Central limit theorem in Hölder spaces

by

Alfredas RAČKAUSKAS Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius, Lithuania

and

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

Version 18/04/98

0. Introduction

The space C[0, 1] of continuous functions is a classical framework for many limit theorems in the theory of stochastic processes. The C[0, 1] -weak convergence of a sequence of stochastic processes ξ_{n} gives useful results about the convergence in distribution of continuous functionals of the paths. In many situations, the processes ξ_{n} are known to have almost surely paths with at least some Hölder regularity and the same happens for the limiting process ξ (see for instance invariance principles for random polygonal lines, perturbed empirical processes, empirical characteristic functions). Looking for limit theorems with respect to some weak Hölder convergence instead of the C[0, 1] one is then a natural question. One important interest of this new functional framework is to provide more continuous functionals of the paths (see Hamadouche (1998) for a list of examples). Moreover when a process ξ is known by its finite dimensional distributions, the question of the existence of a version with almost all paths of some given Hölder regularity has its own interest.

The first result in this direction goes back to the Kolmogorov sufficient condition for the existence of a version of ξ with continuous paths, namely:

(1)  P{|ξ(t + h) − ξ(t)| > λ} ≤ cλ^{−γ}h^{1+δ},

where c, δ > 0 and γ > 1 are constants. In fact the same condition gives a version of ξ with α - Hölder regularity for any 0 < α < δ/γ. Ciesielski (1961) gave sufficient conditions for a Gaussian process to have a version with Hölderian paths. Lamperti (1962) proved for the polygonal line interpolating the Donsker Prohorov partial sums process an α -Hölder invariance principle for any α < 1/2 (in the most favorable case). This result was derived again by Kerkyacharian and Roynette (1991) by another method using Ciesielski (1960) analysis of Hölder spaces by triangular functions. The problem of the existence of a version with Hölderian paths was studied by Ibragimov (1979) and by Nobelis (1981). Recently Hamadouche (1998) extended Lamperti’s invariance principle to dependent variables. Hamadouche (1997) proved the weak α -Hölder convergence of the polygonal uniform empirical process to the Brownian bridge for any 0 < α < 1/4 and the optimality of this bound.

All these results rely on some control of the first differences

Δ_{h}^{1}ξ(t) = ξ(t + h) − ξ(t).

In this paper, we propose to use instead the second differences:

Δ_{h}^{2}ξ(t) = ξ(t + h) + ξ(t − h) − 2ξ(t).

The usefulness of Δ_{h}^{2}ξ(t) in the problem of sample paths differientiability is known (see Cramér-Leadbetter (1967)). From an analytical point of view, there is no loss in working with Δ_{h}^{2}f to study the Hölder regularity of a function f. This observation goes back to Zygmund (1945) who investigating smooth functions noticed that a necessary and sufficient condition that a continuous and periodic function f(x) should satisfy a Hölder condition of order α, 0 < α < 1, is that

Δ_{h}^{2}f(x) = O(h^{α}), as h → +0,

uniformly in x. The role of Δ_{h}^{2} is now well understood in the more general context of Besov spaces (see Peetre (1976)).

This fact stimulated us to study the central limit theorem for processes in Hölder spaces in terms of second differences. It turns out that second difference allows to recognize more processes which admits a version with almost all paths in Hölder space. In particular, as a special case of our Theorem 1.2 we obtain the following version of the Kolmogorov theorem: if there exists δ > 0, γ > 1 and a constant c > 0 such that for all t, t ± h ∈ [0, 1]

(2)  P{|Δ_{h}^{2}ξ(t)| > λ} ≤ cλ^{−γ}h^{1+δ},

then there exists a version of ξ with sample paths in H_{α}^{0}[0, 1], where 0 < α < min(1, δ/γ). To get a feeling of the advantage of Condition (2) over Condition (1), let us test both of them on the following crude benchmark (more interesting examples will be given in Section 1). Let g be a deterministic function with first derivative of Hölder regularity τ ( 0 < τ < 1 ), g(1) = 1 and consider the process:

ξ(t) = Xg(t), t ∈ [0, 1],

where X = ξ(1) is a random variable with finite variance (as we are interested in processes verifying the central limit theorem, this is a natural assumption). We have then the obvious estimates:

P{|Δ_{h}^{i}ξ(t)| > λ} ≤ E X^{2}|Δ_{h}^{i}g(t)|^{2}λ^{−2}, i = 1, 2.

Now |Δ_{h}^{1}g(t)| ≤ ||g′||_{∞}|h| and |Δ_{h}^{2}g(t)| ≤ ||g′||_{τ} |h|^{1+τ} , so Condition (1) detects only a Hölder regularity α < 1/2 for ξ, whereas Condition (2) detects α < min(τ + 1/2, 1).

The paper is organized as follows. In the preliminaries section we collected necessary facts about Hölder spaces with the main references to Ciesielski (1960). The main auxiliary result which gives an estimate for the Hölder modulus of continuity of a process in terms of a second difference is also stated in Section 1. This estimate yields several sufficient conditions that a process ξ = (ξ(t), t ∈ [0, 1]) should have almost all paths in Hölder space. In Section 2 we study the embedding of Hölder spaces. Particularly we prove that if 1 ≥ β > α+1/p then the embedding H_{β}[0, 1] → H_{α}[0, 1] is p -absolutely summing. Section 3 is devoted to the central limit theorem for random processes with paths in Hölder spaces. Particularly we prove that the random process ξ = (ξ(t), t ∈ [0, 1]) satisfies the central limit theorem in the space H_{α}^{0}[0, 1], 0 < α < 1 provided the following conditions are satisfied:

i) for each t ∈ [0, 1] E ξ(t) = 0, E ξ^{2}(t) < ∞; ii) there exists a positive random variable M such that E M^{2} < ∞ and for all t, h ∈ [0, 1]

such that t ± h ∈ [0, 1]

|Δ_{h}^{2}ξ(t)| ≤ Mh^{α} ln^{−β} (\frac{1}{h}),

where β > 3/2.

Central limit theorem is supplied with convergence rate estimates with respect to Prohorov and bounded Lipschitz metrics.

The last section is devoted to the central limit theorem for the empirical characteristic process (e.ch.p.) in Hölder spaces. For the relevance of this process in many statistical problems, we refer to the introduction in Feuerverger and Mureika (1977). The study of the e.ch.p. in the functional framework of C[−1/2, 1/2] was achieved in Marcus (1981) and S. Csörgő (1981). Our Hölderian CLT for the e.ch.p. is a first step in the investigation of convergence of e.ch.p. with a stronger topology than C[−1/2, 1/2] one.

