Hölderian invariance principles for triangular arrays of random variables^{∗}

Alfredas Račkauskas

Vilnius University and Institute of Mathematics and Informatics Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius, Lithuania

Charles Suquet

Université des Sciences et Technologies de Lille Mathématiques Appliquées, F.R.E. CNRS 2222 Bât. M2, U.F.R. de Mathématiques F-59655 Villeneuve d’Ascq Cedex, France

Abstract

This paper extends Hölderian invariance principle by Lamperti (1962) to the case of partial sums processes based on a triangular array of rowwise independent random variables. As an application, necessary and sufficient conditions are obtained for the almost sure (resp. in probability) weak Hölder convergence of partial sums processes based on bootstrapped samples.

Keywords : bootstrap, functional central limit theorem, Hölder space, invariance principle, triangular array.

1 Introduction

The classical invariance principle or functional central limit theorem states the weak convergence of a partial sums process ξ_{n} to the Brownian motion W. When the paths of ξ_{n} are polygonal lines, the usual topological framework for this convergence is the space C[0, 1] of continuous functions on [0, 1], endowed with the supremum norm. Because the paths of the processes ξ_{n} and W have a better regularity than the sole continuity, one may be tempted to use some subspace of C[0, 1] with a stronger topology as a support for the paths. The first case to be considered in this prospect is undoubtedly the ladder of Hölder spaces H_{α}^{o} with regularity exponent 0 < α < 1/2, where the bound 1/2 is clearly induced by the regularity of W (for the definition of H_{α}^{o}, cf. Section 2). The question goes beyond the simple theoretical curiosity. Indeed this strengthening of the topology provides more continuous functionals of the paths, widening the applications area of the invariance principle.This proves useful in some statistical applications, particularly for the sharp detection of epidemic type changes [13, 14].

∗Research support by the cooperation agreement CNRS/LITUANIE (4714)

The first Hölderian invariance principle, which goes back to Lamperti [6], treats the case where ξ_{n} is the polygonal process with vertices (k/n, n^{−1/2}S_{k}) with S_{k} := X_{1} +⋯+X_{k} and the X_{i}’s are centered i.i.d. Assuming the finiteness of E|X_{1}|^{p} for some p > 2, Lamperti establishes the weak convergence of ξ_{n} to W in every H_{α}^{o} with 0 < α < 1/2 − 1/p. In order to test the optimality of his moment condition, Lamperti considers a symmetric random variable X_{1} such that P{X_{1} ≥ u} = cu^{−p(α)}, u ≥ 1 where p(α) is the critical exponent defined by

p(α) := \frac{1}{1/2 − α}.  (1)

Then he shows that the sequence (ξ_{n}) is not tight in H_{α}^{o}.

Lamperti’s theorem was derived again by Kerkyacharian and Roynette [5] who used another method relying on Ciesielski’s results [2] providing an analysis of Hölder spaces as sequence spaces, starting from their decomposition on the basis of triangular functions. Erickson [3] studied the case of partial sums processes indexed by [0, 1]^{d}. Hamadouche [4] investigated the case of some stationary weakly dependent sequences (X_{n}). In [9] the authors extended Lamperti’s invariance principle to the case of an i.i.d. sequence (X_{n}) of random elements in a separable Banach space B, with Hölder spaces of functions [0, 1] → B related to the weight function ρ(h) = h^{α} ln^{β}(1/h). Besides they showed in [10] how to relax considerably the moment assumption by combining self-normalization of the partial sums with some adaptive construction of the polygonal lines. Finally they gave in [12] a necessary and sufficient condition for the weak Hölder convergence of ξ_{n} to a Brownian motion, in the case where the X_{i}’s have values in a separable Banach space B and where the Hölder space under consideration is related to the weight function ρ(h) = h^{α}L(1/h), L being a slowly varying function. In the setting of Lamperti’s theorem, this necessary and sufficient condition reads :

ξ_{n} ⟶_{H_{α}^{o}}^{D} W if and only if lim_{t→∞} t^{p(α)}P(|ξ_{1}| ≥ t) = 0.  (2)

All these works concern the case of identically distributed X_{k}’s. In this paper we tackle for the first time the case of triangular arrays

