Change point detection in dependent sequences: invariance principles for some quadratic statistics

Charles Suquet  Marie-Claude Viano

Laboratoire de Statistique et Probabilités, Bât. M2 Université des Sciences et Technologies de Lille F-59655 Villeneuve d’Ascq Cedex, France

Abstract We consider the change point detection problem for the marginal distribution F of a dependent process. We focus on three Cramr-von Mises type test statistics. Under the hypothesis that no change occurs, we prove their convergence in distribution for either an α-mixing or an associated sequence of observations. Thanks to the quadratic context, the space D[0, 1]^{2} can be replaced by L^{2}[0, 1]^{2} in the preliminary topological tools and the convergence results hold for a significantly larger class of dependent sequences than for the classical Kolmogorov-Smirnov statistics.

AMS 1991 subject classifications: 62G10, 62G20, 60F17. Key words: association, change point, distribution free Cramr von Mises test statistic, perturbed empirical process, strong mixing.

1 Introduction

Let (X_{n})_{n≥1} be a random univariate sequence. Suppose that, having observed (X_{1}, …, X_{n}), we want to test the null hypothesis

H_{0}: The X_{i} (1 ≤ i ≤ n) all have the same continuous distribution function F

against the alternative

H_{1}: There is a change point, that is there exists θ ∈ (0, 1) such that for i ≤ [nθ], the X_{i}’s have the same distribution F, while for i > [nθ], the X_{i}’s have a distribution G ≠ F.

The vast amount of publications on non parametric change point detection and estimation gives evidence of the relevance of these problems. The book [6] presents a good overview of the area. More recently, [1] is rather turned to parametric situations and sequential methods.

Denote by F_{i,j} the empirical distribution function of the (i, j]-indexed observations:

F_{i,j}(x) = \frac{1}{j − i} ∑_{k=i+1}^{j} I_{\{X_{k}≤x\}}.

Non-parametric procedures for testing H_{0} against H_{1} are generally derived from the basic differences F_{0,k} − F_{k,n} of the empirical distributions before and after index k, by taking either the sup norm or any other convenient one. The sup norm leads to the Kolmogorov-Smirnov type statistic

B̂_{n} = sup_{x∈IR} sup_{1≤k≤n−1} \frac{k}{n}(1 − \frac{k}{n})\sqrt{n}|F_{0,k}(x) − F_{k,n}(x)|  (1)

used by several authors like [24], [19] and [18]. Quadratic norms lead to Cramrvon Mises type statistics like

A_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∫_{IR}|F_{0,k} − F_{k,n}|^{2} dF,  (2)

in case of known F, or like (6) and (7) defined below when F is unknown. Intermediate forms such as

Ĉ_{n} = sup_{1≤k≤n−1} \frac{k}{n}(1 − \frac{k}{n}) ∫_{IR}|F_{0,k} − F_{k,n}|^{2} dF_{n}  (3)

are proposed for instance in [18]. In an estimation context [8] lists the properties of the more general so called ”mean dominant norms” providing consistent estimators. More recently procedures proposed in [8] are embedded in a more general framework using random seminorms [13]. It is worth noticing that for example statistics (1), (2) and (3) belongs to the class introduced in [13].

Under H_{0} and in case of independent observations the test statistic (2) has the advantage of presenting a well known and tabulated limiting law [4], non depending on the theoretical distribution F. As an easy consequence of the present paper, the same holds (under independence) for the two other statistics (6) and (7) considered in the sequel. In [14], several other Cramr-von Mises type statistics are studied in the i.i.d. case and the author shows how the limiting distributions can be tabulated.

In this paper we drop the independence condition and we investigate the asymptotic behaviour of three test statistics under H_{0} and in two different kinds of dependence contexts. As the dependence structure of data is always difficult to estimate, weakening the conditions on this dependence is highly desirable. Therefrom comes the main motivation of our interest in quadratic statistics. Some topological comments are needed to explain their advantage from this point of view.

As pointed out in [11] and in [18] the basic tool for studying the limit law of statistics of the above type is the weak convergence under H_{0} of some two parameter process such as

\sqrt{nt}(F_{0,[nt]}(x) − F (x))  (t, x) ∈ [0, 1] × IR.  (4)

The limit law of the statistic is then the distribution of the corresponding functional of the limit process.

It is well known [16] that for i.i.d. X_{j}’s the process (4) weakly converges in D([0, 1] × IR) to the Kiefer process, a Gaussian process with covariance function (t ∧ t′)(F (x ∧ x′) − F (x)F (x′)). In [36] the same result is established, of course with a different kernel, under the condition α(n) = O(n^{−τ}) for any τ > 3.

In case of quadratic statistics, weak convergence in D([0, 1] × IR) of the two parameter underlying empirical process is too strong a result. Obviously weak convergence in a suitable L^{2} space should be sufficient. Note that, for both topologies, the convergence of the two parameter process (4) implies the same type of convergence of the uniform empirical process ξ_{n}:

ξ_{n}(s) = n^{−1/2} ∑_{i=1}^{n}(I_{[0,s]}(X_{i}) − s), s ∈ [0, 1].  (5)

Tightness of {ξ_{n}, n ≥ 1} in D(0, 1) generally follows from controlling the fourth moments of the increments of ξ_{n}, while variances inequalities are sufficient for the L^{2}[0, 1]-tightness. As a result, the dependence assumptions in the L^{2}[0, 1]- setting are weaker than in D(0, 1). For instance the weak convergence of ξ_{n} is proved in L^{2}[0, 1] under the strong mixing assumption ∑ α_{n} < ∞ (see [23]) and in D(0, 1) under α(n) = O(n^{−τ}) with τ > 1 + \sqrt{2} (a recent result of Shao and Yu [27] improving the previous Yoshihara’s [35] condition τ > 3).

The same happens in this paper. Our main contribution is to obtain the limit law under H_{0} of some selected Cramr-von Mises statistics with rather mild conditions about the dependence structure.

The paper is organized as follows. Section 2 is devoted to the weak convergence of a two parameter empirical process, which provides the asymptotic distribution of A_{n} both for strong mixing and for associated observations. When F is unknown, A_{n} is no longer computable from the observations. It then seems reasonable to propose the distribution free analogue obtained by replacing F by the empirical distribution F_{0,n}:

Â_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∫_{IR}|F_{0,k} − F_{k,n}|^{2} dF_{0,n}.  (6)

Section 3 is devoted to the convergence of Â_{n} under the strong mixing condition α(n) = O(n^{−τ}) with τ > 2.25. The Bradley’s reconstruction technique plays here a central rle. Unfortunately such a tool is not available in the case of association. This lead us to introduce a slight modification of Â_{n} based on the perturbed empirical process and rather easier to deal with:

D̂_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∫_{IR}|Φ_{0,k} − Φ_{k,n}|^{2} dF_{0,n},  (7)

where Φ_{i,j} is a smoothed version of F_{i,j}:

Φ_{i,j} = a_{n}^{−1}I_{[0,a_{n}]} ∗ F_{i,j} with a_{n} to be discussed later.

Our only result in the case of associated X′_{i}s with unknown distribution function F concerns D̂_{n}. Its convergence is studied in Section 4. It turns out that D̂_{n} has its own interest even in the strong mixing case. It allows us to relax the mixing rate assumption in α(n) = O(n^{−τ}) with τ > 1.5 (at the cost of some extra assumptions on F).

Section 5 presents a short discussion of our results. Analytical background, definitions and basic properties of mixing and association are gathered in Section 6.

2 Convergence of A_{n}

In this Section, we study the asymptotic distribution of the statistic (2) under H_{0}. The main result is the following.

Theorem 1 Let (X_{n})_{n≥1} be strictly stationary, with continuous marginal distribution function F and assume that either of the following two conditions hold.

(i) The sequence (X_{n})_{n≥1} is strong mixing and verifies

∑_{n≥1} α(n) < ∞;  (8)

(ii) The sequence (X_{n})_{n≥1} is associated and verifies

∑_{n≥1} Cov^{ν/(ν+2)}(X_{1}, X_{n}) < ∞,  (9)

together with the extra assumption of uniform Hölder regularity with exponent ν ∈ (0, 1] for the marginal distribution function F.

Then A_{n} converges in distribution to ∫_{[0,1]^{2}} ζ^{2}(s, t) dsdt where ζ is a centered Gaussian random field with covariance function R defined by :

R((s, t); (s′, t′)) = (1 − s ∧ s′)(s ∨ s′)Γ(t, t′),  (s, t, s′, t′) ∈ [0, 1]^{4}, (10)

where Γ is given by

Γ(t, t′) = t ∧ t′ − tt′ + ∑_{k=2}^{∞} (Γ_{k}(t, t′) + Γ_{k}(t′, t)),  (11)

with Γ_{k}(t, t′) = P(X_{1} ≤ t, X_{k} ≤ t′) − tt′, t, t′ ∈ [0, 1].

2.1 The two parameter underlying process

Since F is supposed continuous, the F (X_{i})’s are [0, 1]-uniformly distributed and by change of variable we have

∫_{bkr} |F_{0,k}(x) − F_{k,n}(x)|^{2} dF (x) = ∫_{0}^{1} |G_{0,k}(t) − G_{k,n}(t)|^{2} dt,

where

G_{i,j}(t) = \frac{1}{j − i} ∑_{k=i+1}^{j} I_{\{F (X_{k})≤t\}}.

Hence we shall assume without loss of generality throughout this Section that the X_{i}’s are [0, 1]-uniformly distributed and identify G_{i,j} with F_{i,j}.

Consider now the sequence of random elements in L^{2}[0, 1]^{2} defined by:

ζ_{n}(s, t) = \sqrt{n} ∑_{k=1}^{n−1} \frac{k}{n}(1− \frac{k}{n})I_{( \frac{k−1}{n} , \frac{k}{n}]}(s)(F_{0,k}(t)−F_{k,n}(t)), (s, t) ∈ [0, 1]^{2}. (12)

Noting that

∫_{[0,1]^{2}} ζ_{n}(s, t)^{2} ds dt = n ∑_{k=1}^{n−1}[\frac{k}{n}(1− \frac{k}{n})]^{2} ∫ \frac{k}{n} ; \frac{k−1}{n} ds ∫_{0}^{1}|F_{0,k}(t)−F_{k,n}(t)|^{2} dt = A_{n},

(13)

the statistic A_{n} appears as a L^{2}[0, 1]^{2} continuous functional of ζ_{n} and its convergence in distribution will follow from the weak L^{2}[0, 1]^{2} convergence of (ζ_{n}).

Theorem 2 Under the assumptions (i) or (ii) of Theorem 1, (ζ_{n})_{n≥2} converges weakly in L^{2}[0, 1]^{2} to the centered Gaussian random field ζ with covariance (10).

To establish the weak L^{2}[0, 1]^{2} convergence of (ζ_{n})_{n≥2} to ζ, we first prove the tightness of (ζ_{n})_{n≥2} (Subsection 2.2) and next the convergence in distribution of the inner products ⟨f, ζ_{n}⟩ to ⟨f, ζ⟩ for any f of a dense subspace of L^{2}[0, 1]^{2} (Subsection 2.3). This convergence reduces to a central limit theorem for triangular array of dependent variables.