1. Preliminaries

Throughout T = [0, 1] and 0 < α < 1. Denote by H_{α} the set of real valued continuous functions x : T → R such that w_{α}(x, 1) < ∞, where

w_{α}(x, δ) = sup_{t,s∈T,0<|t−s|<δ} \frac{|x(t) − x(s)|}{|s − t|^{α}}.

The set H_{α} is a Banach space when endowed with the norm ||x||_{α} = |x(0)| + w_{α}(x, 1). Set

H_{α}^{0} = {x ∈ H_{α} : lim_{δ→0} w_{α}(x, δ) = 0}.

Then H_{α}^{0} is a closed subspace of H_{α}.

Let

Λ(t) = \{ \begin{matrix}2t & if 0 ≤ t ≤ 1/2 \\ 2(1 − t) & if 1/2 < t ≤ 1 \\ 0 & otherwise. \end{matrix}

For n = 2^{j} + k, 0 ≤ k < 2^{j}, j = 0, 1, … set

Λ_{n}(t) = Λ_{jk}(t) = Λ(2^{j}t − k), t ∈ [0, 1].

Define moreover Λ_{−1}(t) = χ_{[0,1]}(t) and Λ_{0}(t) = Λ_{−1,0}(t) = t. Then the family {Λ_{n}, n ≥ −1} is a Schauder basis on the space H_{α}^{0} : each function x ∈ H_{α}^{0} admits the series representation

(3)  x(t) = x(0) + ∑_{n=0}^{∞} λ_{n}(x)Λ_{n}(t), t ∈ [0, 1],

with λ_{−1}(x) = λ_{−2,0}(x) = x(0), λ_{0}(x) = λ_{−1,0}(x) = x(1) − x(0), and for n = 2^{j} + k, j ≥ 0, 0 ≤ k < 2^{j} :

λ_{n}(x) = λ_{jk}(x) = x((k + 1/2)2^{−j}) − \frac{1}{2}\{x(k2^{−j}) + x((k + 1)2^{−j})\}.

By Ciesielski (1960), the norm ||x||_{α} is equivalent to the sequence norm

∥x∥_{α}^{seq} = sup_{j≥−2} 2^{(j+1)α} max_{0≤k<2^{j}} |λ_{jk}(x)|.

For j ≥ 0, let us write E_{j}x for the polygonal line interpolating x between the dyadic points k2^{−j} ( 0 ≤ k ≤ 2^{j} ). We clearly have:

E_{j}x = ∑_{i<j} ∑_{0≤k<2^{i}} λ_{ik}(x)Λ_{ik}

whence

(4)  ∥x − E_{j}x∥_{α}^{seq} = sup_{i≥j} max_{0≤k<2^{j}} |λ_{jk}(x)|.

The following estimates are easily obtained:

(5)  ||E_{j}x||_{α} ≤ 3||x||_{α},
(6)  ||x − E_{j}x||_{α} ≤ 4w_{α}(x, 2^{−j}),
(7)  w_{α}(x, 1) ≤ 2^{j}w_{α}(x, 2^{−j}),
(8)  ||x||_{α} ≤ |x(0)| + 2^{j}w_{α}(x, 2^{−j}).

For our purpose, the norm ∥x∥_{α}^{seq} is more tractable than ||x||_{α}. Ciesielski proved that a continuous function x belongs to H_{α}^{0} if and only if lim_{n→∞} n^{α}λ_{n}(x) = 0. We give here a more quantitative version of this result for further use.

Proposition 1.1. We have the following estimate: for x ∈ H_{α}^{0} and j ≥ 0

(9)  w_{α}(x, 2^{−j}) ≤ |λ_{0}(x)|2^{−j(1−α)} + \frac{8}{2^{α}(2 − 2^{α})} ∥x − E_{0}x∥_{α}^{seq} 2^{(f(j)−j)(1−α)}

+ (\frac{8}{2^{α}(2 − 2^{α})} + \frac{2}{2^{α} − 1}) ∥x − E_{f(j)}x∥_{α}^{seq} ,

where we can choose for f any function N → N, such that f(n) ≤ n and the sequences f(n) and n − f(n) are non decreasing with limit infinity.

Proof of Proposition 1.1. Set

c_{jk} = 2^{(j+1)α}λ_{jk}(x) and c_{j} = max_{0≤k<2^{j}} |c_{jk}|.

Fix s and t in T. From (3) we get:

(10)  x(t) − x(s) = λ_{0}(x)(t − s) + ∑_{j≥0} ∑_{0≤k<2^{j}} λ_{jk}(x)(Λ_{jk}(t) − Λ_{jk}(s))

Now observe that for fixed s and t there are at most two non vanishing terms in the block indexed by j in the series above. As the maximal slope of Λ_{jk} is 2^{j+1}, we have moreover:

|Λ_{jk}(t) − Λ_{jk}(s)| ≤ min(1; 2^{j+1}|t − s|).

Let j_{0} be the integer defined by 2^{−j_{0}−1} ≤ |t − s| < 2^{−j_{0}} and j_{1} = f(j_{0}). The splitting

|x(t) − x(s)| ≤ |λ_{0}(x)(t − s)| + 2 ∑_{j=0}^{j_{1}} c_{j}2^{−(j+1)α}2^{j+1}|t − s|

+ 2 ∑_{j=j_{1}+1}^{j_{0}} c_{j}2^{−(j+1)α}2^{j+1}|t − s|

+ 2 ∑_{j>j_{0}} c_{j}2^{−(j+1)α},

leads after some elementary computations to

\frac{|x(t) − x(s)|}{|t − s|^{α}} ≤ |λ_{0}(x)|2^{−j_{0}(1−α)} + \frac{8}{2^{α}(2 − 2^{α})}2^{(j_{1}−j_{0})(1−α)} sup_{j≤j_{1}} c_{j}

+ \frac{8}{2^{α}(2 − 2^{α})} sup_{j_{1}<j≤j_{0}} c_{j}

+ \frac{2}{2^{α} − 1} sup_{j>j_{0}} c_{j}

and finally to

(11)  \frac{|x(t) − x(s)|}{|t − s|^{α}} ≤ |λ_{0}(x)|2^{−j_{0}(1−α)} + \frac{8}{2^{α}(2 − 2^{α})} sup_{j≥0} c_{j}2^{(j_{1}−j_{0})(1−α)}

+ (\frac{8}{2^{α}(2 − 2^{α})} + \frac{2}{2^{α} − 1}) sup_{j>j_{1}} c_{j}.

Let δ = 2^{−J}. By our assumptions on f, taking the supremum over 0 < |t − s| < δ in the left hand side of (11) and over j_{0} ≥ J on the right hand side gives the desired bound for w_{α}(x, δ).