(X_{n,k}, k = 1, …, k_{n}), n = 1, 2, …,

of row wise independent random variables. We assume in all the sequel that EX_{n,k} = 0, σ_{n,k}^{2} := var(X_{n,k}) = EX_{n,k}^{2} < ∞ and σ_{n,k} > 0 for k = 1, …, k_{n}, n = 1, 2, … Put S_{n}(0) = 0, b_{n}(0) = 0 and

S_{n}(k) = X_{n,1} + ⋯ + X_{n,k}, b_{n}(k) = σ_{n,1}^{2} + ⋯ + σ_{n,k}^{2},

for k = 1, 2, …, k_{n}. We also assume that b_{n}(k_{n}) = 1. Consider the process Ξ_{n}(t), t ∈ [0, 1], built by affine interpolation between the points (b_{n}(k), S_{n}(k)), k = 0, 1, …, k_{n}:

Ξ_{n}(t) = S_{n}(k − 1) + σ_{n,k}^{−2}(t − b_{n}(k))X_{n,k}, b_{n}(k − 1) ≤ t ≤ b_{n}(k),

k = 1, …, k_{n}. We study the convergence in distribution of this process in the functional framework of the Hölder spaces H_{α}^{o} defined as follows. For 0 < α < 1, H_{α}^{o} is the set of continuous functions x : [0, 1] → ℝ such that

lim_{δ→0} w_{α}(x, δ) = 0,

where

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

This set H_{α}^{o} becomes a separable Banach space when endowed with the norm

∥x∥_{α} := |x(0)| + w_{α}(x, 1).

In all the paper we consider only the ladder of spaces H_{α}^{o} with 0 < α < 1/2.

The paper is organized as follows. Section 2 reminds the analysis of the Hölder spaces by the basis of Schauder triangular functions, leading to the tightness theorem in H_{α}^{o}. Section 3 is devoted to the general results whose statements are gathered in the subsection 3.1 while the proofs are deferred to the following subsections. Following the same way of exposition, Section 4 presents some applications to the partial sums processes built on bootstrapped samples.

2 Preliminaries

To establish Hölderian invariance principles, it is convenient to use some dyadic discretization procedure associated to the basis of Faber-Schauder triangular functions. Denote by D_{j} the set of dyadic numbers of level j in [0, 1], i. e.

D_{0} = {0, 1}, D_{j} = {(2l − 1)2^{−j}; 1 ≤ l ≤ 2^{j−1}}, j ≥ 1

and by D the set of all the dyadic numbers in [0, 1], D = ∪_{j=0}^{∞}D_{j}. Put for r ∈ D_{j}, j ≥ 0,

r^{−} := r − 2^{−j}, r^{+} := r + 2^{−j}.

We index each Faber-Schauder triangular function by the abscissa of its vertex. So for r ∈ D_{j}, j ≥ 1, Λ_{r} is the function defined by

Λ_{r}(t) := \{ \begin{matrix}2^{j}(t − r^{−}) & if t ∈ (r^{−}, r]; \\ 2^{j}(r^{+} − t) & if t ∈ (r, r^{+}]; \\ 0 & elsewhere. \end{matrix}

In the special case j = 0, we take

Λ_{0}(t) := 1 − t, Λ_{1}(t) := t, t ∈ [0, 1].

If x is a function [0, 1] → ℝ, let us define its Schauder coefficients λ_{r}(x) by

λ_{r}(x) := x(r) − \frac{1}{2}(x(r^{+}) + x(r^{−})), r ∈ D_{j}, j ≥ 1

and λ_{r}(x) := x(r) when r ∈ D_{0}. It is well known that if x is continuous on [0, 1], it admits an expansion in Schauder series

x = ∑_{r∈D} λ_{r}(x)Λ_{r},  (3)

uniformly convergent on [0, 1]. Ciesielski [2] proved that if x belongs to the separable Hölderian space H_{α}^{o}, the convergence (3) holds for the strong topology of this space. Moreover on H_{α}^{o}, the native Hölderian norm ∥x∥_{α} is equivalent to a sequence Hölderian norm:

∥x∥_{α} ∼ ∥x∥_{α}^{seq} := sup_{j≥0} 2^{jα} max_{r∈D_{j}} |λ_{r}(x)|.  (4)

It follows that (Λ_{r}, r ∈ D) is a Schauder (unconditional) basis of H_{α}^{o}, which with the equivalence of norms (4) provides us the following tightness criterion.

