Weak convergence in L^{p}(0, 1) of the uniform empirical process under dependence

P. E. Oliveira^{2}

Universidade de Coimbra, Portugal

Ch. Suquet

Université des Sciences et Technologies de Lille, France

Abstract: The weak convergence of the empirical process of strong mixing or associated random variables is studied in L^{p}(0, 1). We find minimal rates of convergence to zero of the mixing coefficients or the covariances, in either case, supposing stationarity of the underlying variables. The rates obtained improve, for p not too large, the corresponding results in the classical D(0, 1) framework.

AMS Subject Classification: 60B10, 60F17, 60G30.

Keywords: association, empirical process, strong mixing, tightness.

1 Introduction

Let (U_{n}, n ≥ 1) be a stationary sequence of uniform random variables on [0, 1] and define, as usual, the uniform empirical process ξ_{n} by:

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

When the U_{i} are independent, (ξ_{n}, n ≥ 1) is well known to converge weakly in the Skorokhod space D(0, 1) to the Brownian bridge. This functional central limit theorem has been widely extended to dependent U_{i}’s, giving a limiting process ξ, which is Gaussian centered with covariance

Γ(s, t) = s ∧ t − st + ∑_{k=2}^{∞}(P(U_{1} ≤ s, U_{k} ≤ t) − st) + ∑_{k=2}^{∞}(P(U_{k} ≤ s, U_{1} ≤ t) − st).

In each particular dependence context, results are stated using some “dependence rate”, which can be a mixing coefficient or the covariance Cov(U_{1}, U_{k}) in case of associated U_{i}’s. A natural bound for this dependence rate is clearly the best known rate to obtain the finite dimensional CLT for the random variables 1I_{(s,t]}(U_{i}). Those bounds were obtained under ρ-mixing (see Shao and Yu [12]) and under β-mixing up to a logarithmic factor (see Doukhan, Massart and Rio [6]). As for the strong mixing case or the associated one, this achievement remains an open question. Yoshihara [16] proved the weak D(0, 1) convergence of (ξ_{n}, n ≥ 1) under the strong mixing rate α(n) = O(n^{−a}) with a > 3, which was later improved by Shao [11] requiring only a > 2. For associated U_{i}’s, Yu obtained the weak D(0, 1) convergence of (ξ_{n}, n ≥ 1) under the assumption Cov(U_{1}, U_{n}) = O(n^{−a}) with a > 7.5. Shao and Yu [12] improved this result proving that it is enough to have a > (3 + \sqrt{33})/2 ≃ 4.373.

1Correspondence to: P. E. Oliveira, Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal.

2This author was supported by Junta Nacional de Investigação Científica e Tecnológica, Instituto de Telecomunicações - Pólo de Coimbra and grant PRAXIS/2/2.1/MAT/19/94

The classical framework to study the weak convergence of (ξ_{n}, n ≥ 1) to a Gaussian process is the Skorokhod topology. This approach provides useful results about the asymptotic behaviour of a large class of functionals. ¿From a statistical point of view, one can be interested in weakening the dependence rate assumptions for a class of functionals of paths continuous in a weaker topology than Skorokhod’s one. For instance, the Cramér-von Mises ω^{2} test statistic, the Watson statistic [19], the Shepp statistic [13], the Anderson Darling statistics [1], some von Mises functionals, some functionals of the type ∫ G(t, ξ_{n}(t)) μ(dt) require only the L^{2} or L^{p} topology on the space of paths [5]. In [10], the authors obtained the L^{2}(0, 1) weak convergence of (ξ_{n}) to the same process ξ as above, under the hypotheses:

∑_{n≥1} α(n) < ∞  in the α-mixing stationary case  (2)
∑_{n≥1} Cov^{1/3}(U_{1}, U_{n}) < ∞  in the associated stationary case.  (3)

Let us observe that (2) is simply the condition giving the Ibragimov’s unidimensional CLT for bounded strong mixing random variables [7]. In this case the sequence (1I_{(s,t]}(U_{n}), n ≥ 1) is strong mixing with mixing coefficients dominated by those of (U_{n}, n ≥ 1). In the associated case, the best known dependence rate to obtain the unidimensional CLT for the U_{i}’s is the summability of the sequence Cov(U_{1}, U_{n}). Unfortunately the sequence (1I_{(s,t]}(U_{n}), n ≥ 1) does not inherit the association property from (U_{n}, n ≥ 1) and the control of covariances of indicators explains the extra exponent 1/3.