As a special case of the previous computation we obtain with J = 0 :

(12)  w_{α}(x, 1) ≤ |λ_{0}(x)| + (\frac{16}{2^{α}(2 − 2^{α})} + \frac{2}{2^{α} − 1}) sup_{j≥0} max_{0≤k<2^{j}} 2^{(j+1)α}|λ_{jk}(x)|.

Recall that a Young function ϕ is a convex increasing function on R^{+} such that ϕ(0) = 0 and lim_{t→∞} ϕ(t) = ∞. If Z is a random variable such that Eϕ(Z/c) < ∞ for some c > 0 then we set

∥Z∥_{ϕ} = inf{c > 0 : E ϕ(Z/c) ≤ 1}.

Throughout σ will denote an increasing function on T such that σ(0) = 0. For convenience’s sake, we recall here the basic inequalities used throughout the paper to handle the maxima of random variables. The first of them is Lemma 11.3 p. 303 in Ledoux and Talagrand (1995).

Lemma 1.1. Let (X_{i}) be positive random variables on some probability space (Ω, F, P) such that for all 1 ≤ i ≤ n and all A ∈ F

(13)  ∫_{A} X_{i} dP ≤ c_{i}P(A)ϕ^{−1}(\frac{1}{P(A)}),

where ϕ is some Young function and c_{i} a constant. Then, for every set A ∈ F

∫_{A} max_{1≤i≤n} X_{i} dP ≤ cP(A)ϕ^{−1}(\frac{n}{P(A)}),

with c = max_{1≤i≤n} c_{i}.

The two following lemmas provide practical conditions to verify (13).

Lemma 1.2. Let (X_{i}) be positive random variables on some probability space (Ω, F, P) and ϕ some Young function. Then X_{i} satisfies (13) with c_{i} = ∥X_{i}∥_{ϕ} . In particular,

E max_{1≤i≤n} X_{i} ≤ ϕ^{−1}(n) max_{1≤i≤n} ∥X_{i}∥_{ϕ} .

Proof. By convexity of ϕ and Jensen’s inequality, we have for any b > ||X_{i}||_{ϕ}

∫_{A} X_{i} dP = bP(A) ∫_{A} ϕ^{−1}ϕ(\frac{X_{i}}{b} )P^{−1}(A) dP ≤
≤ bP(A)ϕ^{−1}(P^{−1}(A)E ϕ(X_{i}/b)) ≤ bP(A)ϕ^{−1}(P^{−1}(A)).

Taking the infimum over b > ||X_{i}||_{ϕ}, (13) is verified and we can apply Lemma 1.1.

Lemma 1.3. Let (X_{i}) be positive random variables on some probability space (Ω, F, P) satisfying for some constant 1 < p < ∞ and each i = 1, …, n

a_{i} = sup_{t>0} tP^{1/p}(X_{i} > t) < ∞.

Then X_{i} satisfies (13) with c_{i} = qa_{i} where q = p/(p−1). In particular noting a = max_{1≤i≤n} a_{i},

E max_{1≤i≤n} X_{i} ≤ qan^{1/p}.

Proof. Integrating by parts we have

∫_{A} X_{i} dP = P(A) ∫_{0}^{∞} P\{X_{i} > t|A\} dt ≤ P(A)\frac{p}{p − 1} sup_{t} t P^{1/p}\{X_{i} > t|A\}

≤ P^{1−1/p}(A)\frac{p}{p − 1} sup_{t>0} t P^{1/p}\{X_{i} > t\}.

So we can apply Lemma 1.1 with ϕ(t) = t^{p}.

Theorem 1.1. Assume the process ξ = {ξ(t), t ∈ T} is defined on the probability space (Ω, F, P) and satisfies: for each set A ∈ F and all t, t ± h ∈ T

(14)  ∫_{A} |Δ_{h}^{2}ξ(t)| dP ≤ σ(h)P(A)ϕ^{−1}(\frac{1}{P(A)}).

Then for each A ∈ F, and integers K > J ≥ 0,

(15)  ∫_{A} ∥E_{K}ξ − E_{J}ξ∥_{α}^{seq} dP ≤ 2^{1+α}P(A) ∫_{2^{−K}}^{2^{−J}} \frac{σ(u)}{u^{α+1}} ϕ^{−1}(\frac{1}{P(A)u}) du.

Proof. Set

I_{j} = ∫_{A} max_{0≤k<2^{j}} |λ_{jk}(ξ)| dP, j ≥ 0.

By the condition (14), for each measurable set A,

∫_{A} |λ_{jk}(ξ)| dP ≤ σ(2^{−j−1})P(A)ϕ^{−1}(\frac{1}{P(A)}).

Now Lemma 1.1 yields

(16)  I_{j} ≤ P(A)σ(2^{−j−1})ϕ^{−1}(\frac{2^{j}}{P(A)}).

Putting together (4) and (16) we find that

(17)  ∫_{A} ∥E_{K}ξ − E_{J}ξ∥_{α}^{seq} dP ≤ P(A) ∑_{J≤j≤K} 2^{α(j+1)}σ(2^{−j−1})ϕ^{−1}(\frac{2^{j}}{P(A)}).

The result follows by comparing series and integral.

Theorem 1.1 yields several sufficient conditions that the process ξ = (ξ(t), t ∈ T) should have a version with paths in Hölder space H_{α}. The following result is a version of Kolmogorov theorem, whereas Theorem 1.3 gives a version of the corresponding Ibragimov result.

Theorem 1.2. Let p > 1. Assume that the process ξ = (ξ_{t}, t ∈ T) satisfies the condition: for each t ∈ T, h > 0 such that t ± h ∈ T

(18)  P\{|Δ_{h}^{2}ξ(t)| > λ\} ≤ \frac{c}{λ^{p}} σ^{p}(h).

If

(19)  ∫_{0}^{1} \frac{σ(u)}{u^{α+1+1/p}} du < ∞,

then ξ admits a version with almost all paths in the space H_{α}^{0}.

Proof. By Lemma 1.3, (18) yields

∫_{A} |Δ_{h}^{2}ξ(t)| dP ≤ \frac{pc^{1/p}}{p − 1}P(A)P^{−1/p}(A)σ(h)

and from Theorem 1.1 it is easily seen that (E_{J}ξ) is a.s. a Cauchy sequence in H_{α}^{0} and therefore lim_{J→∞} E_{J}ξ exists a.s. If this limit is denoted by ξ̃ then it is easy to see that

ξ̃(t) = ξ(0) + (ξ(1) − ξ(0))t + ∑_{j=0}^{∞} λ_{jk}(ξ)Λ_{jk}(t).

and is the version of ξ with paths in H_{α}^{0}.