2.2 Tightness of (ζ_{n})_{n≥2}

We equally note ∥ ∥ for the norms of L^{2}[0, 1] and L^{2}[0, 1]^{2}. The context will in all cases dispel initial doubts on the right interpretation. Using Theorem 9 below with the Haar multiresolution analysis (Ṽ_{j})_{j≥0} of L^{2}[0, 1]^{2}, we have to prove ζ_{n} verifies the two conditions:

sup_{n≥2} IE ∥ζ_{n}∥^{2} < +∞,  (14)
lim_{j→+∞} sup_{n≥2} IE ∥ζ_{n} − Ẽ_{j}ζ_{n}∥^{2} = 0,  (15)

where Ẽ_{j} is the orthogonal projector on Ṽ_{j}. Recall that following Oliveira Suquet [23], under (8) or (9) the sequence of uniform empirical processes (ξ_{n}) satisfies the unidimensional version of (14) and (15). It is easily checked that:

ζ_{n}(s, t) = ∑_{k=1}^{n−1} I_{( \frac{k−1}{n} , \frac{k}{n}]}(s)[(\frac{k}{n})^{1/2}ξ_{k}(t) − \frac{k}{n}ξ_{n}(t)].  (16)

Define now the triangular array of processes {ξ_{n,i}, 1 ≤ i ≤ n, n ≥ 2} by

ξ_{n,i}(t) = (\frac{i}{n})^{1/2}ξ_{i}(t) − \frac{i}{n}ξ_{n}(t),  t ∈ [0, 1],  (17)

noting that ξ_{n,n} ≡ 0. Write I_{n,i} for the interval (\frac{i−1}{n}, \frac{i}{n}] and J_{n,i} for its indicator function. With these notations, we have

ζ_{n} = ∑_{i=1}^{n} J_{n,i} ⊗ ξ_{n,i}.  (18)

The triangular array {ξ_{n,i}} inherits the tightness from (ξ_{n}), more precisely:

Lemma 3 The triangular array of processes {ξ_{n,i}, 1 ≤ i ≤ n, n ≥ 2} verifies the one dimensional version of (14) and (15).

This follows from the bounds:

IE ∥ξ_{n,i}∥^{2} ≤ \frac{2i}{n} IE ∥ξ_{i}∥^{2} + \frac{2i^{2}}{n^{2}} IE ∥ξ_{n}∥^{2} ≤ 4 sup_{k≥1} IE ∥ξ_{k}∥^{2}

and

IE ∥ξ_{n,i}−E_{j}ξ_{n,i}∥^{2} ≤ \frac{2i}{n} IE∥ξ_{i}−E_{j}ξ_{i}∥^{2}+\frac{2i^{2}}{n^{2}} IE∥ξ_{n}−E_{j}ξ_{n}∥^{2} ≤ 4 sup_{k≥1} IE∥ξ_{k}−E_{j}ξ_{k}∥^{2}.

Observing now that the functions J_{n,i} (1 ≤ i ≤ n) are orthogonal with norm n^{−1/2}, we have by Lemma 10 of Section 6:

∥ζ_{n}∥^{2} = ∑_{i=1}^{n} ∥J_{n,i}∥^{2} ∥ξ_{n,i}∥^{2} = \frac{1}{n} ∑_{i=1}^{n} ∥ξ_{n,i}∥^{2},

so by Lemma 3 the sequence (ζ_{n}) satisfies (14).

To prove (15), observe first that using (18) and (98) gives the bound

∥ζ_{n} −Ẽ_{j}ζ_{n}∥^{2} ≤ 2∥∑_{i=1}^{n} J_{n,i} ⊗(ξ_{n,i} −E_{j}ξ_{n,i})∥^{2} +2∥∑_{i=1}^{n}(J_{n,i} −E_{j}J_{n,i})⊗(E_{j}ξ_{n,i})∥^{2}.

(19)

We write 2T_{n,j}^{(1)} and 2T_{n,j}^{(2)} for the first and second term in the right hand side of (19). Lemma 10 gives

IE T_{n,j}^{(1)} = ∑_{i=1}^{n} ∥J_{n,i}∥^{2} IE ∥ξ_{n,i} − E_{j}ξ_{n,i}∥^{2} = \frac{1}{n} ∑_{i=1}^{n} IE ∥ξ_{n,i} − E_{j}ξ_{n,i}∥^{2}
≤ sup_{n,i} IE ∥ξ_{n,i} − E_{j}ξ_{n,i}∥^{2},

which goes to zero as j goes to infinity by Lemma 3. The control of sup_{n} IE T_{n,j}^{(2)} is less obvious. Our guideline is the method used in [29] to prove the tightness in L^{p}(0, 1) of the Donsker random step functions.

Recalling that the marginals of the integral kernel E_{j}(s, t) of E_{j} are probability densities on [0, 1], we have the representation

J_{n,i}(x) − E_{j}J_{n,i}(x) = ∫_{0}^{1} E_{j}(x, s)(J_{n,i}(x) − J_{n,i}(s)) ds.

It follows

T_{n,j}^{(2)} = ∫_{0}^{1} dy ∫_{0}^{1} dx(∫_{0}^{1} E_{j}(x, s) ∑_{i=1}^{n}(J_{n,i}(x) − J_{n,i}(s))E_{j}ξ_{n,i}(y) ds)^{2}. (20)

Write A^{2}(y) for the function under the integral with respect to dy in (20). For n = 2^{j}, by the mean value property of the Haar multiresolution analysis, E_{j}J_{n,i} = J_{n,i} whence T_{n,j}^{(2)} = 0. We separately treat the cases n > 2^{j} and n < 2^{j}. \underline{Case n > 2^{j}}: Recasting A^{2}(y) as

A^{2}(y) = ∑_{i=1}^{n} ∫ \frac{i}{n} ; \frac{i−1}{n} dx(∑_{l=1}^{n} ∫ \frac{l}{n} ; \frac{l−1}{n} E_{j}(x, s)(E_{j}ξ_{n,i}(y) − E_{j}ξ_{n,l}(y)) ds)^{2},

we get by convexity

A^{2}(y) ≤ ∑_{i,l=1}^{n}|E_{j}ξ_{n,i}(y) − E_{j}ξ_{n,l}(y)|^{2} ∫ \frac{i}{n} ; \frac{i−1}{n} ∫ \frac{l}{n} ; \frac{l−1}{n} E_{j}(x, s) dx ds.

It follows

T_{n,j}^{(2)} ≤ ∑_{i,l=1}^{n} ∥E_{j}(ξ_{n,i} − ξ_{n,l})∥^{2}a(n, i, l)\frac{2^{j}}{n^{2}} ≤ ∑_{i,l=1}^{n} ∥ξ_{n,i} − ξ_{n,l}∥^{2}a(n, i, l)\frac{2^{j}}{n^{2}} ,

where a(n, i, l) = 1 if the integral of E_{j}(x, s) over the square I_{n,i} × I_{n,l} is positive and a(n, i, l) = 0 else. As a matter of fact, a(n, i, l) = 1 if and only if I_{n,i} × I_{n,l} has a non empty intersection with one of the diagonal squares C_{j,k} = [k2^{−j}, (k + 1)2^{−j})^{2} of the support of E_{j}(x, s). We have then

IE T_{n,j}^{(2)} ≤ \frac{2^{j}}{n^{2}} ∑_{k=0}^{2^{j}−1} ∑_{(i,l)∈Q_{j,k,n}} IE ∥ξ_{n,i} − ξ_{n,l}∥^{2},  (21)

where the square index sets Q_{j,k,n} = \{(i, l) : \frac{k}{2^{j}} − \frac{1}{n} < \frac{i}{n}, \frac{l}{n} < \frac{k + 1}{2^{j}}\},

have side length q_{j,k,n} bounded by: n2^{−j} ≤ q_{j,k,n} ≤ n2^{−j} + 2. Assume for the moment the following bound holds

IE ∥ξ_{n,i} − ξ_{n,l}∥^{2} ≤ C \frac{|l − i|}{n}, for some constant C.  (22)

It follows

IE T_{n,j}^{(2)} ≤ C \frac{2^{j}}{n^{2}} ∑_{k=0}^{2^{j}−1} \frac{q_{j,k,n}^{3}}{n} ≤ C \frac{2^{2j}}{n^{3}} (\frac{n}{2^{j}} + 2)^{3} ≤ \frac{27C}{2^{j}}  (n > 2^{j}).  (23)

\underline{Case n < 2^{j}} In view of the diagonal localization of the kernel E_{j}(x, s), A^{2}(y) can be recast as

A^{2}(y) = ∑_{k=0}^{2^{j}−1} ∫ \frac{k+1}{2^{j}} ; \frac{k}{2^{j}} |∫ \frac{k+1}{2^{j}} ; \frac{k}{2^{j}} 2^{j} ∑_{i=1}^{n}(J_{n,i}(x) − J_{n,i}(s))E_{j}ξ_{n,i}(y) ds|^{2} dx.

If C_{j,k} is included in some I_{n,i}^{2}, the corresponding term in the summation over k above vanishes. Hence we have only to consider the ”small” squares C_{j,k}^{i} which cover the junction of two consecutive ”big” squares, I_{n,i−1}^{2} and I_{n,i}^{2}. Writing C_{j,k}^{i} = [k_{i}2^{−j}, (k_{i} + 1)2^{−j})^{2}, we get

A^{2}(y) = ∑_{i=2}^{n} ∫ \frac{k_{i}+1}{2^{j}} ; \frac{k_{i}}{2^{j}} |∫ \frac{k_{i}+1}{2^{j}} ; \frac{k_{i}}{2^{j}} 2^{j}[(J_{n,i−1}(x) − J_{n,i−1}(s))E_{j}ξ_{n,i−1}(y) +

(J_{n,i}(x) − J_{n,i}(s))E_{j}ξ_{n,i}(y)] ds|^{2} dx  (24)
≤ 2^{j} ∑_{i=2}^{n} ∫_{C_{j,k}^{i}} |(J_{n,i−1}(x) − J_{n,i−1}(s))E_{j}ξ_{n,i−1}(y) +

(J_{n,i}(x) − J_{n,i}(s))E_{j}ξ_{n,i}(y)|^{2} ds dx,  (25)

by Jensen’s inequality. As the function under the integral in (25) vanishes on the traces of I_{n,i−1}^{2} and I_{n,i}^{2} over C_{j,k}^{i}, the integral in (25) is bounded by the area of C_{j,k}^{i} times |E_{j}ξ_{n,i−1} − E_{j}ξ_{n,i}|. It follows

T_{n,j}^{(2)} ≤ \frac{1}{2^{j}} ∑_{i=2}^{n} ∥E_{j}(ξ_{n,i−1} − ξ_{n,i})∥^{2} ≤ \frac{1}{2^{j}} ∑_{i=2}^{n} ∥ξ_{n,i−1} − ξ_{n,i}∥^{2}.

By (22), this leads to

IE T_{n,j}^{(2)} ≤ \frac{C}{2^{j}}  (n < 2^{j}).  (26)

Hence (ζ_{n})_{n≥2} verify (15) by (23) and (26). To complete the proof of the tightness of (ζ_{n})_{n≥2}, it remains to establish (22). For l > i we have

ξ_{n,l}(y) − ξ_{n,i}(y) = \frac{1}{\sqrt{n}} ∑_{k=i+1}^{l}(I_{[0,y]}(X_{k}) − y) − \frac{l − i}{n}ξ_{n}(y).

The control of the second term is obvious recalling that sup_{n} IE ∥ξ_{n}∥^{2} < ∞. The control of the first term reduces to the following moment inequality

IE( ∑_{i<k≤l}(I_{[0,y]}(X_{k}) − y))^{2} ≤ C(l − i) uniformly in y,

which classically follows from our hypothesis (8) in the strong mixing case or (9) in the associated case (cf. Lemma 12 Section 6).

2.3 Inner products convergence

We will now prove the convergence in distribution of

⟨f, ζ_{n}⟩ = ∫_{[0,1]^{2}} f(s, t)ζ_{n}(s, t) ds dt,

for each continuous function f on [0, 1]^{2}. Going back to the definitions of F_{0,k} and F_{k,n}, we can rearrange (12) in the form

ζ_{n}(s, t) = \frac{1}{\sqrt{n}} ∑_{i=1}^{n}[I_{( \frac{i−1}{n} , \frac{n−1}{n}]}(s) − ∑_{k=1}^{n−1} \frac{k}{n}I_{( \frac{k−1}{n} , \frac{k}{n}]}(s)](I_{[0,t]}(X_{i}) − t),

whence

⟨f, ζ_{n}⟩ = \frac{1}{\sqrt{n}} ∑_{i=1}^{n} ∫_{0}^{1}[∫ \frac{n−1}{n} ; \frac{i−1}{n} f(s, t) ds − ∑_{k=1}^{n−1} \frac{k}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} f(s, t) ds](I_{[0,t]}(X_{i}) − t) dt.

