Louhichi [6].Definition 1.1.Let η = (η r ) r≥0 be a real, positive sequence decreasing to zero.We say that a d-dimensional process (ξ i ) i∈Z is Ψ, Λ (1) , η -weakly dependent, if for any r-distant finite sequences i = (i 1 , . . ., i u ) and j = (j 1 , . . ., j v ) and any functions h 1 , h 2 in Λ (1) , defined on R u , R v , respectively, we have For the examples of the d-dimensional processes satisfying the cited weak dependence condition with appropriate Ψ, Λ (1) and η, we refer the reader to Doukhan et al. [7].
Our goal is to show that, under suitable weak dependence assumptions on the process Y = (Y i ), the following convergence is satisfied in D [0, 1] d , d S -the space of cadlag functions endowed with the Skohorod metrics, (1.5) lim for any x ∈ [0, 1] d and any z ∈ R, where B n is an empirical process defined by (1.3) and B is a centered, Gaussian process, such that, for any vectors History of the empirical processes theory dates back to the 1930's and 1940's, when the study of the empirical distribution functions F n (x) and the corresponding empirical processes began.The two basic assertions concerning F n and B n are the Glivenko-Cantelli theorem and the Donsker theorem.The first of the mentioned results states that if X 1 , . . ., X n are i.i.d.real-valued r.v.'s with cdf F , then where F n stands for the corresponding empirical df.
An immediate application of this claim is Kolmogorov's goodness-of-fit test.The latter result -the Donsker theorem -states that B n =⇒ B ≡ U (F ) in D (R, • ∞ ), where U denotes a standard Brownian bridge process on [0,1], i.e., it is a zero-mean Gaussian process with the covariance given by E(U (s) U (t)) = min (s, t) − st.The convergence =⇒ means that for any bounded and continuous function g: D (R, • ∞ ) → R, where d → stands for the convergence in distribution.
In the 1950's and 1960's a need for generalizations of both of the cited theorems has naturally arisen.In particular, it became clear that if a more general sample space Π (such as, e.g., R d or some function spaces) is considered, then the empirical distribution function is not so easy to deal with.Two basic questions have been stated in this context: (i) for what classes C of subsets of the space Π or collections G of real-valued functions on Π does a natural extension of the Glivenko-Cantelli theorem hold?, (ii) for what classes C of subsets of the space Π or collections G of real-valued functions on Π does a natural generalization of the Donsker theorem hold?The most relevant answers to these questions have been given during the 1970's in the papers by Vapnik and Chervonenkis [32] and Dudley [8] with significant contributions in the 1970's, 1980's and 1990's, due to the works by Pollard [23]- [25], Giné and Zinn [15]- [16], Giné [17], Talagrand [29], Gänssler and Stute [13] and Gänssler [14], among others.It is especially seen through the publications of David Pollard that the theory of empirical processes provides a set of powerful tools allowing to prove principal assertions in the field of asymptotic statistics.In view of the importance of the theory of modern empirical processes in statistics, the book of van der Vaart and Wellner [31] is worthwhile to mention as well.In its third chapter, the usefulness of this theory in statistical applications related to, i.a., the M-estimators approach, the Bootstrap methods, the Two-sample problem and Minimax theorems, has been presented in detail.For a comprehensive overview on the theory of empirical processes and their applications (e.g., in the research of asymptotic normality of M-estimators and in penalized least-squares estimation), we refer the reader to the work by van de Geer [30].
The remainder of our work is structured as follows.In Section 2, we precisely state our main result.In Section 3, we prove some lemmas, which are necessary for the proof of our claim.Section 4 contains this proof, whereas in Section 5, some application of the established proposition in terms of copula processes is depicted.Finally, in Section 6, an example of a sequence satisfying the assumptions of Theorem 2.1 is presented.This example is given in the form of Theorem 6.1.

Main result.