Theorem 1. The sequence (Z_{n})_{n≥1} of random elements in the Hölder space H_{α}^{o} is tight if and only if it satisfies both following conditions.

i) lim_{u→∞} sup_{n≥1} P{|Z_{n}(0)| > u} = 0;

ii) for every ε > 0,

lim_{J→∞} lim sup_{n→∞} P{ sup_{j≥J} 2^{jα} max_{r∈D_{j}} |λ_{r}(Z_{n})| > ε} = 0.

For a proof, we refer to [11].

3 General results

3.1 Statements

In the first Hölderian functional central limit theorem of this section, we assume the existence of moments of order q > 2 and generalize Lamperti’s theorem to the case of triangular arrays.

Theorem 2. If

lim_{n→∞} max_{1≤k≤k_{n}} σ_{n,k}^{2} = 0  (5)

and if for some q > p(α), where p(α) is defined by (1),

lim_{n→∞} ∑_{k=1}^{k_{n}} σ_{n,k}^{−2αq}E|X_{n,k}|^{q} = 0,  (6)

then

Ξ_{n} ⟶_{n→∞}^{D} W in the space H_{α}^{o}.  (7)

To check that Theorem 2 generalizes Lamperti’s one, let us consider the special case where k_{n} = n, X_{n,k} = n^{−1/2}X_{k}, the sequence (X_{k}) being i.i.d. centered, with variance EX_{k}^{2} = 1 and E|X_{1}|^{q} < ∞ for some q > 2. Since σ_{n,k}^{2} = n^{−1}, (5) is trivially satisfied and (6) reduces to the convergence to 0 of n^{1−qα−q/2}, which holds automatically because q > 2. So the only constraint is q > p(α). Hence Theorem 2 tells us in this case that Ξ_{n} converges in distribution to W in H_{α}^{o} for each α < 1/2 − 1/q, which is precisely Lamperti’s result.

In order to relax the moment condition (6) in Theorem 2, let us introduce for each τ > 0, the truncated random variables

X_{n,k,τ} := X_{n,k}1{|X_{n,k}| ≤ τσ_{n,k}^{2α}}, k = 1, …, k_{n}; n = 1, 2, …

Theorem 3. Assume that the following conditions are satisfied.

1. For every ε > 0,

lim_{n→∞} ∑_{k=1}^{k_{n}} P(|X_{n,k}| ≥ εσ_{n,k}^{2α}) = 0.  (8)

2. For every ε > 0,

lim_{n→∞} ∑_{k=1}^{k_{n}} EX_{n,k}^{2}1{|X_{n,k}| ≥ ε} = 0.  (9)

3. For some q > p(α),

lim_{τ→0} lim sup_{n→∞} ∑_{k=1}^{k_{n}} σ_{n,k}^{−2αq}E|X_{n,k,τ} |^{q} = 0.  (10)

Then the convergence in distribution (7) holds true.

Conditions (8) and (10) substitute to (6) and are evidently more general. Lindeberg’s condition (9) insures that the triangular array (X_{n,k}) satisfies the central limit theorem. It implies also the infinitesimal negligibility condition (5).

Let us consider now the more specific case of a triangular array whose rows are linear combinations with deterministic coefficients of the same i.i.d. sequence. So let (X_{k}) be a sequence of i.i.d. centered random variables, with variance EX_{k}^{2} = 1. Let

a := (a_{n,k}, 1 ≤ k ≤ k_{n}, n ≥ 1)

be a triangular array of non null real numbers satisfying

∀n ≥ 1, ∑_{k=1}^{k_{n}} a_{n,k}^{2} = 1.  (11)

lim_{n→∞} max_{1≤k≤k_{n}} a_{n,k}^{2} = 0.  (12)

Let us denote by Ξ_{a,n} the polygonal partial sums process built on the triangular array (X_{n,k}), where

X_{n,k} := a_{n,k}X_{k}, k = 1, …, k_{n}, n ≥ 1.

Theorem 4. Assume that the following conditions are satisfied.