Combining the bound (17) with Proposition 1.1 we get:

Corollary 1.1. If the random element ζ in C[0, 1] satisfies (18) and (19) then for each j ≥ 0 :

E w_{α}(ζ, 2^{−j}) ≤ 2^{−j(1−α)}E |ζ(1) − ζ(0)| + C_{α}2^{(f(j)−j)(1−α)}I(1) + C_{α}I(2^{−f(j)})

where C_{α} is a constant depending only on α, f is as in Proposition 1.1. and

I(δ) = ∫_{0}^{δ} \frac{σ(u)}{u^{α+1+1/p}} du.

Theorem 1.3. Let ϕ be a Young function. Assume that the process ξ = (ξ_{t}, t ∈ T) satisfies the condition: for each t ∈ T, h > 0 such that t ± h ∈ T

(20)  ||Δ_{h}^{2}ξ(t)||_{ϕ} ≤ cσ(h).

If

(21)  ∫_{0}^{1} \frac{σ(u)}{u^{α+1}} ϕ^{−1}(\frac{1}{u}) du < ∞,

then the process ξ admits a version with almost all paths in the space H_{α}^{0}.

Proof. One has to note (see Lemma 1.2) that the condition (20) yields

∫_{A} |Δ_{h}^{2}ξ(t)| dP ≤ cP(A)ϕ^{−1}(\frac{1}{P(A)})σ(h)

for all measurable sets A. The proof one completes by the same arguments as above.

Corollary 1.2. If the random element ζ in C[0, 1] satisfies (20) and (21) then for each j ≥ 0 :

E w_{α}(ζ, 2^{−j}) ≤ 2^{−j(1−α)}E |ζ(1) − ζ(0)| + C_{α}2^{(f(j)−j)(1−α)}I′(1) + C_{α}I′(2^{−f(j)})

where C_{α} is a constant depending only on α, f is as in Proposition 1.1. and

I′(δ) = ∫_{0}^{δ} \frac{σ(u)}{u^{α+1}} ϕ^{−1}(\frac{1}{u}) du.

Example 1. Consider symmetric p -stable process (X(t), t ∈ T), 1 ≤ p < 2, with stochastic integral representation

(X(t), t ∈ T) =^{D} ( ∫_{E} f(t, x) M(dx), t ∈ T),

where (E, E) is a measurable space, M is a symmetric p -stable random measure with σ -finite spectral measure m and =^{D} denotes equality in distribution (see Samorodnitsky and Taqqu (1994)). Define

σ^{p}(h) = sup_{t∈T} ∫_{E} |f(t + h, x) + f(t − h, x) − 2f(t, x)|^{p} m(dx).

Corollary 1.3. Let α ∈ (0, 1). If σ is increasing function such that

(22)  ∫_{0}^{1} \frac{σ(u)}{u^{α+1+1/p}} du < ∞,

then (X(t), t ∈ T) admits a version with almost all paths in the space H_{α}^{0}.

Proof. It is well known (see Samorodnitsky and Taqqu (1994)), that

P{|Δ_{h}^{2}X(t)| > λ} ≤ Cλ^{−p}σ^{p}(h).

The result now follows by Theorem 1.2.

The necessary and sufficient conditions for the stable process with index p ∈ (0, 1) to have a version with a.s. paths in H_{α} are found by Nolan (1989). Let us also remark, that after Theorem 1.3 the condition (22) yields that the function f(t), t ∈ T considered as a process defined on (E, E, m) has a version with a.s. paths in H_{α}^{0}.

Example 2. Let (X(t), t ∈ T) be a centered Gaussian process. Under which conditions on its covariance structure is the process Hölderian?

Corollary 1.4. Let

σ^{2}(h) = sup_{t∈T} E |Δ_{h}^{2}X(t)|^{2}.

If σ is increasing function such that

∫_{0}^{1} \frac{σ(u)}{u^{α+1}} | ln u|^{1/2} du < ∞,

then (X(t), t ∈ T) admits a version with almost all paths in the space H_{α}^{0}.

Proof. Apply Theorem 1.3. with the function ϕ_{2} = exp(t^{2}) − 1, t ∈ R, noting that

||Δ_{h}^{2}X(t)||_{ϕ_{2}} ≤ \sqrt{2}σ(h).

We end this section with some tightness conditions related to the above results.

Theorem 1.4. The sequence (ζ_{N}) of random elements in the Hölder space H_{α}^{0} is tight if and only if the following two conditions are satisfied:

i) lim_{b→∞} sup_{N} P{|ζ_{N}(0)| > b} = 0 ;
ii) for each ε > 0 lim_{δ→0} sup_{N} P{w_{α}(ζ_{N}, δ) > ε} = 0.

Proof. The estimates (6) and (8) enable us to use the flat concentration criterion (see Lemma 2.2 p. 40 in Ledoux Talagrand (1991)) to derive the tightness of (ζ_{N}) from i) and ii).

The necessity of i) for the tightness of (ζ_{N}) is obvious. For the necessity of ii), we invoke the following lemma, the proof of which can be found in Suquet (1996).

Lemma 1.4. Let K be a compact family (for the topology of weak convergence) of probability measures on the separable metric space H. Let (F_{j}, j ≥ 1) be a sequence of closed subsets of H decreasing to ∅. Define the functions u_{j} : K → [0, 1], μ ↦ u_{j}(μ) = μ(F_{j}). Then the sequence (u_{j}) uniformly converges to zero on K.

Since the functionals w_{α}(. , δ) are continuous and decreasing to zero in δ on H_{α}^{0}, the choice of the closed sets

F_{j} = {x ∈ H_{α}^{0}; w_{α}(x, δ_{j}) ≥ ε}, δ_{j} ↓ 0,

shows the necessity of ii) for the tightness of (ζ_{N}).

Combining Theorem 1.4 with Corollaries 1.1 and 1.2, we obtain the following sufficient conditions.

Corollary 1.5. The sequence (ζ_{N}) of random elements in Hölder space H_{α}^{0} is tight if : i) lim_{b→∞} sup_{N} P{|ζ_{N}(k)| > b} = 0, k = 0, 1 ; ii) For each t ∈ T, h > 0 such that t ± h ∈ T

sup_{N≥1} P\{|Δ_{h}^{2}ζ_{N}(t)| > λ\} ≤ \frac{c}{λ^{p}} σ^{p}(h),

where p > 1 and the function σ satisfies (19).

Corollary 1.6. The sequence (ζ_{N}) of random elements in Hölder space H_{α}^{0} is tight if : i) lim_{b→∞} sup_{N} P{|ζ_{N}(k)| > b} = 0, k = 0, 1 ; ii) For some Young function ϕ and each t ∈ T, h > 0 such that t ± h ∈ T

sup_{N≥1} ||Δ_{h}^{2}ζ_{N}(t)||_{ϕ} ≤ cσ(h),

where the functions σ and ϕ satisfy (21).