For the functions h 1 , h 2 from Λ (1) -the class of functions defined on R u , R v , respectively, which are bounded by 1 and have finite Lipschitz modulus -we define the following mappings: Our major assertion is the following ASCLT for some empirical process.
an identically distributed, stationary, centered, d-dimensional process, with uniform marginal distributions and a joint cdf F (thus, Y i,j s have the same uniform distribution, for any i ∈ N and j = 1, . . ., d).Let in addition: F n be such as in (1.2) and denote an empirical cdf of Y, and B n be an empirical process defined by (1.3).Moreover, assume that there exists a constant C > 0, such that: (1) , η Y,r -weakly dependent, or (iv) Y is Ψ 4 , Λ (1) , η Y,r -weakly dependent, with the weak dependence coefficient satisfying η Observe that all of the functions Ψ 1 -Ψ 4 may be written in the form where c is some function defined on N 2 and µ is a locally bounded function on R 2 + .Furthermore, it is easy to check that Ψ 1 -Ψ 4 may be bounded by for some s > 0 and some t ∈ [0, 2], since: Therefore, if Y is either Ψ 1 , Λ (1) , η Y,r or Ψ 2 , Λ (1) , η Y,r or Ψ 3 , Λ (1) , η Y,r or Ψ 4 , Λ (1) , η Y,r -weakly dependent, then it is also Ψ 5 , Λ (1) , η Y,r -weakly dependent with This fact allows, e.g., to prove the weak convergence of the corresponding sequence of Ψ 5 , Λ (1) , η Y,r -weakly dependent processes (see, e.g., Proposition 1 in Doukhan and Louhichi [6]).
In addition, it is seen from Lemma 10 in [6] that some classes of Markov chains form the sequences of Ψ 6 , Λ (1) , θ r -weakly dependent sequences with Ψ 6 satisfying and hence, these Markov chains are Ψ 5 , Λ (1) , θ r -weakly dependent as well.
In the subsequent section, we state and prove two lemmas, which are needed for the proof of Theorem 2.1.

Auxiliary results.
The following lemma will be used in the proof of our main result.Lemma 3.1.Under the notations and assumptions of Theorem 2.1, we have for any x ∈ [0, 1] d and any Lipschitz and bounded by 1 function g )) be a d-dimensional, identically distributed, stationary, centered process with uniform marginal cdfs, a common joint cdf F and the corresponding empirical cdf F n (i.e., F , F n are such that F (x) := P (Y i,1 ≤ x 1 , . . ., Y i,d ≤ x d ), for any i ∈ N, where x = (x 1 , . . ., x d ) ∈ R d , and F n is given by (1.2)).Due to a definition of an empirical process where For each k ∈ N, we shall introduce the following notation where Furthermore, we define the term Z (f n , (2k + 1, . . ., 2k + n) , x) by where Put: s i (x) = s i , t i (x) = t i , i = 1, . . ., l, s = (s 1 , . . ., s l ), t = (t 1 , . . ., t l ).
We have, for any l ∈ N, This and a definition of the Lipschitz modulus in (1.4) yield Our aim now is to give a bound for the covariance Cov (g(B k (x)), g(B n (x))), where g is any Lipschitz function bounded by 1.We assume first that Y is Ψ 1 , Λ (1) , η Y,r -weakly dependent with η Y,r = O (r −α ) for some α > d + √ 1 + d 2 , which corresponds to the case (i) in the statement of Theorem 2.1.As, due to the derivation at the beginning of the current proof, Observe that Due to the stationarity of Y, we immediately obtain (3.7) A 3 = 0.
Let us now estimate the component A 1 in (3.6).In view of (3.2)-(3.3)and the definitions of Z and f k in (3.2), we have (1) , η Y,r -weakly dependent process with a dependence coefficient η Y,r satisfying η Y,r = O (r −α ) for some α > d + √ 1 + d 2 , then by the proof of Lemma 2.1 in Doukhan et al. [7] and the fact that g • f l ∈ Λ (1) , we obtain where η k = 3 (η Y,r d) 1/2 which -by an assumption on η Y,r in the statement of Theorem 2.1 -implies that η k ≤ Cr −α for some α > d + √ 1 + d 2 and some C > 0.