As one can see, there is an hiatus for intermediate results allowing applications to a larger set of functionals than L^{2}(0, 1) continuous ones, at a lower cost than D(0, 1)’s conditions. For α-mixing stationary underlying variables U_{i}, Suquet [15] proves the L^{p}(0, 1)-weak convergence of ξ_{n} under the conditions α(n) = O(n^{−a}) with a > p/2. The main tool used in [10] and [15] is some special wavelets multiresolution analysis of L^{p} spaces. In this paper we propose a more classical approach of L^{p}(0, 1) weak convergence allowing a unified treatment of strong mixing and associated cases obtaining a general condition which gives us back the results of [15].

Our results are stated in Section 2 together with some tightness conditions which can have their own interest. Proofs are given in Section 3 and some applications to functionals of paths are presented in Section 4.

2 Results

Theorem 1 Let (ξ_{n}, n ≥ 1) be a sequence of random elements in L^{p}(0, 1), p ≥ 1, verifying:

(i) For some γ > 1, sup_{n≥1} IE ∥ξ_{n}∥_{1}^{γ} < ∞,

(ii) lim_{h→0} sup_{n≥1} IE ∥ξ_{n}(· + h) − ξ_{n}(·)∥_{p}^{p} = 0.

Then (ξ_{n}, n ≥ 1) is tight in L^{p}(0, 1).

Corollary 2 A sufficient condition for the sequence of random functions (ξ_{n}, n ≥ 1) to be tight in L^{p}(0, 1), p ≥ 1, is the existence of positive exponents q ≤ p < r such that:

(a) For some constant c,

IE |ξ_{n}(t)|^{r} ≤ c, t ∈ [0, 1], n ≥ 1;

(b) For some function ε(h) = o(1) as h → 0:

IE |ξ_{n}(t + h) − ξ_{n}(t)|^{q} ≤ ε(h), 0 ≤ h < 1, 0 ≤ t ≤ 1 − h, n ≥ 1.

In the next theorem, we suppose that p ≥ 2, {U_{n}, n ≥ 1} is a stationary sequence of [0, 1]-uniformly distributed random variables and ξ_{n} denotes the uniform empirical process defined by (1).

Theorem 3 Assume the sequence {U_{n}, n ≥ 1} is either strong mixing with rate

∑_{n≥1} α(n) < ∞  if p = 2,  (4)
α(n) = O(n^{−a}) for some a > p/2, if p > 2;  (5)

either associated with

∑_{n≥1} Cov^{1/3}(U_{1}, U_{n}) < ∞  if p = 2,  (6)
Cov(U_{1}, U_{n}) = O(n^{−a}) for some a > 3p/2, if p > 2.  (7)

Then the uniform empirical process ξ_{n} weakly converges in L^{p}(0, 1) to a centered Gaussian process ξ with a.s. continuous paths and covariance Γ:

Γ(s, t) = s ∧ t − st + ∑_{k=2}^{∞}(P(U_{1} ≤ s, U_{k} ≤ t) − st) + ∑_{k=2}^{∞}(P(U_{k} ≤ s, U_{1} ≤ t) − st).

3 Proofs

Proof of Theorem 1: We shall use a convolution approximation. To avoid notational complications, we define ξ_{n}(t) = 0 for any t outside [0, 1]. Let κ be a probability density with support [−1, 1]. We suppose moreover that κ is Lipschitz, that is:

∥κ∥_{Lip} := sup_{s≠t} \frac{|κ(t) − κ(s)|}{|t − s|} < ∞.

For any positive integer j, define κ_{j}(t) = jκ(jt). The sequence (κ_{j}, j ≥ 1) is an approximate identity. Using the classical trick:

κ_{j} ∗ ξ_{n}(x) − ξ_{n}(x) = ∫_{|t|≤1/j}{ξ_{n}(x − t) − ξ_{n}(x)}κ_{j}(t) dt

and the Jensen inequality with respect to the probability measure κ_{j}(t) dt, we easily obtain:

IE ∫_{IR} |κ_{j} ∗ ξ_{n}(x) − ξ_{n}(x)|^{p} dx ≤ IE ∫_{|t|≤1/j} κ_{j}(t) dt ∫_{IR} |ξ_{n}(x − t) − ξ_{n}(x)|^{p} dx
≤ sup_{|t|≤1/j} IE ∥ξ_{n}(·) − ξ_{n}(· + t)∥_{p}^{p}.