(27)

Our first step is to approximate ⟨f, ζ_{n}⟩ by S_{n}(f) defined as

S_{n}(f) = \frac{1}{\sqrt{n}} ∑_{i=1}^{n} ∫_{0}^{1}(∫ 1_{\frac{i}{n}} f(s, t) ds − ∫_{0}^{1} sf(s, t) ds)(I_{[0,t]}(X_{i}) − t) dt, (28)

Using the continuity of f we easily obtain the bound

IE|⟨f, ζ_{n}⟩ − S_{n}(f)| ≤ \frac{4∥f∥_{∞}}{\sqrt{n}} ∫_{0}^{1} IE|I_{[0,t]}(X_{1}) − t| dt,

so ⟨f, ζ_{n}⟩ and S_{n}(f) will have the same limit in distribution. Now from (28) we derive

S_{n}(f) = \frac{1}{\sqrt{n}} ∑_{i=1}^{n} H(\frac{i}{n}, X_{i}),  (29)

where

H(u, v) = ∫_{[0,1]^{2}}(I_{[0,s]}(u) − s)(I_{[0,t]}(v) − t)f(s, t) ds dt, (u, v) ∈ [0, 1]^{2}.

So the problem is reduced to a central limit theorem for a triangular array of zero mean dependent variables. It is worth noticing here that H is absolutely continuous in each variable and verifies

|\frac{∂}{∂v} H(u, v)| ≤ ∥f∥_{∞},  (u, v) ∈ [0, 1]^{2}.  (30)

To identify the covariance structure of the limiting process, we need the following result which can be obtained by elementary computations (details can be found in subsection 2.4 of [31]):

lim_{n→∞} IE S_{n}(f)^{2} = σ^{2}(f) = ∫_{[0,1]^{4}}(1 − s ∧ s′)(s ∨ s′)Γ(t, t′)f(s, t)f(s′, t′) dsds′dtdt′

(31)

Let ζ be a centered Gaussian field on [0, 1] with covariance function R given by (10). Clearly σ^{2}(f) = IE⟨f, ζ⟩^{2}. If σ^{2}(f) = 0, then ⟨f, ζ⟩ = 0 almost surely and by (31) the convergence in distribution of ⟨f, ζ_{n}⟩ to ⟨f, ζ⟩ is obvious. From now on, σ^{2}(f) is assumed to be positive. \underline{Strong mixing case}: Consider the triangular array \{Y_{n,i}, 1 ≤ i ≤ n\} where Y_{n,i} = H(i/n, X_{i}). As the sequence (α_{n}) is non increasing and ∑ α_{n} < ∞, α_{n} = o(n^{−1}). Then by Theorem 2.1 (A) combined with Theorem 1.2 and Proposition 2.1 (b) of Withers [33], S_{n}(f) will converge in distribution to N(0, σ^{2}(f)) if

sup_{n≥1} max_{a<b≤n} \frac{1}{b − a} IE( ∑_{a<i≤b} Y_{n,i})^{2} < ∞.  (32)

For notational convenience, we set g(x, y) = I_{[0,x]}(y) − x, (x, y) ∈ [0, 1]^{2} and

Γ_{i,j}(t, t′) = IE(g(t, X_{i}), g(t′, X_{j})), and Γ_{k} = Γ_{1,k}.

The moment bound (32) easily follows from our hypothesis (8) by combining the integral representation

IE Y_{n,i}Y_{n,j} = ∫_{[0,1]^{4}} g(s, \frac{i}{n})g(s′, \frac{j}{n})Γ_{i,j}(t, t′)f(s, t)f(s′, t′) dsds′dtdt′

with the obvious estimate

|Γ_{i,j}(t, t′)| = |P(X_{i} ≤ t, X_{j} ≤ t′) − P(X_{i} ≤ t) P(X_{j} ≤ t′)| ≤ α(|j − i|).

\underline{Associated case}: The proof of the central limit theorem for \{Y_{n,i}, 1 ≤ i ≤ n\} follows the same steps as Lemma 3 and Theorem 3 in Birkel [3]. Recall the Cox and Grimmett coefficient u(n) of the stationary associated sequence (X_{n}) is:

u(n) = 2 ∑_{j=n+1}^{∞} Cov(X_{1}, X_{j}), n ≥ 1.

Let us define

s_{n}^{2} = Var(∑_{i=1}^{n} X_{i}) and s^{2} = lim_{n→∞} n^{−1}s_{n}^{2} = Var(X_{1}) + u(1).

We now decompose Z_{n} = ∑_{i=1}^{n} Y_{n,i} in blocks of length p = p(n):

Z_{n,l} = ∑_{j=lp+1}^{(l+1)p} Y_{n,j}, 0 ≤ l < k := [n/p], Z_{n,k} = ∑_{j=kp+1}^{n} Y_{n,j}.

Denote by W_{n,l} (0 ≤ l ≤ k) the same blocks with X_{j} instead of Y_{n,j} and write

σ_{n}^{2} = Var(Z_{n}), and τ_{n}^{2} = ∑_{l=0}^{k} Var(Z_{n,l}).

Using Newman’s inequality (Theorem 13 Section 6), we have for any t ∈ IR:

|IE exp(itσ_{n}^{−1}Z_{n}) − ∏_{l=0}^{k} IE exp(itσ_{n}^{−1}Z_{n,l})| ≤ \frac{2t^{2}}{σ_{n}^{2}} ∑_{1≤l<m≤k} Cov(Z̄_{n,l}, Z̄_{n,k}), (33)

where

Z̄_{n,l} = ∑_{j=lp+1}^{(l+1)p} Ȳ_{n,j}, with Ȳ_{n,j} = ∫_{0}^{X_{j}} |\frac{∂}{∂v} H(\frac{i}{n}, v)| dv.

Write Δ_{n} for the left hand side of (33). Lemma 19 together with (30) gives

Δ_{n} ≤ 2t^{2}∥f∥_{∞}^{2}σ_{n}^{−2} ∑_{1≤l<m≤k} Cov(W_{n,l}, W_{n,k})

By (31) we have σ_{n}^{2} = nσ_{n}^{2}(f)(1 + o(1)), hence by stationarity of (X_{n}), there is some constant C such that:

Δ_{n} ≤ C[\frac{1}{n} Var(∑_{i=1}^{n} X_{i}) − \frac{k}{n} Var(∑_{i=1}^{p} X_{i})].

As p = p(n) goes to infinity with n, and (p+1)^{−1} < k/n ≤ p^{−1}, the convergence to zero of Δ_{n} follows. So the first condition of Lemma 3 of Birkel [3] is satisfied. Now following the verification of the condition (1) in the proof of Theorem 3 in [3] and using Lemma 19 again, we obtain

|1 − σ_{n}^{−2}τ_{n}^{2}| ≤ 2∥f∥_{∞}^{2}σ_{n}^{−2}k ∑_{j=1}^{p} u(j) ≤ 2∥f∥_{∞}^{2}(nσ_{n}^{−2})p^{−1} ∑_{j=1}^{p} u(j).

Under (9), u(n) goes to zero as n goes to infinity and so by Csaro theorem |1 − σ_{n}^{−2}τ_{n}^{2}| converges to zero. Next, the same line of arguments as in [3] p. 70 leads to

σ_{n}^{−2} ∑_{l=0}^{k} IE Z_{n,l}^{2}I_{{|Z_{n,l}|≥εσ_{n}}} ≤ p^{2}σ_{n}^{−2} ∑_{j=1}^{n} IE Y_{n,j}^{2}I_{{|Y_{n,j}|≥εσ_{n}/p}}

From the asymptotical behaviour of the variances σ_{n}^{2} and s_{n}^{2}, for each positive ε, there is a δ > 0 independent of n such that εσ_{n} > δs_{n} for n large enough. Using the bound |Y_{n,j}| ≤ ∥f∥_{∞}|X_{j}|, and the stationarity of (X_{n}), we get finally

p^{2}σ_{n}^{−2} ∑_{j=1}^{n} IE Y_{n,j}^{2}I_{{|Y_{n,j}|≥εσ_{n}/p}} ≤ C′p^{2} IE X_{1}^{2}I_{{|X_{1}|≥δ′n/p}},

for some constant C′ and some δ′ > 0. Adjusting the rate of convergence of p = p(n) to infinity like in Lemma 4 of [3] makes this last bound converge to zero. So the triangular array {Y_{n,i}, 1 ≤ i ≤ n} satisfies all the requirements of Lemma 3 in [3] for the central limit theorem.

In the sequel, F is no longer known. The aim is to provide conditions on the dependence structure of the underlying stationary process (X_{n})_{n≥1} under which the limit law of Â_{n} and D̂_{n} are the same as that of A_{n} in Theorem 1.

3 Limit law of Â_{n} under mixing

Suppose that (X_{n})_{n≥1} is strictly strong mixing and that the unknown distribution function F is continuous. In this case, the statistics Â_{n} respectively based on the X_{j}’s and on the F (X_{j})’s are almost surely equal, so that F shall be taken as the identity on [0, 1] throughout this section. We shall prove the following.

Theorem 4 If F is continuous and if α(n) = O(n^{−τ}) for some τ > 2.25, then Â_{n} converges in distribution to the limit provided by Theorem 1.

Clearly it is enough to prove that IE |A_{n} − Â_{n}| → 0, as n → ∞. The proof consists in two steps. Subsection 3.1 reduces the convergence of IE |A_{n} − Â_{n}| to that of Q_{n} defined below. Subsection 3.2 gives the proof of the convergence of Q_{n}. It relies on two lemmas whose proofs are deferred in Subsection 3.3. Before proceeding, we recall the elementary asymptotic estimate for Riemann sums multiply used in the sequel: for a > −3, b > −3 and a + b ≠ −1

∑_{k=1}^{n−1} k^{a}(n − k)^{b}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∼ n^{a+b+1} ∫_{0}^{1} t^{a+2}(1 − t)^{b+2} dt.  (34)

In this section, the letter c denotes a constant whose value may differ at each occurrence.

3.1 Reduction of the problem

For notational convenience, we write:

F̃_{i,j} = F_{i,j} − IE F_{i,j}.  (35)

Define Q_{n} by

Q_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2}(I_{n}(k) + J_{n}(k)),  (36)

where I_{n}(k) = ∫_{0}^{1} F̃_{0,k}^{2}(x)dF̃_{k,n}(x) and J_{n}(k) = ∫_{0}^{1} F̃_{k,n}^{2}(x^{−})dF̃_{0,k}(x). We be-

gin by proving the approximation result

IE |Â_{n} − A_{n} − Q_{n}| ≤ \frac{c}{n},  (37)

which reduces the study of Â_{n} − A_{n} to that of Q_{n}. Now

Â_{n} − A_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∫_{0}^{1}|F̃_{0,k} − F̃_{k,n}|^{2}dF̃_{0,n},  (38)

where the integral can easily be rewritten as

∫_{0}^{1}|F̃_{0,k} − F̃_{k,n}|^{2}dF̃_{0,n} = \frac{k}{n} ∫_{0}^{1}|F̃_{0,k} − F̃_{k,n}|^{2}d(F̃_{0,k} − F̃_{k,n}) + ∫_{0}^{1} F̃_{k,n}^{2}dF̃_{k,n}
+ ∫_{0}^{1} F̃_{0,k}^{2}dF̃_{k,n} − 2 ∫_{0}^{1} F̃_{0,k}F̃_{k,n}dF̃_{k,n}.

Applying (i) of Lemma 22 in Section 6 with successively H_{1} = F̃_{0,k} − F̃_{k,n} and H_{1} = F̃_{k,n}, then (ii) of the same lemma with H_{1} = F̃_{0,k} and H_{2} = F̃_{k,n} leads after a few calculations to:

∫_{0}^{1}|F̃_{0,k} − F̃_{k,n}|^{2}dF̃_{0,n} − I_{n}(k) − J_{n}(k)

= −\frac{1}{3k(n − k)} + (\frac{1}{n − k} − \frac{1}{n(n − k)}) ∑_{i=k+1}^{n}(F_{0,k}(X_{i}) − F_{k,n}(X_{i}))

= \frac{1}{6k(n − k)} + \frac{1}{n}(\frac{1}{k^{2}} + \frac{1}{(n − k)^{2}} ) \begin{matrix}∑ \\ k<i,j≤n \\ i≠j \end{matrix} I_{\{X_{i}=X_{j}\}} − \frac{1}{nk(n − k)} ∑_{k<i,j≤n}I_{\{X_{i}=X_{j}\}}.