∀ε > 0, lim_{n→∞} ∑_{k=1}^{k_{n}} P(|X_{1}| ≥ ε|a_{n,k}|^{2α−1}) = 0.  (13)

M := sup_{t>0, n≥1} t^{p(α)} ∑_{k=1}^{k_{n}} P(|X_{1}| ≥ t|a_{n,k}|^{2α−1}) < ∞.  (14)

Then

Ξ_{a,n} ⟶_{n→∞}^{D} W in the space H_{α}^{o}.  (15)

Besides, Condition (13) is necessary for the convergence (15).

Corollary 5. Assume that the triangular array a satisfies (11), that the sequence (k_{n})_{n≥1} is non decreasing, such that

sup_{n≥1} \frac{k_{n+1}}{k_{n}} < ∞  (16)

and

∀n ≥ 1, ∀k = 1, …, k_{n},  \frac{β}{k_{n}} ≤ a_{n,k}^{2} ≤ \frac{γ}{k_{n}} ,  (17)

for some positive constants β, γ. Then Ξ_{a,n} converges in distribution to W in the space H_{α}^{o} if and only if

lim_{t→∞} t^{p(α)}P(|X_{1}| ≥ t) = 0.  (18)

This corollary extends the necessary and sufficient condition (2).

3.2 Proof of Theorem 2

First we note that Condition (5) gives the infinitesimal negligibility of (X_{n,k}). As q > p(α) > 2, we get the elementary inequality

∑_{k=1}^{k_{n}} EX_{n,k}^{2}1\{|X_{n,k}| ≥ ε\} ≤ \frac{1}{ε^{q−2}} ∑_{k=1}^{k_{n}} E|X_{n,k}|^{q}.  (19)

Accounting (5), Condition (6) gives immediatly the convergence to zero of ∑_{k=1}^{k_{n}} E|X_{n,k}|^{q}, which in turn, implies together with (19) the Lindeberg’s condition:

∀ε > 0, lim_{n→∞} ∑_{k=1}^{k_{n}} EX_{n,k}^{2}1{|X_{n,k}| ≥ ε} = 0.

Hence the triangular array (X_{n,k}) satisfies the central limit theorem and the classical functional central limit theorem (cf. [1]), namely

Ξ_{n} ⟶_{n→∞}^{D} W in the space C[0, 1].

As a result this provides the convergence of the finite dimensional distributions of the process Ξ_{n}.

To prove the tightness in H_{α}^{o} of (Ξ_{n})_{n≥1} , we use Theorem 1. Since Ξ_{n}(0) = 0, we only have to check the second condition of this theorem. To this aim we have to prove that for every ε > 0,

lim_{J→∞} lim sup_{n→∞} P (J, n, ε) = 0,  (20)

where

P (J, n, ε) := P {sup_{j≥J} 2^{αj} max_{r∈D_{j}} |Ξ_{n}(r) − Ξ_{n}(r^{−})| ≥ ε} .

The handling of the differences |Ξ_{n}(r^{+}) − Ξ_{n}(r)| is of course completely similar and will be omited.

Define for t ∈ [0, 1],

u(t) := max{k ≥ 0 : b_{n}(k) ≤ t},

which provides the inequalities

∀t ∈ [0, 1], b_{n}(u(t)) ≤ t < b_{n}(u(t) + 1).  (21)

For notational simplicity, we put

Y_{n} := max_{1≤k≤k_{n}} σ_{n,k}^{−2α}|X_{n,k}|.

For t ∈ [0, 1], we have

Ξ_{n}(t) = S_{n}(u(t)) + \frac{t − b_{n}(u(t))}{b_{n}(u(t) + 1) − b_{n}(u(t))}X_{n,u(t)+1}.  (22)

Let r ∈ D_{j}. To control |Ξ_{n}(r)−Ξ_{n}(r^{−})|, we shall consider the three possible configurations of r^{−} and r, relatively to the intervals generated by the b_{n}(k)’s. The points r^{−} and r are either in the same interval (case 1), either in contiguous intervals (case 2), or separated by at least one interval (case 3). Case 1, u(r) = u(r^{−}). Clearly then 2^{−j} ≤ σ_{n,u(r)+1}^{2} and

|Ξ_{n}(r) − Ξ_{n}(r^{−})| = σ_{u(r)+1}^{−2}(r − r^{−})|X_{n,u(r)+1}| ≤ Y_{n}2^{−jα}.