Remark. In all the results of this section σ was supposed increasing. In fact this hypothesis was used only in the comparison between series and integrals starting from (17). If we drop the assumption of monotonicity, all the results remain valid provided the conditions involving integrals of σ ( like (19), (21), (22)) are replaced by the corresponding ones involving series.

2. Embeddings maps between Hölder spaces

In this section we prove some useful properties of embedding operator between Hölder spaces. Set

H_{σ} = \{x : T → R : ||x||_{σ} = |x(0)| + sup_{s≠t} \frac{|x(t) − x(s)|}{σ(|t − s|)} < ∞\}

and

H_{σ}^{0} = \{x ∈ H_{σ} : lim_{δ→0} sup_{|s−t|<δ} \frac{|x(t) − x(s)|}{σ(|t − s|)} = 0\}.

Recall that a continuous linear operator u : E → F between Banach spaces E and F is of type p, 1 < p ≤ 2 if there exists a constant c > 0 such that for any finite sequence (x_{i}) ⊂ E

E || ∑_{i} ε_{i}ux_{i}|| ≤ c( ∑_{i} ||x_{i}||^{p})^{1/p},

where (ε_{i}) denotes a Rademacher sequence.

Lemma 2.1. Let 1 < p ≤ 2 and q = p/(p − 1). If

(23)  ∫_{0}^{1} \frac{σ(s)}{s^{1+α}} ln^{1/q} s^{−1} ds < ∞,

then the embedding map H_{σ} → H_{α}^{0} is of type p.

Proof. It is more convenient to use here the sequential norm. Fix the finite sequence (x_{i}) ∈ H_{σ} and let (ε_{i}) be a Rademacher sequence. Let ϕ_{q}(x) = exp(x^{q}) − 1. It is known (see Lemma 4.3 p. 93 in Ledoux and Talagrand (1991)) that there exists a constant c_{p} > 0 such that for each finite sequence (a_{i}) of real numbers

(24)  || ∑_{i} ε_{i}a_{i}||_{ϕ_{q}}^{p} ≤ c_{p} ∑_{i} |a_{i}|^{p}.

Observe that |λ_{−2,0}(x)| = |x(0)| ≤ ||x||_{σ} and for j ≥ −1, |λ_{jk}(x)| ≤ σ(2^{−j−1})||x||_{σ}. Let us start with the special case j = −2. Using the elementary inequality E |X| ≤ ϕ^{−1}(1)||X||_{ϕ} between L^{1} and Orlicz norms and (24) we get

(25)  E |∑_{i} ε_{i}x_{i}(0)| ≤ (ln 2)^{1/q}c_{p}^{1/p}(∑_{i} |x_{i}(0)|^{p})^{1/p} ≤ (ln 2)^{1/q}c_{p}^{1/p}(∑_{i} ||x||_{σ}^{p})^{1/p}.

Now for j ≥ −1 and 0 ≤ k < 2^{j} we have

(26)  ∥λ_{jk}( ∑_{i} ε_{i}x_{i})∥_{ϕ_{q}} ≤ c_{p}^{1/p}(∑_{i} |λ_{jk}(x_{i})|^{p})^{1/p} ≤ c_{p}^{1/p}σ(2^{−j−1})( ∑_{i} ||x_{i}||_{σ}^{p})^{1/p}

Lemma 1.2 gives then

(27)  E max_{1≤i≤n}|λ_{jk}( ∑_{i} ε_{i}x_{i})| ≤ c_{p}^{1/p}σ(2^{−j−1}) ln^{1/q}(2^{j})( ∑_{i} ||x_{i}||_{σ}^{p})^{1/p}.

Finally since

E ∥∑_{i} ε_{i}x_{i}∥_{α}^{seq} ≤ E |∑_{i} ε_{i}x_{i}(0)| + ∑_{j≥−1} E |λ_{jk}( ∑_{i} ε_{i}x_{i})|,

the conclusion follows from (23), (25) and (27).

Recall that an operator u : E → F is said to be p -absolutely summing, if there exists a constant c > 0 such that for any finite sequence (x_{i}) ⊂ E

∑_{i} ||ux_{i}||^{p} ≤ c sup{∑_{i} |f(x_{i})|^{p} : f ∈ E^{∗}, ||f|| ≤ 1}.

As usual E^{∗} denotes the topological dual of E.

Lemma 2.2. Let p > 1. If 1 ≥ β > α + 1/p, then the embedding operator H_{β} → H_{α} is p -absolutely summing.

Proof. Working with the sequential norms in H_{β} and H_{α}, we shall use the well known Pietsch (1980) theorem. Consider for j ≥ −2, 0 ≤ j < 2^{j} the linear functionals λ_{jk}^{∗} : H_{β} → R defined by

λ_{jk}^{∗}(x) = 2^{β(j+1)}λ_{jk}(x).

Obviously K_{β} = {λ_{jk}^{∗}; j ≥ −2, 0 ≤ j < 2^{j}} is a norming set, that is

∥x∥_{β}^{seq} = sup_{f∈K_{β}} |f(x)|.

Define the measure

μ = ∑_{j=−2}^{∞} 2^{p(α−β)j} ∑_{0≤k<2^{j}} δ_{λ_{jk}^{∗}},

where δ_{a} is Dirac measure. It is a bounded measure supported by K_{β} since the series ∑_{j} 2^{(p(α−β)+1)j} converges. Now we have

∫_{K_{β}} |f(x)|^{p} μ(df) = ∑_{j=−2}^{∞} 2^{p(α−β)j} ∑_{0≤k<2^{j}} |λ_{jk}^{∗}(x)|^{p} =

= ∑_{j=−2}^{∞} 2^{pαj} ∑_{0≤k<2^{j}} |λ_{jk}(x)|^{p} ≥ (∥x∥_{α}^{seq})^{p}.

Hence the embedding map is p -absolutely summing by Pietsch theorem.

Remark. From the last step of the proof of Lemma 2.2 with p = 2, one easily see that the embedding I_{βα} : H_{β} → H_{α} factorizes through Hilbert space

H = {u = (u_{n}) : ||u||^{2} = ∑_{j=0}^{∞} 2^{2αj} ∑_{k=0}^{2^{j}−1} u_{2^{j}+k}^{2} < ∞}.

If we denote the corresponding factorization by I_{λ}I_{β}, where I_{β} : H_{β} → H and I_{α} : H → H_{α} then the operator I_{α} is γ -radonifying in the sense that if ν denotes the canonical Gaussian measure on H, then νI_{α}^{−1} is a Gaussian Radon measure on H_{α}.

3. CLT in Hölder spaces

For a random element ξ ∈ H_{α}^{0} we denote by ξ_{1}, …, ξ_{N} independent copies of ξ and

ζ_{N} = N^{−1/2} ∑_{k=1}^{N} ξ_{k}.