Hence by (ii),

lim_{j→∞} sup_{n≥1} IE ∥κ_{j} ∗ ξ_{n} − ξ_{n}∥_{p}^{p} = 0.  (8)

Now it is easily checked that

IE ∥κ_{j} ∗ ξ_{n}∥_{∞} ≤ ∥κ_{j}∥_{∞} sup_{i≥1} IE ∥ξ_{i}∥_{1}.

and for any 0 ≤ s < t ≤ 1:

IE |κ_{j} ∗ ξ_{n}(t) − κ_{j} ∗ ξ_{n}(s)|^{γ} ≤ ∥κ_{j}∥_{Lip}^{γ} sup_{i≥1} IE ∥ξ_{i}∥_{1}^{γ} |t − s|^{γ}.

It follows (see for instance Theorem 12.3, p. 95 in [3]) that for each fixed j the sequence (κ_{j} ∗ ξ_{n}, n ≥ 1) is tight in C(0, 1) and hence also in L^{p}(0, 1).

For fixed η > 0, define η_{i} = 2^{−i}η (i ≥ 1) and choose a sequence of positive ε_{i} decreasing to 0. By (8) and the Markov inequality, we can extract a subsequence of indexes j_{i} such that:

P (∥ξ_{n} − κ_{j_{i}} ∗ ξ_{n}∥_{p} > ε_{i}) < η_{i}, n ≥ 1, i ≥ 1.  (9)

By the L^{p}(0, 1)-tightness of the sequence (κ_{j_{i}} ∗ ξ_{n}, n ≥ 1), there is some compact C_{i} in L^{p}(0, 1) such that

P (κ_{j_{i}} ∗ ξ_{n} ∉ C_{i}) < η_{i}, n ≥ 1, i ≥ 1.  (10)

¿From (9) and (10) we deduce:

P (ξ_{n} ∈ A) > 1 − 2η, n ≥ 1,

where

A = {f ∈ L^{p}(0, 1), κ_{j_{i}} ∗ f ∈ C_{i} and ∥κ_{j_{i}} ∗ f − f∥_{p} ≤ ε_{i}, i ≥ 1}.

Clearly for any fixed δ > 0, A can be recovered by a finite number of balls with radius 2δ. As L^{p}(0, 1) is complete, the compacity of A follows and (ξ_{n}, n ≥ 1) is tight in L^{p}(0, 1).

Proof of Corollary 2: If q = p it is evident that (b) implies (ii). Suppose now that q < p and denote by 1/u and 1/v the barycentric coordinates of p in the segment (q, r), that is:

p = \frac{1}{u}q + \frac{1}{v} r, \frac{1}{u} + \frac{1}{v} = 1, u, v > 0.

The Hölder inequality applied to any non negative random variable Y writes:

IE Y^{p} ≤ IE^{1/u} Y^{q} IE^{1/v} Y^{r}.

Choosing Y = |ξ_{n}(t + h) − ξ_{n}(t)| and integrating with respect to t, we get:

IE ∥ξ_{n}(· + h) − ξ_{n}(·)∥_{p}^{p} ≤ ∫ IE^{1/u} |ξ_{n}(t + h) − ξ_{n}(t)|^{q} IE^{1/v} |ξ_{n}(t + h) − ξ_{n}(t)|^{r} dt
≤ (2^{r−1}c)^{1/v}ε(h)^{1/u},

using (a) and (b). Hence IE ∥ξ_{n}(· + h) − ξ_{n}(·)∥_{p}^{p} converges to zero, uniformly in n, as h goes to zero and the hypotheses of Theorem 1 are satisfied.

Proof of Theorem 3: We first check the tightness of (ξ_{n}, n ≥ 1). \underline{Case p = 2}: By stationarity of (U_{i}, i ≥ 1), we have

IE |ξ_{n}(t) − ξ_{n}(s)|^{2} = g_{1}(s, t) + 2 ∑_{k=2}^{n}(1 − \frac{k}{n})g_{k}(s, t),

where

g_{k}(s, t) = Cov(1I_{(s,t]}(U_{1}), 1I_{(s,t]}(U_{k})), s, t ∈ [0, 1], k ≥ 1.