Now as each of the double sums above is less than ∑_{i,j=1}^{n} I_{{X_{i}=X_{j}}}:

IE|∫ |F̃_{0,k} − F̃_{k,n}|^{2}dF̃_{0,n} − I_{n}(k) − J_{n}(k)| ≤
\frac{1}{6n}(\frac{1}{k} + \frac{1}{n − k} ) + \frac{c}{n}(\frac{1}{k} + \frac{1}{n − k} )^{2} ∑_{i,j=1}^{n} P \{X_{i} = X_{j}\}. (39)

In order to conclude, remark that the hypothesis α(n) = O(n^{−τ}) with τ > 2.25 implies ∑ α(n)^{2/3} < ∞ which in turn via Corollary 18 (Section 6) of Bradley’s theorem, leads to ∑_{j≥1} P {X_{1} = X_{j}} < ∞. This together with (39), (38) and (36) implies (37).

3.2 Convergence of Q_{n}

Hence it remains to prove that

IE(Q_{n}) → 0.  (40)

It should be obvious that nearly the same arguments lead to the convergence to zero of the two summands coming from (36). We only check the first one, namely that

∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2}[IE(I_{n}^{2}(k))]^{1/2} → 0.  (41)

Now IE I_{n}^{2}(k) can be decomposed into:

IE(I_{n}^{2}(k)) = \frac{1}{(n − k)^{2}} ( ∑_{i,j=k+1}^{n} Δ_{i,j} + ∑_{i,j=k+1}^{n} δ_{i,j}) − \frac{2}{n − k} ∑_{i=k+1}^{n} Δ_{i}, (42)

with

Δ_{i,j} = IE(F̃_{0,k}^{2}(X_{i})F̃_{0,k}^{2}(X_{j}) − ∫_{[0,1]^{2}} F̃_{0,k}^{2}(x)F̃_{0,k}^{2}(y)dP_{i,j}(x, y)),
δ_{i,j} = IE(∫_{[0,1]^{2}} F̃_{0,k}^{2}(x)F̃_{0,k}^{2}(y)dP_{i,j}(x, y) − ∫_{[0,1]^{2}} F̃_{0,k}^{2}(x)F̃_{0,k}^{2}(y)dxdy),
Δ_{i} = IE(F̃_{0,k}^{2}(X_{i}) ∫_{[0,1]} F̃_{0,k}^{2}(x)dx − ∫_{[0,1]^{2}} F̃_{0,k}^{2}(x)F̃_{0,k}^{2}(y)dxdy).

In these expressions P_{i,j} denotes the distribution of (X_{i}, X_{j}). It is easily seen that, should the X_{j}’s be independent, Δ_{i,j}, δ_{i,j} and Δ_{i} all would be zero.

Moreover they can be rewritten as

Δ_{i,j} = IE(F̃_{0,k}^{2}(X_{i})F̃_{0,k}^{2}(X_{j}) − F̃_{0,k}^{2}(X_{i}^{∗})F̃_{0,k}^{2}(X_{j}^{∗})),  (43)

where (X_{i}^{∗}, X_{j}^{∗}) is distributed according P_{i,j} and is independent of (X_{1}, …, X_{k}),

δ_{i,j} = IE(U(X_{i}, X_{j}) − U(X_{i}, X_{j}^{∗})),  (44)

where U(x, y) = IE(F̃_{0,k}^{2}(x)F̃_{0,k}^{2}(y)) and X_{j}^{∗} is uniformly distributed on [0, 1] and is independent of X_{i},

Δ_{i} = ∫_{0}^{1} IE[F̃_{0,k}^{2}(x)(F̃_{0,k}^{2}(X_{i}) − F̃_{0,k}^{2}(X_{i}^{∗}))]dx,  (45)

where X_{i}^{∗} is uniformly distributed on [0, 1] and is independent of (X_{1}, …, X_{k}). Let us first turn to Δ_{i,j}. From (43) it easily follows that

|Δ_{i,j}| ≤ IE^{1/4}|F̃_{0,k}(X_{i}) − F̃_{0,k}(X_{i}^{∗})|^{4} IE^{1/2} F̃_{0,k}^{4}(X_{j})
×[IE^{1/4} F̃_{0,k}^{4}(X_{i}) + IE^{1/4} F̃_{0,k}^{4}(X_{i}^{∗})]
+ IE^{1/4}|F̃_{0,k}(X_{j}) − F̃_{0,k}(X_{j}^{∗})|^{4} IE^{1/2} F̃_{0,k}^{4}(X_{i})
×[IE^{1/4} F̃_{0,k}^{4}(X_{j}) + IE^{1/4} F̃_{0,k}^{4}(X_{j}^{∗})].  (46)

In the right hand side of (46), three types of fourth moments appear. The technical tools for bounding them are Lemmas 5 and 6 below whose proofs are deferred to Subsection 3.3.

Lemma 5 In (43), (X_{i}^{∗}, X_{j}^{∗}) can be chosen such that if ∑_{n≥1} nα(n)^{γ/(4+γ)} < ∞,

IE |F̃_{0,k}(X_{l}) − F̃_{0,k}(X_{l}^{∗})|^{4} ≤ c[α(i ∧ j − k)^{8/9} + \frac{1}{k^{2}} α(i ∧ j − k)^{8/9(4+γ)}], l = i, j.

As ∑ nα(n) < ∞, Yokoyama’s theorem (Theorem 14 below) directly leads to IE(F̃_{0,k}^{4}(x)) ≤ ck^{−2}. By independence of X_{i}^{∗} and F̃_{0,k}, integrating both sides of this inequality with respect to the Lebesgue measure on [0, 1] yields

IE(F̃_{0,k}^{4}(X_{i}^{∗})) ≤ \frac{c}{k^{2}} .  (47)

Changing x for X_{i} in IE(F̃_{0,k}^{4}(x)) means changing x for X_{i} in the indicators I{X_{ℓ} ≤ x} and therefore causes a real problem. Lemma 6 proves that the order of the fourth moment’s bound in the Yokoyama’s theorem is only damaged up to a logarithmic coefficient.

Lemma 6 Let (X_{n})_{n≥1} be a stationary strong mixing sequence of uniformly [0, 1]-distributed random variables such that α(n) = O(n^{−τ}) with τ > 2.25.

IE F̃_{0,k}^{4}(X_{i}) ≤ c(ln k)^{3}k^{−2}, i ≥ k + 1.  (48)

From Lemmas 5 and 6, (46) and (47) follows:

|Δ_{i,j}| ≤ ck^{−3/2}(ln k)^{9/4}[(i ∧ j − k)^{−2τ/9} + k^{−1/2}(i ∧ j − k)^{−2τ/9(4+γ)}].

Let d_{0} be in (0, 1). Notice that for every m ≥ 1 and every d ≥ d_{0},

∑_{l=1}^{m} \frac{1}{l^{d}} ≤ ∑_{l=1}^{m} \frac{1}{l^{d_{0}}} ≤ 2m^{1−d_{0}}.  (49)

Of course this estimate is crude for d ≥ 1. But as we are looking for a result with range of validity τ ∈ (τ_{0}, ∞), (49) avoids a useless discussion. From (49) it easily follows that for d ≥ d_{0},

∑_{i,j=k+1}^{m} \frac{1}{(i ∧ j − k)^{d}} ≤ 5(n − k)^{2−d_{0}}.  (50)

Let now τ > 9/4 and choose ε > 0 small enough such that τ ≥ 9/4 + ε and 1/2 + 2ε/9 < 1. Applying (50) successively with

d_{0} = \frac{1}{2} + \frac{2ε}{9} and d′_{0} = \frac{1}{4 + γ} (\frac{1}{2} + \frac{2ε}{9})

leads to

\frac{1}{(n − k)^{2}} ∑_{i,j=k+1}^{n} |Δ_{i,j}| ≤ \frac{c(ln k)^{9/4}}{k^{3/2}} (\frac{1}{(n − k)^{d_{0}}} + \frac{1}{k^{1/2}(n − k)^{d′_{0}}} ).  (51)

The third term Δ_{i} in (42) is treated similarly, leading to the same bound. As for the intermediate term δ_{i,j} in (42), for fixed x, y and z,

|U(x, y) − U(x, z)| ≤ IE^{1/2} F̃_{0,k}^{4}(x)[IE^{1/4} F̃_{0,k}^{4}(y) + IE^{1/4} F̃_{0,k}^{4}(z)]
IE^{1/4}|F̃_{0,k}(y) − F̃_{0,k}(z)|^{4}

and

IE|F̃_{0,k}(y) − F̃_{0,k}(z)|^{4} ≤ c|y − z|^{4} + c IE(\frac{1}{k} ∑_{l=1}^{k} I_{\{|X_{l}−y|≤|y−z|\}}) ≤ c|y − z|.

This, together with the moment’s bound already used for (47) yields :

|U(x, y) − U(x, z)| ≤ ck^{−3/2}|y − z|^{1/4}.

According to Corollary 18, it is possible to choose X_{j}^{∗} independent of X_{i}, uniformly distributed on [0, 1] and such that

IE |X_{j} − X_{j}^{∗}|^{1/4} ≤ cα(|i − j|)^{1/3}.  (52)

Replacing then x, y and z respectively by X_{i}, X_{j} and X_{j}^{∗} gives the estimate |δ_{i,j}| ≤ ck^{−3/2}α(|i − j|)^{1/3}. Imposing the extra condition ε < 3/4 in the choice of ε above, we can apply (49) with d′′_{0} = 3/4 + ε/3 to obtain

\frac{1}{(n − k)^{2}} ∑_{i,j=k+1}^{n} |δ_{i,j}| ≤ \frac{c}{k^{3/2}(n − k)^{d′′_{0}}} .  (53)

In order to conclude it remains firstly to choose γ = 32 in Lemma 5 and then to gather (51) and (53) in (42) and to remark that using (34), the convergence (41) comes from the fact that IE I_{n}^{2}(k) is bounded above by a finite sum of terms of the form k^{a}(n − k)^{b} with a > −6, b > −6 and a + b < −2. Hence (40) is proved.

3.3 Proofs of Lemmas 5 and 6

Proof of Lemma 5: For fixed x and y, |I_{{X_{l}≤x}} − I_{{X_{l}≤y}}| ≤ I_{{|X_{l}−x|≤|x−y|}} and |F̃_{0,k}(x) − F̃_{0,k}(y)| ≤ 2 so that for every q ∈ (0, 1],

|F̃_{0,k}(X_{i}) − F̃_{0,k}(X_{i}^{∗})| ≤ c[I_{\{|X_{i}−X_{i}^{∗}|>q\}} + q + \frac{1}{k} ∑_{l=1}^{k} I_{\{|X_{l}−X_{i}^{∗}|≤q\}}].

Let s(q, x) = P (|X_{l} − x| ≤ q). Noticing that s(q, x) ≤ 2q, we get

|F̃_{0,k}(X_{i}) − F̃_{0,k}(X_{i}^{∗})| ≤ c[I_{\{|X_{i}−X_{i}^{∗}|>q\}} + q + |\frac{1}{k} ∑_{l=1}^{k} I_{\{|X_{l}−X_{i}^{∗}|≤q\}} − s(q, X_{i}^{∗})|].

whence

IE |F̃_{0,k}(X_{i}) − F̃_{0,k}(X_{i}^{∗})|^{4} ≤ c[P (|X_{i} − X_{i}^{∗}| > q) + q^{4}

+ IE(\frac{1}{k} ∑_{l=1}^{k} I_{\{|X_{l}−X_{i}^{∗}|≤q\}} − s(q, X_{i}^{∗}))^{4}].