Recall, that a random element ξ satisfies the central limit theorem (denoted ξ ∈ CLT (H_{α}^{0}) ), if the sequence (ζ_{N}) converges in distribution in H_{α}^{0}.

Theorem 3.1. Assume that the random element ξ in H_{α}^{0} satisfies the following conditions: i) E ξ(t) = 0 and E ξ^{2}(t) < ∞ for all t ∈ T; ii) there exists a positive random variable M and an increasing function σ : R_{+} → R, σ(0) =

0 such that E M^{2} < ∞ and

|Δ_{h}^{2}ξ(t)| ≤ Mσ(h), for all t, t ± h ∈ T,

where

∫_{0}^{1} \frac{σ(s)}{s^{1+α}} \sqrt{ln s^{−1}} ds < ∞.

Then ξ ∈ CLT (H_{α}^{0}) and E ∥ξ∥_{α}^{2} < ∞.

Proof. Consider the Rademacher sequence (ε_{l}) which is independent on (ξ_{k}). It may be thought as constructed on another probability space Ω′. By Theorem 10.14 in Ledoux-Talagrand (1991) it suffices to prove that for almost every ω of the probability space Ω supporting the ξ_{l}, the sequence of random elements ζ̃_{N}^{ω} of H_{α}^{0} defined on Ω′ by

ζ̃_{N}^{ω}(t) = N^{−1/2} ∑_{l=1}^{N} ε_{l}ξ_{l}(ω, t), t ∈ T

converges in distribution.

For the convergence of finite dimensional distributions, fix t_{1} < … < t_{m} in T and note that for any scalars a_{1}, …, a_{m}

E_{ε} (∑_{i=1}^{m} a_{i}ζ̃_{N}^{ω}(t_{i}))^{2} = ∑_{i,j=1}^{m} a_{i}a_{j} \frac{1}{N} ∑_{l=1}^{N} ξ_{l}(ω, t_{i})ξ_{l}(ω, t_{j}).

Using i) and the strong law of large numbers we see that the factor of a_{i}a_{j} in the above formula converges to E_{ξ}(ξ(t_{i})ξ(t_{j})) for almost every ω ∈ Ω. Hence the convergence of finite dimensional distributions of ζ̃_{N}^{ω} holds for almost every ω by the finite-dimensional CLT. To check the tightness, we invoke Corollary 1.6 whose Condition i) is a simple by product of the case m = 1 above. Using (24) with p = q = 2, we get

||Δ_{h}^{2}ζ̃_{N}^{ω}(t)||_{ϕ_{2}}^{2} = \frac{1}{N} ∥∑_{l=1}^{N} ε_{l}Δ_{h}^{2}ξ_{l}(ω, t)∥_{ϕ_{2}}^{2} ≤ \frac{C}{N} ∑_{l=1}^{N} |Δ_{h}^{2}ξ_{l}(ω, t)|^{2} ≤ \frac{C}{N} ∑_{l=1}^{N} M_{l}^{2}(ω)σ^{2}(h),

where M_{1}, …, M_{N} are independent copies of the random variable M. By the strong law of large numbers, for almost every ω ∈ Ω, sup_{N≥1} N^{−1} ∑_{l=1}^{N} M_{l}^{2}(ω) is finite and so ζ_{N}^{ω} satisfies Condition ii) of Corollary 1.6, which ends the proof.

Theorem 3.2. Let σ : R^{+} → R be an increasing function with σ(0) = 0. Let p ≥ 2. Assume that the process ξ = (ξ_{t}, t ∈ T) satisfies the following conditions:

i) E ξ(t) = 0 and E ξ^{2}(t) < ∞ for all t ∈ T; ii) for each t ∈ T, h > 0 such that t ± h ∈ T

E |Δ_{h}^{2}ξ(t)|^{p} ≤ σ^{p}(h).

and

∫_{0}^{1} \frac{σ(u)}{u^{α+1+1/p}} du < ∞.

Then ξ ∈ CLT (H_{α}^{0}).

Proof. From condition i) follow the convergence of finite dimensional distributions of (ζ_{N}) and the condition i) of Corollary 1.5. To complete the proof of the tightness, observe that by Rosenthal L_{p} -inequality

E |Δ_{h}^{2}ζ_{N}(t)|^{p} ≤ cE |Δ_{h}^{2}ξ(t)|^{p} ≤ cσ^{p}(h),

from which ii) of Corollary 1.5 is verified.

We supplement the central limit theorem in Hölder spaces with results on a convergence rate. Recall that the Prohorov metric on the space of probability measures on the Banach space B is defined by

π(μ, ν) = inf{ε > 0 : μ(F) ≤ ν(F^{ε}) + ε for all closed sets F ⊂ B},

where F^{ε} = {x ∈ B : inf_{y∈F} ||x − y|| < ε}. The bounded Lipschitz metric is defined by

d_{BL}(μ, ν) = sup {| ∫ f dμ − ∫ f dν| : f ∈ L_{1}(B)},

where L_{1}(B) denotes the set of functions f : B → R that satisfy the condition

sup_{x} |f(x)| + sup_{x≠y} \frac{|f(x) − f(y)|}{||x − y||} ≤ 1.

Let π^{α}, and d_{BL}^{α} denote the Prohorov and bounded Lipschitz metrics in the space of probability measures on H_{α} respectively.

Theorem 3.3. Assume that the process ξ = (ξ(t), t ∈ T) satisfies the following conditions: i) E ξ(t) = 0 and E ξ^{2}(t) < ∞ for all t ∈ T; ii) there exist a positive random variable M and an increasing function σ : R_{+} → R, σ(0) = 0

such that E M^{3} < ∞ and

|Δ_{h}^{2}ξ(t)| ≤ Mσ(h), for all t, t ± h ∈ T.

If for 1 ≥ β > 1/2

∫_{0}^{1} \frac{σ(s)}{s^{1+β}} \sqrt{ln s^{−1}} ds < ∞

then for each 0 ≤ α < β − 1/2

π^{α}(μ_{N}, μ) ≤ C(E M^{3})^{1/4}N^{−1/8}, d_{BL}^{α}(μ_{N}, μ) ≤ C(E M^{3})^{1/3}N^{−1/6},

where μ_{N} is the distribution of ζ_{N} whereas μ is the distribution of the limiting Gaussian process ζ.