Additionally, in view of (3.4) and the fact that g is Lipschitz with some Lipschitz coefficient L, we get This, the fact that k < n and derivation (3.8) imply we may write that (3.9) By the similar reasoning as in the estimation of the expression in the case (i), it is easy to verify that in the case (ii), i.e., when Y is Ψ 2 , Λ (1) , η Y,r -weakly dependent with η Y,r = O (r −α ) for some α > d + in the case (iii), i.e., when Y is Ψ 3 , Λ (1) , η Y,r -weakly dependent with in the case (iv), i.e., when Y is Ψ 4 , Λ (1) , η Y,r -weakly dependent with η Y,r = O (r −α ) for some α > d + √ 1 + d 2 > 2, we get (3.12) By (3.9)-(3.12),we have the following estimate for A 1 in (3.6) Thus, it remains to give the bound for the penultimate component in (3.6).
Since g is a bounded by 1 and Lipschitz function, we obtain This and the stationarity of Y imply Furthermore, observe that assumption (2.1) is equivalent to the following relation and the requirements of Theorem 1 in Doukhan and Louhichi [6] are fulfilled with (X n ) = (I (Y n ≤ x) − F (x)), q = 2 and C r,q = C r,2 .Consequently, by virtue of the mentioned theorem, we get that there exists a positive constant M , not depending on k, such that The relations in (3.14)-( Thus, a desired relation in (3.1) follows from (3.5)-(3.7),(3.13) and (3.16).
The following claim will also be employed in the proof of Theorem 2.1.
Lemma 3.2.Under the notations (in particular for B n and B) and assumptions of Theorem 2.1, we have where D → denotes the convergence in distribution.
Proof.Let: f , g be the functions in Λ (1) , defined on R u , R v , respectively, i = (i 1 , . . ., i u ), j = (j 1 , . . ., j v ) be the sequences of natural indices, and s = (s 1 , . . ., s d ), t = (t 1 , . . ., t d ) stand for the elements in [0, 1] d .With reference to a d-dimensional process Y, we define: By Lemma 2.1 in Doukhan et al. [7], we have that if Y satisfies at least one of the weak dependence conditions (i)-(iv) from Theorem 2.1, then the following property is satisfied where Recall that the condition η Y,r ≤ C 2 r −2α 9d , for some α > d+ This and the relation in (3.18) imply that all the conditions of Theorem 1 in Doukhan et al. [7] are fulfilled and the convergence in (3.17) straightforwardly follows from this assertion.
We are now in a position to prove our main assertion.As has already been mentioned, the results stated as Lemmas 3.1-3.2are intensively used in its proof.

4.
Proof of the main result.The objective of this section is to present the proof of Theorem 2.1.
Proof.First, we will show that the following convergence holds true By a well-known principle in the theory of the pointwise central limit theorem (see, e.g., Lacey and Philipp [19] and Berkes and Csáki [1]), in order to prove (4.1), it is enough to show that (4.2) lim for any Lipschitz function g bounded by 1.By virtue of Lemma 3.1 in Csáki and Gonchigdanzan [5], in order to prove the convergence in (4.2), it is sufficient to show that for some ε > 0.
Observe that Clearly, (4.5) Thus, it remains to estimate the component 2 in (4.4).In view of Lemma 3.1, we have for k < n where the last relation follows from the fact that N n=k+1 Hence, we get (4.6) In view of (4.4)-(4.6),we have Thus, the relations in (4.3) and (4.2) are fulfilled and consequently, (4.1) holds true.Finally, the convergence in (4.1), Lemma 3.2 and a regularity property of logarithmic mean imply the result stated in Theorem 2.1.
In the last part of our work, we give some applications of our main result, which refer to copula processes.
2, . . . is an identically distributed, stationary, centered, d-dimensional process, with uniform marginal distributions and a joint cdf F (thus, Y i,j s have the same uniform distribution, for any i ∈ N and j = 1, ..., d).Let in addition F n be such as in (1.2) and denote an empirical cdf of Y, and B n be an empirical process defined by(1.3).Furthermore, assume that: D 2 (S n,j ) 2 , and that (ζ n ) forms a sequence of i.i.d.r.v.'s, such that for any j = 1, . . ., d, the distribution of Y 1,j is independent of (ζ n ) and that, there exists a measurable function G satisfying the following conditions: 1≤j≤d E (S n,j − ES n,j ) 2+δ = O n 1+δ/2 ,as n → ∞, for some δ > 0, where: S n,j := Y 1,j + Y 2,j + • • • + Y n,j , σ n,j :=