Case 2, u(r) = u(r^{−}) + 1. Then b_{n}(u(r^{−})) ≤ r^{−} < b_{n}(u(r)) ≤ r < b_{n}(u(r) + 1). In this case, (22) enables us to write :

Ξ_{n}(r) − Ξ_{n}(r^{−}) = \frac{r − b_{n}(u(r))}{σ_{n,u(r)+1}^{2}}X_{n,u(r)+1} + \frac{b_{n}(u(r)) − r^{−}}{σ_{n,u(r)}^{2}} X_{n,u(r)}.  (23)

Let us note that under this configuration, the ratio (r − b_{n}(u(r))/σ_{n,u(r)+1}^{2} belongs to [0, 1] and its numerator r−b_{n}(u(r)) is bounded from above by r−r^{−} = 2^{−j}. As the exponent α is less than 1, it follows

\frac{r − b_{n}(u(r))}{σ_{n,u(r)+1}^{2}} ≤ (\frac{r − b_{n}(u(r))}{σ_{n,u(r)+1}^{2}})^{α} ≤ 2^{−jα}σ_{n,u(r)+1}^{−2α}.  (24)

Similarly we get

\frac{b_{n}(u(r)) − r^{−}}{σ_{n,u(r)}^{2}} ≤ 2^{−jα}σ_{n,u(r)}^{−2α}.  (25)

Gathering (23), (24) and (25) we obtain for the case 2 the upper bound :

|Ξ_{n}(r) − Ξ_{n}(r^{−})| ≤ 2^{1−jα}Y_{n}.

Case 3, u(r) > u(r^{−}) + 1. Using again (24) and (25) and triangular inequality gives

|Ξ_{n}(r) − Ξ_{n}(r^{−})| ≤ |S_{n}(u(r)) − S_{n}(u(r^{−}) + 1)| + 2^{1−jα}Y_{n}.

This last upper bound being also valid in the cases 1 et 2, we finally get (with the convention ∑_{k∈∅} ≡ 0):

∀r ∈ D_{j}, |Ξ_{n}(r) − Ξ_{n}(r^{−})| ≤ | ∑_{k=u(r^{−})+2}^{u(r)} X_{k}| + 2^{1−jα}Y_{n}.  (26)

This leads us to introduce the splitting

P (J, n, ε) ≤ P_{1}(J, n, ε) + P_{2}(n, ε),

where

P_{1}(J, n, ε) := P{ sup_{j≥J} 2^{αj} max_{r∈D_{j}} | ∑_{k=u(r^{−})+2}^{u(r)} X_{k}| > ε/2},  (27)
P_{2}(n, ε) := P{|Y_{n}| ≥ ε/4}.  (28)

So, noting that Condition (6) gives the convergence in probability of Y_{n} to zero, the proof of (20) reduces to that of the convergence

lim_{J→∞} lim sup_{n→∞} P_{1}(J, n, ε) = 0.  (29)

By Rosenthal’s inequality, we get with some constant c = c(q),

E| ∑_{i=u(r^{−})+2}^{u(r)} X_{n,k}|^{q} ≤ c( ∑_{i=u(r^{−})+2}^{u(r)} EX_{n,k}^{2})^{q/2} + c ∑_{i=u(r^{−})+2}^{u(r)} E|X_{n,k}|^{q}.

Recalling that by definition of u(r), we have

∑_{k=u(r^{−})+2}^{u(r)} σ_{n,k}^{2} ≤ r − r^{−} = 2^{−j},

we easily arrive to

P_{1}(J, n, ε) ≤ \frac{c}{ε^{q}} ∑_{j≥J} 2^{(qα+1−q/2)j} + \frac{2^{q}c}{ε^{q}} ∑_{j≥J} 2^{αqj} ∑_{r∈D_{j}} ∑_{i=u(r^{−})+2}^{u(r)} E|X_{n,k}|^{q}.