Proof. The remark after the proof of Lemma 2.2 and results in Bentkus and Račkauskas (1985) (see also Paulauskas and Račkauskas (1989)) yields

π^{α}(μ_{N}, μ) ≤ C(E ||ξ||_{β}^{3} + E ||ζ||_{β}^{3})^{1/4}N^{−1/8}, d_{BL}^{α}(μ_{N}, μ) ≤ C(E ||ξ||_{β}^{3} + E ||ζ||_{β}^{3})^{1/3}N^{−1/6},

where the constant C depends only on the covariance operator of the Gaussian random element ζ. Evidently, condition ii) and Ciesielski (1960) yield

E ||ξ||_{β}^{3} ≤ CE sup_{j} 2^{3βj} max_{0≤k<2^{j}} |λ_{kj}(ξ)|^{3} ≤ CE M^{3}.

By Theorem 1.1 and tail behavior of Gaussian random elements,

E ||ζ||_{β} ≤ ∑_{j} 2^{βj}\sqrt{j} max_{0≤k<2^{j}}(E |λ_{jk}(ζ)|^{2})^{1/2}.

According to the condition i), E λ_{jk}^{2}(ζ) = E λ_{jk}^{2}(ξ). Finally, since E ||ζ||_{β}^{3} ≤ C(E ||ζ||_{β})^{3}, invoking the condition ii) we obtain E ||ζ||_{β}^{3} ≤ CE M^{3}.

Let L denotes the Levy metric in the space of distribution functions. We shall write L(ξ, η) for L(F_{ξ}, F_{η}), where F_{ξ} denotes the distribution function of ξ. Recall that

L(ξ, η) = inf{ε > 0 : F_{ξ}(x) ≤ F_{η}(x + ε) + ε and
F_{η}(x) ≤ F_{ξ}(x + ε) + ε for all x ∈ R}.

Theorem 3.4. Assume that the conditions of Theorem 3.3 are satisfied and 0 < α < β − 1/2. Let F : H_{α}^{0} → R satisfy

sup_{x≠y} \frac{|F (x) − F (y)|}{||x − y||_{α}} = C < ∞.

Then

L(F (ζ_{N}), F (ζ)) ≤ C(E M^{3})^{1/4}N^{−1/8}.

Proof. This is direct corollary of Lemma 2.2 and Theorem 5.3.6 and Proposition 5.3.3 in Paulauskas and Račkauskas (1989) (see p. 124 and p. 123 respectively).

Of a particular interest is the function F (x) equal to q -variation norm, where q ≥ 1/α.

4. CLT for empirical characteristic process

Let X, X_{1}, X_{2}, …, be independent identically distributed random variables with probability distribution function F (x) and characteristic function

c(t) = ∫_{−∞}^{∞} e^{itx} dF (x).

Let F_{N}(x) denote the empirical distribution function based on the first N observations X_{1}, …, X_{N}, i.e.

F_{N}(x) = \frac{1}{N} ∑_{k=1}^{N} 1I_{[X_{k},∞)}(x),

where 1I_{A} is the indicator function of the set A. The empirical characteristic function corresponding to F_{N} is

c_{N}(t) = ∫_{−∞}^{∞} e^{itx} dF_{N}(x) = \frac{1}{N} ∑_{k=1}^{N} e^{itX_{k}}

and empirical characteristic process is

Y_{N}(t) = \sqrt{N}(c_{N}(t) − c(t)), t ∈ R.

It is well known that c_{N}(t) converges to c(t) uniformly on each finite interval (more strong convergence is proved in Csörgő (1981)). Marcus (1981) has found necessary and sufficient conditions that the process (Y_{N}(t), t ∈ [−1/2, 1/2]) converges in distribution in the space C[−1/2, 1/2] of continuous complex functions on [−1/2, 1/2] endowed with usual supremum norm. Given the experience of the Hölderian central limit theorem it seems natural to assert that Y_{N} ⟶^{D} Y in a complex H_{α}^{0}.

From now H_{α} and H_{α}^{0} are the spaces of complex valued continuous functions x : T → C such that ω_{α}(x, 1) < ∞ and lim_{δ→0} ω_{a}(x, δ) = 0 respectively. Both spaces are endowed with the norm || · ||_{α}.

Evidently Y_{N}(t) = N^{−1/2} ∑_{k=1}^{N} ξ_{k}(t), where

ξ_{k}(t) = e^{itX_{k}} − c(t), k = 1, …, N.

Hence Y_{N} is a random element in H_{α}^{0} if and only if the deterministic function c(t) is itself in H_{α}^{0}.

An elementary computation gives

(28)  Δ_{h}^{2}ξ(t) = −4e^{itX} sin^{2}(hX/2) + 4E (e^{itX} sin^{2}(hX/2)).

This leads us to express our hypotheses in terms of the moments of sin^{2}(hX/2), introducing the functions

τ_{k}(u) = ∫_{−∞}^{∞} sin^{2k}(ux) dF (x), k ≥ 1.

This choice avoids to put conditions on the moments of the X_{i}.

By the multidimensional central limit theorem, the finite-dimensional distributions of (Y_{N}) converge to those of a complex Gaussian process Y with E Y (t) = 0 and covariance E Y (t)\overline{Y (s)} = c(t − s) − c(t)c(−s), s, t ∈ T. The process Y can be represented in the form

Y (t) = ∫_{−∞}^{+∞} e^{itx}dW (F (x)) − W (1)c(t),

where (W (t)) is a standard Wiener process.

Theorem 4.1. Let 0 < α < 1. If

(29)  ∑_{j=1}^{∞} 2^{αj}\sqrt{j}τ_{2}^{1/2}(2^{−j}) < ∞,

then (Y_{N}) converges in distribution in the space H_{α}^{0}.

Proof. Let us observe first that

|Δ_{h}^{2}c(t)| ≤ 4τ_{1}(h/2) ≤ 4τ_{2}^{1/2}(h/2),

so using the sequential norm, it is clear from (29) that c belongs to H_{α}^{0} and (Y_{N}) can be considered as a sequence of random elements in H_{α}^{0}.

We use the same randomization method (and the same notations) as in the proof of Theorem 3.1, the only difference being in the verification of condition ii) of Corollary 1.6. Write

Ỹ_{N}^{ω}(t) = \frac{1}{\sqrt{N}} ∑_{k=1}^{N} ε_{k}ξ_{k}(t), t ∈ T.

By the series version of Corollary 1.6 (see the remark at the end of Section 1), it suffices to check that for almost every ω ,

sup_{N} ∑_{j≥1} 2^{(j+1)α}ϕ_{2}^{−1}(2^{j}) sup_{t} ∥Δ_{2^{−j−1}}^{2} Ỹ_{N}^{ω}(t)∥_{ϕ_{2}} < ∞.