Take q = α(i∧j −k)^{2/9}. Choosing (X_{i}^{∗}, X_{j}^{∗}) according to Theorem 17 such that

P (|X_{i} − X_{i}^{∗}| + |X_{j} − X_{j}^{∗}| > q) ≤ \frac{cα(i ∧ j − k)}{\sqrt{q}}.

leads to

IE |F̃_{0,k}(X_{i}) − F̃_{0,k}(X_{i}^{∗})|^{4} ≤ c[α(i ∧ j − k)^{8/9}
+ IE(\frac{1}{k} ∑_{l=1}^{k} I_{\{|X_{l}−X_{i}^{∗}|≤q\}} − s(q, X_{i}^{∗}))^{4}].(54)

Applying Corollary 15 to the variables I_{{|X_{l}−x|≤q}} − s(q, x) yields:

IE(\frac{1}{k} ∑_{l=1}^{k} I_{\{|X_{l}−x|≤q\}} − s(q, x))^{4} ≤ \frac{cq^{4/(4+γ)}}{k^{2}} = \frac{cα(i ∧ j − k)^{8/(9(4+γ))}}{k^{2}} .

Integrating this inequality with respect to the Lebesgue measure on [0, 1] leads, via (54), to the desired result.

Proof of Lemma 6: For notational convenience, let us write Ĩ{X_{l} ≤ x} = I{X_{l} ≤ x} − x. The proof runs in two steps. Firstly we prove that if 0 ≤ l < m are such that m − l ≤ i − m, then

IE( ∑_{r=l+1}^{m} Ĩ_{{X_{r}≤X_{i}}})^{4} ≤ c(m − l)^{2}.  (55)

Secondly we split {1, …, k} in subintervals {l + 1, m} with m − l ≤ i − m and use (55) for each of them. First step: Introducing into (55) a random variable X_{i}^{∗} with the same distribution as X_{i} and independent of (X_{l+1}, …, X_{m}) gives the estimate

IE( ∑_{r=l+1}^{m} Ĩ_{{X_{r}≤X_{i}}})^{4} ≤ c[IE( ∑_{r=l+1}^{m} Ĩ_{{X_{r}≤X_{i}^{∗}}})^{4}

+ IE( ∑_{r=l+1}^{m} Ĩ_{{X_{r}≤X_{i}}} − Ĩ_{{X_{r}≤X_{i}^{∗}}})^{4}].

For any γ such that ∑ nα(n)^{γ/(4+γ)} < ∞, the same argument as in the proof of Lemma 5 ensures that for a suitable choice of X_{i}^{∗},

IE( ∑_{r=l+1}^{m} Ĩ_{\{X_{r}≤X_{i}\}} − Ĩ_{\{X_{r}≤X_{i}^{∗}\}})^{4} ≤ c(m − l)^{4}[α(i − m)^{8/9} + \frac{α(i − m)^{8/(9(4+γ))}}{(m − l)^{2}} ]

whence

IE( ∑_{r=l+1}^{m} Ĩ_{{X_{r}≤X_{i}}})^{4} ≤ c(m − l)^{4}[α(i − m)^{8/9} + (m − l)^{−2}].

Now, i−m ≥ m−l and τ > 9/4 imply α(i−m)^{8/9} ≤ c(m−l)^{−2} so (55) follows.

Second step: If i ≥ 2k, the inequality k ≤ i−k authorizes the application of (55) with l = 0, m = k and (48) follows directly. The case k < i < 2k needs a little more work. To use (55), we define inductively the finite decreasing sequence of integers m_{j} by:

m_{1} = k, m_{j} − m_{j+1} = i − m_{j},  1 ≤ j ≤ s(k) + 1,  (56)

that is explicitly m_{j} = i − 2^{j−1}(i − k). Here s(k) is determined by: m_{s(k)+1} < 0 ≤ m_{s(k)} ≤ … ≤ m_{1} = k, so it satisfies:

\frac{i}{i − k} < 2^{s(k)} ≤ \frac{2i}{i − k} .  (57)

Dividing the interval [1, k] in subintervals (m_{j+1}, m_{j}], we get by convexity:

IE|∑_{l=1}^{k} Ĩ_{{X_{l}≤X_{i}}}|^{4} ≤ c{s(k)^{3} ∑_{j=1}^{s(k)−1} IE| ∑_{l=m_{j+1}+1}^{m_{j}} Ĩ_{{X_{l}≤X_{i}}}|^{4} + IE| ∑_{l=1}^{m_{s(k)}} Ĩ_{{X_{l}≤X_{i}}}|^{4}}

where the last term is zero if m_{s(k)} = 0. Now (56) for 1 ≤ j < s(k) and m_{s(k)} < i − m_{s(k)} legitimize the use of (55):

IE|∑_{l=1}^{k} Ĩ_{{X_{l}≤X_{i}}}|^{4} ≤ c{s(k)^{3} ∑_{j=1}^{s(k)−1}(m_{j} − m_{j+1})^{2} + m_{s(k)}^{2}}

As m_{j} − m_{j+1} = 2^{j−1}(i − k) and m_{s(k)} ≤ i/2,

IE|∑_{l=1}^{k} Ĩ_{{X_{l}≤X_{i}}}|^{4} ≤ c{s(k)^{3}2^{s(k)}(i − k)^{2} + i^{2}}.

Now from (57) and the fact that k < i < 2k follows:

IE|∑_{l=1}^{k} Ĩ_{{X_{l}≤X_{i}}}|^{4} ≤ c(ln k)^{3}k^{2}

and the proof is over.

4 Limit law of D̂_{n} under mixing or association

We turn now to the smoothed statistic D̂_{n} defined by (7). As the smoothing by convolution does not commute with the change of variables Y = F (X), we no longer suppose the initial observations X_{i} uniformly [0, 1]-distributed. Our aim in this section is to give sufficient conditions assuring that under H_{0}, D̂_{n} has the same limit law as A_{n}. Some notations are needed.

4.1 Notations and results

Let (a_{n})_{n≥1} be a sequence of positive (deterministic) real numbers decreasing to 0 at a rate to be specified later. Let us write:

η = I_{[0,+∞)},  η_{n} = a_{n}^{−1}I_{[0,a_{n}]} ∗ η, n ≥ 1.  (58)

Now the empirical distribution functions F_{i,j} and empirical processes ξ_{i,j} can be rewritten as:

F_{i,j}(x) = \frac{1}{j − i} ∑_{i<l≤j} η(x − X_{l}), x ∈ IR,

ξ_{i,j}(x) = \sqrt{j − i}F̃_{i,j}(x) = \frac{1}{\sqrt{j − i}} ∑_{i<l≤j}(η(x − X_{l}) − F (x)), x ∈ IR.

Observe that by Fubini’s theorem:

IE η_{n}(x − X_{l}) = a_{n}^{−1}I_{[0,a_{n}]} ∗ IE η(x − X_{l}) = (a_{n}^{−1}I_{[0,a_{n}]} ∗ F )(x).

The smooth versions of the empirical distribution functions and processes are based on η_{n} instead of the indicator η. For 0 ≤ i < j ≤ n:

Φ_{i,j}(x) = (a_{n}^{−1}I_{[0,a_{n}]} ∗ F_{i,j})(x) = \frac{1}{j − i} ∑_{i<l≤j} η_{n}(x − X_{l}),  (59)
Φ̃_{i,j}(x) = Φ_{i,j}(x) − IE Φ_{i,j}(x) = (a_{n}^{−1}I_{[0,a_{n}]} ∗ F̃_{i,j})(x),
χ_{i,j}(x) = (a_{n}^{−1}I_{[0,a_{n}]} ∗ ξ_{i,j})(x) = \frac{1}{\sqrt{j − i}} ∑_{i<l≤j}(η_{n}(x − X_{l}) − IE η_{n}(x − X_{l})).

It should be noticed here that unlike ξ_{i,j} the notations Φ_{i,j},…,χ_{i,j} are not intrinsic (i.e. depending on i,j exclusively). Notations like Φ_{n,i,j} or χ_{n,i,j} instead would be more correct, but are not retained to avoid a useless burdening. As IE Φ_{i,j}(x) = IE η_{n}(x − X_{1}) does not depend on (i, j), we have:

Φ_{0,k} − Φ_{k,n} = Φ̃_{0,k} − Φ̃_{k,n} = \frac{1}{\sqrt{k}} χ_{0,k} − \frac{1}{\sqrt{n − k}} χ_{k,n}.  (60)

Finally we introduce the auxiliary smooth statistic

D_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∫_{IR}|Φ_{0,k} − Φ_{k,n}|^{2} dF.  (61)

Theorem 7 Let (X_{n})_{n≥1} be strictly stationary α-mixing, with mixing rate:

α(n) = O(n^{−τ} ),  (62)

and a marginal distribution verifying:

P (|X_{1}| > t) = O(t^{−β}), t → +∞.  (63)

Suppose moreover that for some positive δ we have:

β > 1 + δ,  (64)
τ > \frac{3}{2}(1 + \frac{1}{δ}).  (65)

Then if a_{n} goes to 0 and a_{n}^{−1} = o(\sqrt{n}), the statistic D̂_{n} converges in distribution to ∫_{[0,1]^{2}} ζ^{2}(s, t) dsdt where ζ is a centered Gaussian random field with covariance function R given by (10).

Theorem 8 Let (X_{n})_{n≥1} be strictly stationary associated with

Cov(X_{1}, X_{n}) = O(n^{−τ} ).  (66)

Assume the marginal distribution F is Hölderian with exponent 0 < ν ≤ 1:

|F (t) − F (s)| ≤ C|t − s|^{ν}, s, t ∈ IR,

and verifies the tail condition (63). Suppose moreover that for some positive δ:

β > 1 + δ,  (67)
τ > \frac{3λ}{2}(1 + \frac{1}{δ}) where λ = \frac{ν + 2}{ν}.  (68)

Then if a_{n} goes to 0 and a_{n}^{−1} = o(\sqrt{n}), the statistic D̂_{n} converges in distribution to ∫_{[0,1]^{2}} ζ^{2}(s, t) dsdt where ζ is a centered Gaussian random field with covariance function R given by (10).

In both theorems, hypotheses on F and on the dependence rate play antagonistic roles. In the strong mixing setting, one extremal case is β = +∞ (that is P (|X_{1}| > t) decreasing faster than any power of t^{−1}). The minimal dependence rate required here is only τ > 1.5. On the other hand if δ ≤ 1 in (64) and (65), the result obtained is weaker than for Â_{n} since the corresponding dependence rate is at least τ > 3 (with an extra assumption on F). The true domain of interest of Theorem 7 is then obtained for δ_{0} ≤ δ ≤ +∞ (where 1.5(1 + 1/δ_{0}) = 2.25) allowing dependence rates 1.5 < τ ≤ 2.25. To our knowledge, the result obtained in the case of associated variables is the only available one for change point testing quadratic statistics. Here the better rate of dependence is τ > 4.5 corresponding to distributions with a two-sided tail rapidly decreasing and F uniformly Lipschitz. The factor λ = ν/(ν + 2) in (68) comes from some covariance inequality between Cov(I_{{X_{1}≤x}}, I_{{X_{n}≤y}}) and Cov(X_{1}, X_{n}) given in Lemma 12. Note that in the strong-mixing case the covariance of indicators above is bounded by α(n).

4.2 A sketch of the proof

For an easier reading we only give here the basic arguments which are common to the proofs of Theorems 7 and 8. Technical details related to moment inequalities under mixing or association are deferred to the following subsections. First we prove the convergence to 0 of IE |D_{n}−A_{n}| which only needs variance inequalities. Then IE |D_{n} − D̂_{n}| is shown to converge to 0. The main technique used here is some integration by parts which reduces the problem to control third order moments of (smoothed) empirical processes and their derivatives.

Let us start with the convergence of IE |D_{n} − A_{n}|. Using the first equality in (60) we get:

|D_{n} − A_{n}| ≤ ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2}|∫_{IR}\{(Φ̃_{0,k} − Φ̃_{k,n})^{2} − (F̃_{0,k} − F̃_{k,n})^{2}\}dF |.