In the strong mixing case, the bound |g_{k}(s, t)| ≤ α(k − 1) follows obviously from the definition of the mixing coefficients. In the associated case, Lemma 4.5 of Yu [18] provides the uniform bound:

|g_{k}(s, t)| ≤ 16 Cov^{1/3}(U_{1}, U_{k}), s, t ∈ [0, 1].  (11)

So under (4) or (6),

IE |ξ_{n}(t) − ξ_{n}(s)|^{2} ≤ g_{1}(s, t) + 2 ∑_{k=2}^{∞} |g_{k}(s, t)|  (12)

where the right hand side is a normally convergent series of continuous functions. Its sum, say g(s, t), is uniformly continuous on the square [0, 1]^{2} and vanishes on the diagonal. This gives the uniform estimate

IE |ξ_{n}(t + h) − ξ_{n}(t)|^{2} ≤ sup_{0≤s≤1} g(s, s + h) = ε(h), t ∈ [0, 1], n ≥ 1.  (13)

So we have condition (b) of Corollary 2 and hence (ii) of Theorem 1 is verified. Condition (i) with γ = 2 is easily obtained choosing s = 0 in (12). \underline{Case p > 2}: We use Corollary 2. From (13), condition (b) is satisfied with q = 2.

By Theorem 2 of Yokoyama [17], condition (a) holds in the strong mixing case if the sequence (n^{r/2−1}α(n), n ≥ 1) is summable. Choosing p < r < 2a, this follows from our assumption (5).

In the associated case, the random variables 1I_{[0,t]}(U_{i}) are associated as non increasing functions of associated variables. So combining the moment inequality of Birkel [4] with (11), we obtain condition (a) of Corollary 2 by choosing p < r < 2a/3.

Once the tightness of (ξ_{n}) established, to prove the weak L^{p}(0, 1) convergence of ξ_{n} to ξ, it suffices to verify the convergence in distribution of ∫_{0}^{1} f(t)ξ_{n}(t) dt to ∫_{0}^{1} f(t)ξ(t) dt for each f in L^{p′}(0, 1) the dual space of L^{p}(0, 1). Observe now that:

∫_{0}^{1} f(t)ξ_{n}(t) dt = \frac{1}{\sqrt{n}} ∑_{i=1}^{n}(∫_{U_{i}}^{1} f(t) dt − IE ∫_{U_{i}}^{1} f(t) dt).

As L^{p′}(0, 1) ⊂ L^{1}(0, 1), the function F (x) = ∫_{x}^{1} f(t) dt is absolutely continuous. In the strong mixing case, Ibragimov’s central limit theorem [7] for strong mixing bounded random variables applies to the F (U_{i})’s and gives the desired convergence and the covariance function of ξ. In the associated case, the same conclusion is obtained from the following theorem.

Theorem 4 (Newman, [9]) Let (Y_{n})_{n≥1} be strictly stationary associated random variables and X_{n} =

F (Y_{n}), where F is an absolutely continuous function. Put X̄_{n} = F̄(Y_{n}) where F̄(t) = ∫_{[0,t]} |F′(u)| du. If

∫_{[0,1]^{2}} F′(x)F′(y)Γ(x, y) dx dy < ∞,  (14)

Γ being defined by:

Γ(x, y) = Γ_{1}(x, y) + ∑_{k≥2}(Γ_{k}(x, y) + Γ_{k}(y, x)),

where

Γ_{k}(x, y) = P(Y_{1} > x, Y_{k} > y) − P(Y_{1} > x) P(Y_{k} > y),

then X̄_{1} is square integrable and n^{−1/2} ∑_{j=1}^{n} (X_{j} − IE X_{j}) converges weakly to a centered Gaussian random

variable with variance

σ^{2} = ∫_{[0,1]^{2}} F′(x)F′(y)Γ(x, y) dx dy.

4 Some functionals of paths

As an application, we consider now some examples of functionals of paths T (ξ_{n}) whose convergence in distribution follows from the L^{p}(0, 1) weak convergence of the empirical process ξ_{n}.

Example 1: Watson statistic.

Watson [19] proposed the following statistic for testing the “goodness of fit” on a circle in the i.i.d. case:

W_{n}^{2} = \frac{1}{2} ∫_{0}^{1} ∫_{0}^{1}[ξ_{n}(t) − ξ_{n}(s)]^{2}ds dt = ∫_{0}^{1} ξ_{n}(t)^{2} dt − (∫_{0}^{1} ξ_{n}(t) dt)^{2} .

Clearly this functional of paths is continuous in the L^{2}(0, 1) topology. Example 2: Weighted Mallows statistics.