Using (28), we obtain with absolute constants a, b

∥Δ_{2^{−j−1}}^{2} Ỹ_{N}^{ω}(t)∥_{ϕ_{2}} ≤ \frac{a}{N} ∑_{l=1}^{N} |Δ_{2^{−j−1}}^{2}ξ_{l}(ω, t)|^{2} ≤ \frac{b}{N} ∑_{l=1}^{N} sin^{4}(\frac{X_{l}(ω)}{2^{j+2}}) + bE^{2} sin^{2}(\frac{X_{1}}{2^{j+2}} ).

Using (29) and τ_{1} ≤ τ_{2}^{1/2} to treat the contribution of E^{2} sin^{2}(X_{1}2^{−j−2}) reduces the problem to prove that for almost every ω,

(30)  sup_{N} \frac{1}{N} ∑_{l=1}^{N} ∑_{j=1}^{∞} 2^{jα}\sqrt{j} sin^{4}(\frac{X_{l}(ω)}{2^{j+2}}) < ∞.

Write Z_{l}(ω) for the sum of the series indexed by j in (30) and observe that by (29),

E Z_{l} = ∑_{j=1}^{∞} 2^{jα}\sqrt{j}τ_{2}(2^{−j−2}) ≤ ∑_{j=1}^{∞} 2^{jα}\sqrt{j}τ_{2}^{1/2}(2^{−j−2}) < ∞.

Hence (30) follows from the strong law of large numbers for the i.i.d. sequence (Z_{l}) and the proof is complete.

The following result supplements Corollary 1 in Csörgő (1981).

Corollary 4.1. Assume that the distribution function F has a nonempty monotonicity interval. Let

F_{1}(x) = ∫_{0}^{1} |x(t)|^{2} dt; F_{2}(x) = ∫_{0}^{1}(ℜ x(t))^{2} dt; F_{3}(x) = ∫_{0}^{1}(ℑ x(t))^{2} dt.

Then there exists a constant C > 0 such that

sup_{r≥0} |P{F_{i}(Y_{N}) < r} − P{F_{i}(Y) < r}| ≤ CN^{−1}.

holds for each i = 1, 2, 3.

Proof. The result follows easily from Bentkus and Götze (1997) if to check that the limiting random element Y considered in the space L_{2}(0, 1) is at least of dimension d ≥ 9. By the representation (20) it is enough to show that the random element Y_{0} = ∫_{−∞}^{+∞} e^{itx}dW (F (x)) is at least of dimension d ≥ 9. This is ensured by monotonicity of the distribution function F in certain nonempty interval. Indeed, assume that F is monotone on the interval (a, b) and consider functions f_{1}, …, f_{M} ∈ L_{2}(0, 1) such that g_{1}, …, g_{M} have non intersecting supports on (a, b), where g_{k}(x) = ∫_{0}^{1} e^{itx}f_{k}(t)dt, x ∈ R, k = 1, …, M. Then, the random variables ∫_{0}^{1} Y_{0}(t)f_{k}(t)dt = ∫_{0}^{1} g_{k}(x)dW (F (x)), k = 1, …, M are independent. This yields, that Y is at least M dimensional in L_{2}(0, 1).

Consider the following functionals on the space H_{α}^{0} :

F_{4}(x) = ∫_{0}^{1} ∫_{0}^{1} \frac{|x(t) − x(s)|^{2}}{|t − s|^{2α}} dtds; F_{5}(x) = ∫_{0}^{1} ∫_{0}^{1} \frac{|ℜ(x(t) − x(s))|^{2}}{|t − s|^{2α}} dtds;

F_{6}(x) = ∫_{0}^{1} ∫_{0}^{1} \frac{|ℑ(x(t) − x(s))|^{2}}{|t − s|^{2α}} dtds;

Corollary 4.2. Assume that the distribution function F has a nonempty monotonicity interval and

∫_{0}^{1} ∫_{0}^{1} |t − s|^{−4α}τ_{2}(t − s) dtds < ∞.

Then there exists a constant C > 0 such that

sup_{r≥0} |P{F_{i}(Y_{N}) < r} − P{F_{i}(Y) < r}| ≤ CN^{−1}.

holds for each i = 4, 5, 6.

Proof. The proof is much the same as that of Corollary 4.1.

References

V. Bentkus, F. Götze, Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces,

Probab. Theory Relat. Fields 109 (1997), 367–416. V. Bentkus and A. Račkauskas, Estimates of the distance between sums of independent random elements in Banach

spaces, Theor. Probab. Appl. 29(1) (1985), 50–65. Z. Ciesielski, On the isomorphisms of the spaces H_{α} and m , Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8 (1960),

217–222. Z. Ciesielski, Hölder conditions for realizations of Gaussian processes, Trans. Amer. Math. Soc. 99 (1961), 403–413. H. Cramér and M. R. Leadbetter, Stationary and Related Stochastic Processes, Wiley, New York London Sydney,

1967. S. Csörgő, Limit behavior of the empirical characteristic function, Ann. Probab. 9 (1981), 130–144. A. Feuerverger and R. A. Mureika, The empirical characteristic function and its applications, Ann. Stat. 5 (1981),

88–97. D. Hamadouche, Weak convergence of smoothed empirical process in Hölder spaces, Stat. Probab. Letters 36 (1998),

393–400. D. Hamadouche, Invariance principles in Hölder spaces (preprint), Pub. IRMA Lille 44-VII (1998). I. Ibragimov, On smoothness conditions for trajectories of random functions, Theor. of Probab. and Appl. XXVIII,

240–262. G. Kerkyacharian and B. Roynette, Une démonstration simple des théorèmes de Kolmogorov, Donsker et Ito-Nisio,

C. R. Acad. Sci. Paris 312, Série I (1991), 877–882. J. Lamperti, On convergence of stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 430–435. M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, Berlin, Heidelberg, 1991. M. Marcus, Weak convergence of the empirical characteristic function, Ann. Probab. 9 (1981), 194–201. Ph. Nobelis, Fonctions aléatoires lipschitziennes, Lecture Notes Math. 850 (1981), 38–43. J. P. Nolan, Continuity of stable processes, J. Multivar. Anal. 29 (1989), 84–93. V. Paulauskas and A. Račkauskas, Approximation Theory in the Central Limit Theorem. Exact results in Banach

spaces, Kluwer Academic Publishers, The Netherlands, 1989. J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Ser., 1976. A. Pietsch, Operator ideals, North-Holland Publishing company, Amsterdam – New York – Oxford, 1980. G. Samorodnitsky, M. S. Taqqu, Stable Non–Gaussian Random Processes. Stochastic Models with Infinite Variance,

Chapman & Hall, New York, London, 1994. Ch. Suquet, Tightness in Schauder decomposable Banach spaces, Translations of A.M.S., Proceedings of the St

Petersburg Math. Soc. 5 (1996). A. Zygmund, Smooth functions, Duke-Math.-J. 12 (1945), 47–76.