Factorizing the expression under the integral, expanding its expectation and applying the Cauchy-Schwarz inequality in L^{2}(Ω), we have to bound eight terms whose the first is

T_{1} = IE^{1/2}(Φ̃_{0,k}(x) − F̃_{0,k}(x))^{2} IE^{1/2} Φ̃_{0,k}^{2}(x).  (69)

Using moment inequalities (see Subsections 4.3 and 4.4), one can find some constant C such that:

T_{1} ≤ C \frac{ε(a_{n})}{k}, uniformly in x ∈ IR,  (70)

where ε(a_{n}) goes to 0 as n goes to ∞ (recall that Φ̃_{0,k} depends on n). This bound being uniform in x, integration with respect to dF on IR will change nothing and the contribution of T_{1} to the upper bound for IE |D_{n} − A_{n}| is then:

Cε(a_{n}) ∑_{k=1}^{n−1} \frac{1}{k} [\frac{k}{n}(1 − \frac{k}{n})]^{2} ∼ Cε(a_{n}) ∫_{0}^{1} t(1 − t)^{2} dt.

The contributions of the seven other terms T_{2}, …, T_{8} can be treated in the same way. In each case we use the asymptotic estimates (34) for Riemann sums.

Let us turn to IE |D̂_{n} − D_{n}|. Using (60) we have:

D̂_{n} − D_{n} = ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2} ∫_{IR}|\frac{1}{\sqrt{k}} χ_{0,k} − \frac{1}{\sqrt{n − k}} χ_{k,n}|^{2} \frac{dξ_{0,n}}{\sqrt{n}}

= ∑_{k=1}^{n−1}[\frac{k}{n}(1 − \frac{k}{n})]^{2}(\frac{T′_{1}}{k\sqrt{n}} − 2\frac{T′_{2}}{\sqrt{k(n − k)n}} + \frac{T′_{3}}{(n − k)\sqrt{n}})

where

T′_{1} = ∫_{IR} χ_{0,k}^{2} dξ_{0,n}, T′_{2} = ∫_{IR} χ_{0,k}χ_{k,n} dξ_{0,n}, T′_{3} = ∫_{IR} χ_{k,n}^{2} dξ_{0,n}.

Now it suffices to prove that for a suitable choice of (a_{n}), each IE |T′_{i}| is o(\sqrt{n}) uniformly in k. Again we only explicit the treatment of T′_{1}. Using integration by parts (cf. Lemmas 20 and 21 in Section 6), we get:

∫_{IR} χ_{0,k}^{2} dξ_{0,n} = −2 ∫_{IR} χ_{0,k}(x)χ′_{0,k}(x)ξ_{0,n}(x) dx.

By Hölder inequality in L^{3}(IR) and in L^{3}(Ω) it follows:

IE |T′_{1}| ≤ 2 IE^{1/3}∫_{IR} |χ_{0,k}(x)|^{3} dx IE^{1/3}∫_{IR} |χ′_{0,k}(x)|^{3} dx IE^{1/3}∫_{IR} |ξ_{0,n}(x)|^{3} dx.

(71)

Using third moment inequalities (see Subsections 4.3 and 4.4), one can find some positive constants C and ρ such that uniformly in i, j:

IE |ξ_{i,j}(x)|^{3} ≤ C[F (x)(1 − F (x))]^{ρ}, x ∈ IR.  (72)

Now classical inequality for convolutions gives

IE^{1/3}∫_{IR} |χ_{i,j}(x)|^{3} dx = IE^{1/3}∥a_{n}^{−1}I_{[0,a_{n}]} ∗ ξ_{i,j}∥_{L^{3}(IR)}^{3}
≤ IE^{1/3} (∥a_{n}^{−1}I_{[0,a_{n}]}∥_{L^{1}(IR)}^{3}∥ξ_{i,j}∥_{L^{3}(IR)}^{3})
= IE^{1/3}∫_{IR} |ξ_{i,j}(x)|^{3} dx.  (73)

For 0 ≤ i < j ≤ n, we have χ′_{i,j}(x) = a_{n}^{−1}(ξ_{i,j}(x) − ξ_{i,j}(x − a_{n})). Hence:

IE^{1/3}∫_{IR} |χ′_{i,j}(x)|^{3} dx ≤ 2a_{n}^{−1} IE^{1/3}∫_{IR} |ξ_{i,j}(x)|^{3} dx.  (74)

Finally from (71), (72), (73) and (74), we obtain:

IE |T′_{1}| ≤ C′a_{n}^{−1} ∫_{IR}[F (x)(1 − F (x))]^{ρ} dx,  (75)

uniformly in k. The treatment of T′_{2} and T′_{3} being perfectly similar, the convergence to 0 of IE |D̂_{n} − D_{n}| follows from a suitable condition on the tail of F and the choice of a_{n}^{−1} = o(\sqrt{n}).

4.3 Proof of Theorem 7

To complete the proof of Theorem 7, it suffices to check (70) and (72) when the sequence (X_{i})_{i≥1} is strictly stationary, with α-mixing rate given by (65) and marginal distribution satisfying the tail condition (64).

To prove (70) and the inequalities of the same type related to T_{2}, …, T_{8}, observe first that for 0 ≤ i < j ≤ n:

IE |ξ_{i,j}(x)|^{2} ≤ Var I_{\{X_{1}≤x\}} + 2 ∑_{1≤k<j−i}(1 − \frac{k}{j − i})α(k) ≤ 1 + 2 ∑_{k=1}^{∞} α(k).

So under (65), we have:

IE |F̃_{i,j}(x)|^{2} ≤ \frac{C}{j − i}, uniformly in x.  (76)

The same bound holds for IE |Φ̃_{i,j}(x)|^{2} via Jensen’s inequality:

IE |Φ̃_{i,j}(x)|^{2} = IE|\frac{1}{a_{n}} ∫_{x−a_{n}}^{x} F̃_{i,j}(y) dy|^{2}
≤ \frac{1}{a_{n}} ∫_{x−a_{n}}^{x} IE |F̃_{i,j}(y)|^{2} dy ≤ \frac{C}{j − i}.  (77)

To control IE |Φ̃_{i,j}(x) − F̃_{i,j}(x)|^{2}, consider a stationary α-mixing centered sequence (Y_{k})_{k≥1}. From Davydov’s inequality (cf. Theorem 11 in Section 6) with q = r and \frac{1}{p} = 1 − \frac{2}{r} follows the straightforward variance estimate:

\frac{1}{m} IE(∑_{k=1}^{m} Y_{k})^{2} ≤ IE Y_{1}^{2} + 8 ∑_{k=1}^{m−1} α(Y_{1}, Y_{k})^{1−2/r} IE^{2/r} |Y_{1}|^{r}.  (78)

Applying (78) to Y_{k} = (η − η_{n})(x − X_{k}) − IE(η − η_{n})(x − X_{k}), and using again Jensen’s inequality, we obtain

IE |Y_{1}|^{r} = IE|\frac{1}{a_{n}} ∫_{x−a_{n}}^{x}\{I_{(t,x]}(X_{1}) − (F (x) − F (t))\} dt|^{r}
≤ \frac{1}{a_{n}} ∫_{x−a_{n}}^{x} IE|I_{(t,x]}(X_{1}) − (F (x) − F (t))|^{r} dt  (79)

For each t ∈ [x − a_{n}, x],

IE|I_{(t,x]}(X_{1}) − (F (x) − F (t))|^{r} ≤ 2(F (x) − F (t)) ≤ 2(F (x) − F (x − a_{n})).

The same holds for IE Y_{1}^{2}, so going back to (79) we obtain for any 0 ≤ i < j ≤ n:

IE |Φ̃_{i,j}(x) − F̃_{i,j}(x)|^{2} ≤ \frac{C}{j − i} ω(F, a_{n})^{2/r}, uniformly in x,  (80)

where ω(F, a_{n}) denotes the modulus of continuity of F. This last estimate requires the convergence of the series ∑_{k≥1} α(k)^{1−2/r}. By our hypothesis α(k) = O(k^{−τ}) with τ > 3/2, the summability of the series requires only 2/r < 1 − 1/τ and hence any r > 6 is suitable. Now (70) and the inequalities of the same type follow from (76), (77) and (80).

To obtain (72), Corollary 15 of Yokoyama theorem (see (105) in Section 6) applied to Z_{k} = I{X_{k} ≤ x} − F (x) gives:

IE |ξ_{i,j}(t)|^{3} ≤ A IE^{3/(3+γ)}|I_{{X_{1}≤x}} − F (x)|^{3+γ}
≤ 8A[F (x)(1 − F (x))]^{3/(3+γ)},  (81)

under the condition:

∑_{j=1}^{+∞} j^{1/2}α(j)^{γ/(3+γ)} < ∞.  (82)

Recalling our assumptions P (|X_{1}| > t) = O(t^{−β}) and α(n) = O(n^{−τ}), the mixing rate needed here is given by

\frac{1}{2} − \frac{γτ}{3 + γ} < −1,  (83)

while the integrability condition on the tail of F required by (75) writes:

\frac{−3β}{3 + γ} < −1.  (84)

Putting γ = 3δ and solving (83), (84) lead to the hypotheses (64) and (65) of Theorem 7 whose proof is now complete.

4.4 Proof of Theorem 8

To complete the proof of Theorem 8, it suffices to verify (70) and (72) when (X_{i})_{i≥1} is associated, strictly stationary with marginal distribution function F of Hölder regularity ν and satisfies (67) and (68). The additional Hölder regularity assumption enables us to control the covariance of indicator functions. By Lemma 12 in Section 6, there exists a constant K only depending on F and ν such that:

0 ≤ Cov(I_{{X_{1}≤x}}, I_{{X_{l}≤x}}) ≤ K Cov^{ν/(ν+2)}(X_{1}, X_{l}),  (85)

To prove (70) a first application of (85) gives for 0 ≤ i < j ≤ n:

IE |ξ_{i,j}(x)|^{2} = \frac{1}{j − i} ∑_{k,l=i}^{j} Cov(I_{\{X_{k}≤x\}}, I_{\{X_{l}≤x\}})

≤ Var I_{{X_{1}≤x}} + 2 ∑_{l=2}^{∞} Cov(I_{{X_{1}≤x}}, I_{{X_{l}≤x}})

≤ 1 + 2K ∑_{l=2}^{∞} Cov^{ν/(ν+2)}(X_{1}, X_{l}).  (86)

The argument leading from (76) to (77) remains valid and hypothesis (68) assures the convergence of the series in (86). Hence there is a constant C such that:

IE |Φ̃_{i,j}(x)|^{2} ≤ IE |F̃_{i,j}(x)|^{2} ≤ \frac{C}{j − i}, uniformly in x.  (87)

To control IE |Φ̃_{i,j}(x) − F̃_{i,j}(x)|^{2}, we start from:

IE |χ_{i,j}(x) − ξ_{i,j}(x)|^{2} ≤ Var Y_{1} + 2 ∑_{l=2}^{∞} Cov(Y_{1}, Y_{l}),  (88)

with (η − η_{n})(x − X_{l}) = \frac{1}{a_{n}} ∫_{x−a_{n}}^{x} I_{(s,x]}(X_{l}) ds. By Fubini’s theorem:

Cov(Y_{1}, Y_{l}) = \frac{1}{a_{n}^{2}} ∫_{[x−a_{n},x]^{2}} Cov(I_{(s,x]}(X_{1}), I_{(t,x]}(X_{l})) ds dt.