The convergence of the Mallows statistic M_{n} = ∫_{0}^{1} |ξ_{n}(t)| dt needs only the L^{1}(0, 1) weak convergence of ξ_{n}. More generally, for any non negative weight function ψ in L^{p′}(0, 1), the dual space of L^{p}(0, 1), we have the convergence in distribution of

M_{n,ψ} = ∫_{0}^{1} ψ(t)|ξ_{n}(t)| dt,

under L^{p}(0, 1) weak convergence of ξ_{n}. Example 3: Some Anderson Darling statistics.

Anderson and Darling [1] studied the asymptotic behaviour of the statistics:

A_{n,ψ}^{2} = ∫_{0}^{1} ξ_{n}(t)^{2}ψ(t) dt

for non negative weight functions ψ. The choice of the suitable L^{p}(0, 1) space depends on ψ. Indeed to check the continuity of the functional T : f ↦ ∫_{0}^{1} f^{2}ψ dt, Hölder inequality provides the bound:

|T f − T g| ≤ ∥ψ∥_{r} · ∥f − g∥_{2s} · ∥f + g∥_{2s}, \frac{1}{r} + \frac{1}{s} = 1.

So if ψ belongs to L^{r}(0, 1) we can take p = 2s. For instance, choosing the weight ψ(t) = t^{−a}(1 − t)^{−a}, with a < 1, we obtain the convergence in distribution of A_{n,ψ}^{2} under the L^{p}(0, 1) weak convergence of ξ_{n} for p > 2/(1 − a). The case a = 1 seems out of reach of the L^{p}(0, 1) method. Its study would require more knowledge about the regularity of the limiting process ξ. Example 4: Hölder functions and local time.

Define the local time of ξ_{n} ([14] p. 398) by:

L_{n}(x) = n^{−1/2}{number of times ξ_{n}(t) = x, 0 ≤ t ≤ 1}

For any suitable Borel function f we have:

∫_{0}^{1} f(ξ_{n}(t)) dt = ∫_{−∞}^{+∞} f(x)L_{n}(x) dx.

Suppose now f ∈ Ċ^{p}(IR), p ∉ IN, the homogeneous Hölder space of order p ([8] p. 177), so we have the representation:

f(x) = ∑_{k=0}^{[p]} \frac{f^{(k)}(x_{0})}{k!}(x − x_{0})^{k} + R(x_{0}, x), x_{0}, x ∈ IR,

where |R(x_{0}, x)| ≤ C|x − x_{0}|^{p}, together with the estimate:

|f^{([p])}(x_{0})| ≤ C′(f^{([p])}(0) + |x_{0}|^{p−[p]}).

It follows that |f^{(k)}(x_{0})| ≤ C′′(1 + |x_{0}|^{p−k}) for 0 ≤ k ≤ [p], so the existence and continuity on L^{p}(0, 1) of the functional T : h ↦ ∫_{0}^{1} f(h(t)) dt is easily checked.

Consider now the occupation measure μ of ξ defined by

μ(B) = λ{t ∈ [0, 1] : ξ(t) ∈ B}

for any Borel subset B of IR (λ denotes the Lebesgue measure). Following Berman ([2], Ex. 3.2), a sufficient condition for μ to be absolutely continuous with respect to λ is

∫_{[0,1]^{2}} Q(s, t)^{−1/2} ds dt < ∞, where Q(s, t) = Γ(s, s) + Γ(t, t) − 2Γ(s, t).

Define analogously Q_{1}(s, t) with Γ_{1}(s, t) = s ∧ t − st and Q̃(s, t) with Γ̃(s, t) = Γ(s, t) − Γ_{1}(s, t). Then Q̃ is non negative on [0, 1]^{2}, so the integrability of Q^{−1/2} follows from integrability of Q_{1}^{−1/2} that is from:

∫_{[0,1]^{2}} |t − s|^{−1/2}(1 − |t − s|)^{−1/2} ds dt < ∞,

which is easily checked. So the occupation measure μ has a density L (the local time of ξ) and for each f ∈ Ċ^{p}(IR)

∫_{0}^{1} f(ξ(t)) dt = ∫_{0}^{1} f(t) μ(dt) = ∫_{IR} f(x)L(x) dx.

Finally by L^{p}(0, 1) continuity of the functional T for any f ∈ Ċ^{p}(IR), the moments ∫_{IR} f(x)L_{n}(x) dx of the local time L_{n} converge in distribution to the corresponding moments of the local time L of ξ.

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