The first of these sums does not depend on n and goes to zero when J goes to infinity because q > p(α). Write I(J, n, q) for the second sum in this upper bound (without the constant 2^{q}cε^{−q}). Exchanging summations leads to

I(J, n, q) = ∑_{k=1}^{k_{n}} E|X_{n,k}|^{q} ∑_{j≥J} 2^{αqj} ∑_{r∈D_{j}} 1{u(r^{−}) + 1 < k ≤ u(r)}.

Now consider for fixed k the inequalities

u(r^{−}) + 1 < k ≤ u(r).  (30)

Assume there is some r ∈ D_{j} satisfying (30) and let r′ be another dyadic number in D_{j}. If r′ < r, then r′ < r^{−} whence by increasingness of u, u(r′) < u(r^{−})+1 < k, which forbids r′ to satisfy (30). If r′ > r, then r^{′−} > r whence k ≤ u(r) ≤ u(r^{′−}) < u(r^{′−}) + 1, which forbids r′ to satisfy (30). So for fixed k, there is at most one r ∈ D_{j} satisfying (30). When this r do exist, the definition of u provides the inequalities

r^{−} < ∑_{i=1}^{u(r^{−})+1} σ_{n,i}^{2} < ∑_{i=1}^{k} σ_{n,i}^{2} ≤ ∑_{i=1}^{u(r)} σ_{n,i}^{2} ≤ r,

from which we immediatly obtain σ_{n,k}^{2} ≤ 2^{−j}. Therefore

∀k = 1, …, k_{n}, ∑_{r∈D_{j}} 1{u(r^{−}) + 1 < k ≤ u(r)} ≤ 1{2^{−j} ≥ σ_{n,k}^{2}},

which gives

∑_{j≥J} 2^{αqj} ∑_{r∈D_{j}} 1\{u(r^{−}) + 1 < k ≤ u(r)\} ≤ \frac{2^{qα}}{2^{qα} − 1}σ_{n,k}^{−2αq}.  (31)

From this we finally deduce that

∀J ≥ 1, ∀n ≥ 1, I(J, n, q) ≤ C ∑_{k=1}^{k_{n}} σ^{−2αq}E|X_{n,k}|^{q},

with some constant C = C(α, q) and we conclude using Condition (6).

3.3 Proof of Theorem 3

The convergence of the finite dimensional distributions of Ξ_{n} follows from Lindeberg’s condition as in the proof of Theorem 2. It remains to establish the tightness, whose proof reduces as previously to the asymptotic control of the quantities P_{1}(J, n, ε) and P_{2}(n, ε) defined by (27) and (28). First we note that

P(Y_{n} ≥ \frac{ε}{4}) ≤ ∑_{k=1}^{k_{n}} P(|X_{n,k}| ≥ \frac{ε}{4}σ_{n,k}^{2α}),  (32)

which due to (8), implies the convergence of P_{2}(n, ε) to 0.

Write A_{n} := { max_{1≤k≤k_{n}} |X_{n,k}| ≤ τσ_{n,k}^{2α}} = {Y_{n} ≤ τ}. Noting that on this event X_{n,k} = X_{n,k,τ} for k = 1, …, k_{n}, we may write

P_{1}(J, n, ε) ≤ P′_{1}(J, n, ε, τ) + P(A_{n}^{c}),

where

P′_{1}(J, n, ε, τ) := P\{ sup_{j≥J} 2^{αj} max_{r∈D_{j}}| ∑_{k=u(r^{−})+2}^{u(r)} X_{n,k,τ} | ≥ \frac{ε}{2}\}.

By (32) and (8), P(A_{n}^{c}) converges to 0. It remains to study the asymptotic behavior of P′_{1}(J, n, ε, τ). Before using Rosenthal’s inequality as in the previous proof, it is worth noticing that unlike the X_{n,k}’s, the X_{n,k,τ} ’s are not centered.

To control the contribution of the EX_{n,k,τ} ’s to P′_{1}, we use repetedly Cauchy-Schwarz inequality to write

| ∑_{k=u(r^{−})+2}^{u(r)} EX_{n,k,τ} | ≤ ∑_{k=u(r^{−})+2}^{u(r)} (EX_{n,k}^{2})^{1/2}P^{1/2}(|X_{n,k}| ≥ τσ_{n,k}^{2α})

≤ [ ∑_{k=u(r^{−})+2}^{u(r)} σ_{n,k}^{2}]^{1/2}[ ∑_{k=1}^{n} P(|X_{n,k}| ≥ τσ_{n,k}^{2α})]^{1/2}.