For 0 < γ < 1/2, it follows:

| Cov(Y_{1}, Y_{l})| ≤ \frac{1}{a_{n}^{2}} ∫_{[x−a_{n},x]^{2}} Var^{γ/2} I_{(s,x]}(X_{1}) Var^{γ/2} I_{(t,x]}(X_{l})
|Cov(I_{(s,x]}(X_{1}), I_{(t,x]}(X_{l}))|^{1−γ} ds dt. (89)

Using the Hölder regularity of F, there is a constant C_{ν,γ} such that

Var^{γ/2} I_{(s,x]}(X_{1}) ≤ (F (x) − F (s))^{γ/2} ≤ C_{ν,γ}a_{n}^{νγ/2}  (90)

and clearly the same bound holds for Var^{γ/2} I_{(t,x]}(X_{l}). On the other hand:

|Cov(I_{(s,x]}(X_{1}), I_{(t,x]}(X_{l}))|^{1−γ} ≤ Cov^{1−γ}(I_{{X_{1}≤s}}, I_{{X_{l}≤t}})
+ Cov^{1−γ}(I_{{X_{1}≤s}}, I_{{X_{l}≤x}})
+ Cov^{1−γ}(I_{{X_{1}≤x}}, I_{{X_{l}≤t}})
+ Cov^{1−γ}(I_{{X_{1}≤x}}, I_{{X_{l}≤x}}). (91)

From (85), (88), (89), (90) and (91), if

∑_{l=2}^{∞} Cov(X_{1}, X_{l})^{(1−γ)ν/(ν+2)} < ∞,  (92)

there is a constant C′ such that uniformly in 0 ≤ i < j ≤ n and in x ∈ IR:

IE |Φ̃_{i,j}(x) − F̃_{i,j}(x)|^{2} ≤ \frac{C′a_{n}^{νγ}}{j − i} .  (93)

By our assumption (68) this condition is satisfied if \frac{3}{2}(1 − γ) ≥ 1, so any γ in (0, \frac{1}{3}] is suitable. The verification of (70) now follows from (87) and (93).

To obtain the bound (72), we apply Theorem 16 of Birkel (see Section 6) to Y_{l} = I_{{X_{l}≤x}} − F (x). Here the required dependence rate is:

u(n, x) = O(n^{−(3+δ)/2δ}),  (94)

where

u(n, x) = ∑_{l>n} Cov(I_{{X_{1}≤x}}, I_{{X_{l}≤x}}).  (95)

Under (94) we have uniformly in 0 ≤ i < j ≤ n:

IE |ξ_{i,j}|^{3} ≤ B_{3}.

Here the explicit bound for B_{3} given in (110) takes the form:

B_{3} ≤ max{40 IE |Ĩ_{{X_{1}≤x}}|^{3}, b_{δ} IE^{1/(1+δ)} |Ĩ_{{X_{1}≤x}}|^{3+δ}, b′_{δ}u(0, x)^{3(3+2δ)/2(1+δ)}}.

The first two terms in this maximum are bounded by C_{δ}|F (x)(1−F (x))|^{1/(1+δ)} where C_{δ} does not depend on x. The same bound can be obtained for the third term in the following way. For any 0 < γ′ < 1:

u(0, x) ≤ |F (x)(1 − F (x))|^{γ′} ∑_{l=1}^{∞} Cov^{1−γ′}(I_{{X_{1}≤x}}, I_{{X_{l}≤x}}).  (96)

The choice of γ′ = 2/(3(3 + 2δ)) leads to the expected bound, subject to the uniform convergence of the series in (96). Using (85), this last requirement will follow from:

τ(1 − γ′)\frac{ν}{ν + 2} > 1.  (97)

On the other hand, condition (94) will be satisfied if Cov(X_{1}, X_{n})^{ν/(ν+2)} = O(n^{−1−(3+δ)/2δ}) which reduces to:

τ > \frac{ν + 2}{ν} \frac{3(1 + δ)}{2δ},

which is exactly hypothesis (68). Now from our choice of γ′:

\frac{1}{1 − γ′} = 1 + \frac{2}{7 + 6δ} < \frac{3}{2},

so (97) is always satisfied under (68). Finally, the bound (72) holds with ρ = 1/(1 + δ) and taking account of (63) the integrability condition required by (75) is exactly (67). The proof of Theorem 8 is now complete.

5 Conclusion

Several important questions are not tackled in this paper. Firstly a comparison analysis of the proposed statistics and for instance of the Kolmogorov-Smirnov one (1) should be useful: in fact we do not know which of these statistics is best to use in which situation. Even in the independent case only few authors study the power of change-point testing procedures (see for instance [14] and [32]). For long memory sequences of observations there is an attempt in [18] to study the behaviour of Kolmogorov-Smirnov type tests along converging alternatives. But to our knowledge almost everything remains to be done in weak dependence contexts. Secondly nothing is said about estimating the change-point when H_{0} is rejected. Interesting results can be found in [8]. The authors prove the convergence of a family of estimators under the very weak condition of ergodicity. Finally there are all the questions linked with the limit law, that is the law of ∫_{[0,1]^{2}} ζ^{2}(s, t) dsdt. For instance the covariance function of the centered Gaussian random field ζ as defined in (10) and (11) clearly depends on the distributions of all the pairs (X_{1}, X_{k}). This means that in a dependence situation the quadratic statistics loose the nice advantage of having a known and tabulated limiting law (this drawback is common to all statistics whose asymptotic behaviour is linked to the empirical limiting process ξ with covariance (11)). Obtaining an estimation of this limit law is the subject of a foregoing work.

As was pointed out in the introduction, this paper is rather an attempt for obtaining the above-mentioned limiting law for as large as possible families of dependent sequences. From this point of view, our most achieved result is Theorem 1 concerning the statistic A_{n} (remark that the condition ∑ α(n) < ∞ providing the convergence of A_{n} is just the one required by Ibragimov’s CLT for bounded strong mixing X_{i}’s). Results on the distribution free statistic Â_{n} seem less satisfactory. On one hand our condition α(n) = O(n^{−τ}) with τ > 2.25 for the convergence of Â_{n} is weaker than the rate τ > 1 + \sqrt{2} which provides the convergence of the (unidimensional) uniform empirical process ξ_{n} in D(0, 1). On the other hand we have no result at all for this statistic in the associated case. To our knowledge this is an open question. As for D̂_{n} which belongs to the family of statistics built from the perturbed empirical process, the reason for introducing it mainly was to get round the difficulty met with Â_{n}. In fact we obtain its convergence for mixing as well as associated sequences. The fact that we need for D̂_{n} hypotheses both on the dependence rate and on the distribution function F is not surprising since it is now well known (see for instance [28]) that smoothed versions of F_{0,n} may perform better than F_{0,n} in estimating the underlying distribution function F, at least when F is smooth enough.

6 Auxiliary results

6.1 Tightness in L^{2}[0, 1]^{2}

Our main tool to prove the tightness of {ζ_{n}, n ≥ 2} in L^{2}[0, 1]^{2} is the following theorem which generalizes Prohorov’s Theorem (Th. 1.13 [25]).

Theorem 9 (Suquet [29]) Let X be a Banach space having the decomposition X = ⨁_{j∈IN} G_{j} where each G_{j} is of finite dimension and the direct sum is topological. Denote by E_{j} the continuous projection from X onto V_{j} = ⨁_{i≤j} G_{j}. A sequence (ξ_{n})_{n≥1} of random elements in X is tight if:

(i) ∃α > 0, sup_{n≥1} IE ∥ξ_{n}∥^{α} < +∞,

(ii) ∃β > 0, lim_{j→+∞} sup_{n≥1} IE ∥ξ_{n} − E_{j}ξ_{n}∥^{β} = 0.

This theorem apply with α = β = 2 either to X = L^{2}[0, 1] or to X = L^{2}[0, 1]^{2} with (V_{j})_{j≥0} the Haar multiresolution analysis. We give now the properties of the Haar multiresolution analysis which are needed in this paper. \underline{Unidimensional case}: Here V_{j} is the subspace of functions constant on each interval [k2^{−j}, (k + 1)2^{−j}) (0 ≤ k < 2^{j}) and E_{j} is the orthogonal projector on V_{j}. Note that E_{j}f is merely the approximation of f ∈ L^{2}[0, 1] by a step function equal to its mean value over each [k2^{−j}, (k + 1)2^{−j}). Let us write E_{j}(s, t) for the integral kernel of E_{j}, that is:

(E_{j}f)(s) = ∫_{0}^{1} E_{j}(s, t)f(t) dt, f ∈ L^{2}[0, 1], s ∈ [0, 1].

It is easily verified that:

E_{j}(s, t) = ∑_{0≤k<2^{j}} 2^{j}I_{C_{j,k}}(s, t) where C_{j,k} = [k2^{−j}, (k + 1)2^{−j})^{2}.

So E_{j}(s, t) is a probability density on [0, 1]^{2} which is constant on the “diagonal squares” C_{j,k} of its support. Moreover each marginal of E_{j}(s, t) is the uniform density on the dyadic interval which contains the fixed variable. \underline{Bidimensional case}: To avoid confusions with the unidimensional case, we will note (Ṽ_{j}) for the multiresolution analysis and Ẽ_{j} for the associated orthogonal projectors. Here Ṽ_{j} is the subspace of functions constant on each square

C_{j,k,l} = [k2^{−j}, (k + 1)2^{−j}) × [l2^{−j}, (l + 1)2^{−j}) 0 ≤ k < 2^{j}, 0 ≤ l < 2^{j}.

For any h ∈ L^{2}[0, 1]^{2}, Ẽ_{j}h is the step function equal to the mean value of h on each C_{j,k,l}. The integral kernel of Ẽ_{j} implicitly defined by

Ẽ_{j}h(x, y) = ∫_{[0,1]^{2}} Ẽ_{j}(x, y, s, t)h(s, t) ds dt, h ∈ L^{2}[0, 1]^{2}, (x, y) ∈ [0, 1]^{2} can easily be expressed as

Ẽ_{j}(x, y, s, t) = 2^{2j} ∑_{0≤k,l<2^{j}} I_{C_{j,k,l}}(x, y)I_{C_{j,k,l}}(s, t), (x, y, s, t) ∈ [0, 1]^{4},

or equivalently

Ẽ_{j}(x, y, s, t) = 2^{2j} ∑_{0≤k,l<2^{j}} I_{C_{j,k,k}}(x, s)I_{C_{j,l,l}}(y, t) = E_{j}(x, s)E_{j}(y, t).

As a result we have the convenient formula

Ẽ_{j}(f ⊗ g) = (E_{j}f) ⊗ (E_{j}g)  f, g ∈ L^{2}[0, 1],  (98)

where f ⊗ g is defined by f ⊗ g(x, y) = f(x)g(y). As ζ_{n} is a sum of such tensor products, the following lemma whose verification is straightforward is useful:

Lemma 10 Let the f_{i}, 1 ≤ i ≤ n be orthogonal functions in L^{2}[0, 1]. Then for any system {g_{i}, 1 ≤ i ≤ n} of functions in L^{2}[0, 1]

∥∑_{i=1}^{n} f_{i} ⊗ g_{i}∥^{2} = ∑_{i=1}^{n} ∥f_{i}∥^{2} ∥g_{i}∥^{2},

where ∥ ∥ denotes the Hilbertian norms of the spaces L^{2}[0, 1]^{2} and L^{2}[0, 1].

6.2 Dependent variables

In this paper two types of dependence are considered: strong mixing and association. The strong mixing coefficient of two σ-fields A and B is defined by

α(A, B) = sup{|P(A ∩ B) − P(A) P(B)| , A ∈ A, B ∈ B}.

As a special case, for random variables or vectors X, Y, we write α(X, Y) for α(σ(X), σ(Y)), where σ(X) denotes the σ-field generated by X. The sequence (X_{n})_{n≥1} is called strong mixing (or α-mixing) if the coefficients

α(n) = sup{α(F_{0}^{k+1}, F_{n+k}^{∞}), k ∈ IN}

converge to 0 as n → ∞, where F_{j}^{l} stands for the σ-field generated by the variables (X_{i}, j < i < l). For the properties of strong mixing, the reader is referred to [12] where an abundant bibliography can be found.

On the other hand the random variables (X_{n})_{n≥1} are called associated if,

Cov (f(X_{1}, …, X_{k}), g(X_{1}, …, X_{k})) ≥ 0,

for each k ≥ 1, each finite sequence (X_{1}, …, X_{k}) and each pair of coordinatewise non decreasing functions f, g on IR^{k} for which the covariance above exists. The basic properties of association can be found in [15] whereas [21] provides a review of the limit theorems for this type of dependence.

6.3 Covariance inequalities

We recall the covariance inequality under α-mixing:

Theorem 11 (Davydov [10]) Let X and Y be centered IR-valued random variables with finite variances. For any p, q, r ≥ 1 and \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1,

| Cov(X, Y)| ≤ 8α(X, Y)^{1/p} IE^{1/q} |X|^{q} IE^{1/r} |Y |^{r}.  (99)

In the case of associated variables, a key point is the control of covariances of indicators.

Lemma 12 Let X, Y be associated random variables having a common distribution function F with Hölder regularity of exponent 0 < ν ≤ 1, that is:

∥F ∥_{C^{ν}} = \begin{matrix}sup \\ x,y∈IR \\ x≠y \end{matrix} \frac{|F (x) − F (y)|}{|x − y|^{ν}} < +∞.  (100)

Then for any x, y ∈ IR:

0 ≤ Cov(I_{{X≤x}}, I_{{Y ≤y}}) ≤ K Cov^{ν/(ν+2)}(X, Y),  (101)

where the constant K only depends on ν and ∥F ∥_{C^{ν}}.

Proof : Put h_{0} = I_{[0,+∞)} and define it’s smooth approximate h_{ε} by: h_{ε}(x) = ε^{−1}(x + ε)I_{[−ε,0)}(x) + h_{0}(x). Then following the proof of Lemma 4.5 in [37], we easily get:

0 ≤ Cov(I_{{X≤x}}, I_{{Y ≤y}})
≤ \frac{1}{ε^{2}} Cov(X, Y) + IE(h_{ε}(x − X) − h_{0}(x − X)) IE h_{ε}(y − Y)
+ IE(h_{ε}(y − Y) − h_{0}(y − Y)) IE h_{0}(x − X)
≤ \frac{1}{ε^{2}} Cov(X, Y) + (F (x) − F (x − ε)) + (F (y) − F (y − ε))
≤ \frac{1}{ε^{2}} Cov(X, Y) + 2∥F ∥_{C^{ν}} ε^{ν} = \frac{a}{ε^{2}} + bε^{ν} say.  (102)

The choice of ε = (\frac{a}{bν} )^{1/(ν+2)} gives the minimum in the right hand side of (102). Hence

0 ≤ Cov(I_{{X≤x}}, I_{{Y ≤y}}) ≤ a^{ν/(ν+2)}b^{2/(ν+2)}[ν^{2/(ν+2)} + 2ν^{−ν/(ν+2)}]

which gives the announced inequality.

While strong mixing is preserved under mapping by measurable functions, under association the same result holds only for coordinatewise monotone functions. Nevertheless, one can prove central limit theorems for absolutely continuous functions of associated variables. The keystone of such results is the Newman inequality presented below. Following Newman, for f and f_{1} complex functions on IR^{m}, we write f << f_{1} if f_{1} − Re(e^{iθ}f) is coordinatewise non decreasing for all real θ. In the following statement, f̄ is a local notation (the same as in [21]) having nothing to do with the complex conjugate of f.

Theorem 13 (Newman [21]) Suppose the X_{j}’s are associated and define Y_{j} = f_{j}(X_{1}, X_{2}, …, X_{n}), Ȳ_{j} = f̄_{j}(X_{1}, X_{2}, …, X_{n}). Suppose further that f_{j} << f̄_{j} for each j. Then we have

|IE exp(i ∑_{j=1}^{l} r_{j}Y_{j}) − ∏_{j=1}^{l} IE exp(ir_{j}Y_{j})| ≤ 2 ∑_{1≤j<k≤l}|r_{j}r_{k} Cov(Ȳ_{j}, Ȳ_{k})|.

6.4 Moments inequalities

The following theorem gives the moment bounds for sums under α-mixing:

Theorem 14 (Yokoyama [34]) Let (Y_{j})_{j≥1} be a strictly stationary sequence of α-mixing centered random variables verifying IE |Y_{1}|^{r+γ} < ∞ for some r > 2 and γ > 0. If moreover

∑_{j=1}^{∞}(j + 1)^{r/2−1}α(j)^{γ/(γ+r)} < ∞,  (103)

there is a constant K such that:

IE|∑_{j=1}^{n} Y_{j}|^{r} ≤ Kn^{r/2}, n ≥ 1.  (104)

The constant K depends on the sequence (α(j))_{j≥1}, r, γ and IE |Y_{1}|^{r+γ}. A close examination of Yokoyama’s proof allows us to take K = A max(1; IE |Y_{1}|^{r+γ}), where A only depends on (α(j))_{j≥1} and r. It follows:

Corollary 15 If (Z_{j})_{j≥1} is a strictly stationary sequence of α-mixing centered random variables verifying IE |Z_{1}|^{r+γ} < ∞ for some r > 2 and γ > 0 and with mixing coefficients α(j) satisfying (103), then

IE|∑_{j=1}^{n} Z_{j}|^{r} ≤ An^{r/2}(IE |Z_{1}|^{r+γ})^{r/(r+γ)}, n ≥ 1,  (105)

with A depending only on (α(j))_{j≥1} and r.

Proof : It suffices to apply Theorem 14 with the Y_{j} = (IE |Z_{1}|^{r+γ})^{−1/(r+γ)}Z_{j} whose mixing coefficients are the same as for (Z_{j})_{j≥1}.

In the case of associated variables, moment bounds of the same type as in (104) are due to Birkel. Here the required dependence rate condition involves the asymptotic behaviour of the Cox and Grimmett coefficient:

u(n) = sup_{k∈IN^{∗}} ∑_{|j−k|≥n} Cov(Y_{j}, Y_{k}), n ∈ IN.  (106)

In the stationary case, u(n) = 2 ∑_{j>n} Cov(Y_{1}, Y_{j}).

Theorem 16 (Birkel [2]) Let {Y_{j}, j ∈ IN^{∗}} be a sequence of associated random variables satisfying IE Y_{j} = 0 and

sup_{j∈IN^{∗}} IE |Y_{j}|^{r+δ} < ∞, for some r > 2 and δ > 0.  (107)

Assume

u(n) = O(n^{−(r−2)(r+δ)/2δ}).  (108)

Then there is a constant B_{r} not depending on n such that for all n ∈ IN^{∗}:

sup_{m∈IN} IE| ∑_{j=m+1}^{m+n} X_{j}|^{r} ≤ B_{r}n^{r/2}.  (109)

Computing all the constants used in the proof of Birkel’s result is rather tedious and gives a recursive formula expressing B_{r} in terms of B_{r−1}, r, δ, the constant involved in the O of (108) and of moments of order r and r + δ of the Y_{j}’s. Fortunately in the special case of a stationary process {Y_{j}, j ∈ IN^{∗}} and r = 3, an explicit computation of B_{3} can be achieved and the result takes the form:

B_{3} ≤ max{40 IE |Y_{1}|^{3}, b_{δ} IE^{1/(1+δ)} |Y_{1}|^{3+δ}, b′_{δ}u(0)^{3(3+2δ)/2(1+δ)}}  (110)

where

b_{δ} = 40[\frac{2^{7}}{2^{3/2} − 1}]^{(3+2δ)/(1+δ)}b^{δ/(1+δ)}, b′_{δ} = 40[\frac{2^{4}}{2^{3/2} − 1}]^{(3+2δ)/(1+δ)},

with b = sup_{n≥1} n^{(3+δ)/2δ}u(n).

6.5 Reconstruction

Here is the special case of Bradley’s reconstruction theorem used in Section 3.

Theorem 17 (Bradley [5]) Suppose X and Y are random variables taking their values on S and IR^{d} respectively, where S is a Borel space; suppose U is a uniform-[0, 1] random variable independent of (X, Y); and suppose q is a positive number such that q ≤ ∥Y ∥_{∞} < ∞, where ∥Y ∥_{∞} = ess sup∥Y ∥. Then there exists a real-valued random variable Y^{∗} = f(X, Y, U), where f is a measurable function from S × IR^{d} × [0, 1] into IR^{d}, such that (i) Y^{∗} is independent of X, (ii) Y^{∗} and Y have the same distribution, and (iii) P (|Y^{∗} − Y | ≥ q) ≤ 18(∥Y ∥_{∞}/q)^{1/2}α(X, Y). From this theorem some technical results can be derived: Corollary 18 Let X and Y be uniformly distributed on [0, 1]. Then

(i) P (X = Y) ≤ cα(X, Y)^{2/3}, and for every r > 0 there exists Y^{∗}, uniformly distributed on [0, 1] and independent of X such that: (ii) IE |Y − Y^{∗}|^{r} ≤ cα(X, Y)^{2r/2r+1}, Proof : Let q = α(X, Y)^{2/3} and Y^{∗} be as in Theorem 17. Then

P (X = Y) ≤ P (|Y − Y^{∗}| > q) + P (|X − Y^{∗}| ≤ q)
≤ c[q^{−1/2}α(X, Y) + q]
≤ cα(X, Y)^{2/3}.

As for (ii), note that for any Z uniformly [0, 1]-distributed and independent of X,

IE |Y − Z|^{r} ≤ c[q^{r} + P (|Y − Z| > q)].

Choosing q = α(X, Y)^{2r/2r+1} and Z = Y^{∗} accordingly gives (ii). Remark: Since 0 ≤ |X − Y | ≤ 1 a.s., Rio’s inequality IE |X − Y | ≤ cα(X, Y) (see [26]) leads to

IE |X − Y |^{r} ≤ IE |X − Y | ≤ cα(X, Y), when r ≥ 1,

and

IE |X − Y |^{r} ≤ (IE |X − Y |)^{r} ≤ c^{r}α(X, Y)^{r}, when 0 < r < 1.

As the mixing coefficient α is smaller than one, the bound α^{2r/2r+1} given by (ii) is better only for r < 1/2. Recall we used (ii) with r = 1/4 in (52).

6.6 Integration by parts formulas

Integration by parts gives the useful representation formula for the covariance: Lemma 19 Let X, Y be random variables and f, g be absolutely continuous functions such that f(X) and g(Y) have finite second moments. Then

Cov(f(X), g(Y)) = ∫_{IR_{2}} f′(x)g′(y) Cov(I_{{X≤x}}, I_{{Y ≤y}}) dx dy.

We now state the integration by parts formula relative to the empirical process.

Lemma 20 ([31]) Let f be a continuous function IR → IR verifying:

(i) lim_{x→−∞} f(x) = 0

(ii) f has a right derivative (denoted here by f′) in each x ∈ IR and f′ belongs to L^{1}(IR).

Then we have:

∫_{IR} f(x) dξ_{0,n}(x) = − ∫_{IR} f′(y)ξ_{0,n}(y) dy.  (111)

The application of Lemma 20 to random functions f related to the smoothed empirical process needs the following result.

Lemma 21 ([31]) If the distribution F has a finite expectation, the smoothed empirical process χ_{i,j} = a_{n}^{−1}I_{[0,a_{n}]} ∗ξ_{i,j} has a right derivative χ′_{i,j} in each x ∈ IR, χ′_{i,j} ∈ L^{1}(IR) and:

χ′_{i,j}(x) = a_{n}^{−1}(ξ_{i,j}(x) − ξ_{i,j}(x − a_{n})), x ∈ IR.

Finally we give a result which is familiar when the involved functions H_{1} and H_{2} are C^{1}. It’s proof in the general case is straightforward by successive applications of Fubini’s theorem and shall be omitted.

Lemma 22 Let H_{1} and H_{2} be two differences of distribution functions with discontinuity points x_{1}, x_{2}, … Then

(i) ∫_{IR} H_{1}^{2}(x) dH_{1}(x) =

∑_{i}(H_{1}(x_{i})−H_{1}(x_{i}^{−}))^{2}H_{1}(x_{i})−\frac{1}{3} ∑_{i}(H_{1}(x_{i})−H_{1}(x_{i}^{−}))^{3},

(ii) 2 ∫_{IR} H_{1}(x)H_{2}(x) dH_{2}(x) =

∫_{IR} H_{2}(x^{−})^{2} dH_{1}(x)+∑_{i}(H_{2}(x_{i})−H_{2}(x_{i}^{−}))^{2}H_{1}(x_{i}).

Acknowledgments We would like to express our gratitude to Professor A. Antoniadis who suggested this work and to Professor P. Doukhan for useful discussions.

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