From the definition of u(r) we get

∑_{k=u(r^{−})+2}^{u(r)} σ_{n,k}^{2} ≤ r − r^{−} = 2^{−j}.  (33)

Therefore

sup_{j≥J} 2^{αj} max_{r∈D_{j}}| ∑_{k=u(r^{−})+2}^{u(r)} EX_{n,k,τ} | ≤ 2^{−J(1/2−α)}[ ∑_{k=1}^{n} P(|X_{n,k}| ≥ τσ_{n,k}^{2α})]^{1/2}.

This upper bound is less than ε/4 for n ≥ n_{0}, where n_{0} depends on ε and τ but not on J. So for n ≥ n_{0}, we have P′_{1}(J, n, ε, τ) ≤ P′′_{1}(J, n, ε, τ), where

P′′_{1}(J, n, ε, τ) := P{ sup_{j≥J} 2^{αj} max_{r∈D_{j}}| ∑_{k=u(r^{−})+2}^{u(r)}(X_{n,k,τ} − EX_{n,k,τ})| ≥ ε/4}.

We can now apply Rosenthal’s inequality which, together with the elementary bound varX_{n,k,τ} ≤ EX_{n,k}^{2} and accounting (33), gives :

E| ∑_{k=u(r^{−})+2}^{u(r)}(X_{n,k,τ} − EX_{n,k,τ})|^{q} ≤ c2^{−jq/2} + c ∑_{k=u(r^{−})+2}^{u(r)} E|X_{n,k,τ} |^{q}.

It follows

P′′_{1}(J, n, ε) ≤ \frac{4^{q}c}{ε^{q}} ∑_{j≥J} 2^{(qα+1−q/2)j} + \frac{4^{q}c}{ε^{q}} ∑_{j≥J} 2^{αqj} ∑_{r∈D_{j}} ∑_{k=u(r^{−})+2}^{u(r)} E|X_{n,k,τ} |^{q}.

Recalling that q > p(α) and hence qα + 1 − q/2 < 0 and using (31), we obtain

P′′_{1}(J, n, ε) ≤ C_{1}2^{(qα+1−q/2)J} + C_{2} ∑_{k=1}^{n} σ_{n,k}^{−2αq}E|X_{n,k,τ} |^{q},

with some constants C_{1} and C_{2} depending α, q and ε. Let us observe that this upper bound holds for every τ > 0. In view of the successive reductions in the study of P (J, n, ε) we deduce that

∀τ > 0, lim sup_{J→∞} lim sup_{n→∞} P (J, n, ε) ≤ C_{2} lim sup_{n→∞} ∑_{k=1}^{n} σ_{n,k}^{−2αq}E|X_{n,k,τ} |^{q}.

Letting finally τ go to 0, we obtain due to (10),

lim_{J→∞} lim sup_{n→∞} P (J, n, ε) = 0,

which ends the proof.

3.4 Proof du Theorem 4

To establish the convergence (15), we check that the random variables X_{n,k} = a_{n,k}X_{k} fulfill the conditions of Theorem 3. Note that (13) is nothing else than the translation of (8) in the special case of the processes Ξ_{n} = Ξ_{a,n}.

Lindeberg’s condition (9) follows from (11), (12) and of the finiteness of EX_{1}^{2} via the inequality

∑_{k=1}^{k_{n}} a_{n,k}^{2}EX_{1}^{2}1{|X_{1}| ≥ ε|a_{n,k}|^{−1}} ≤ EX_{1}^{2}1{|X_{1}| ≥ εd_{n}^{1/2}},

where

d_{n} := min_{1≤k≤k_{n}} a_{n,k}^{−2} = ( max_{1≤k≤k_{n}} a_{n,k}^{2})^{−1}.  (34)

To check (10), let us first note that for every positive random variable Y and every constant c > 0,

EY^{q}1{Y ≤ c} = ∫_{0}^{c} qt^{q−1}P(c ≥ Y ≥ t) dt ≤ ∫_{0}^{c} qt^{q−1}P(X ≥ t) dt. (35)

The truncated variables are here X_{n,k,τ} = a_{n,k}X_{k}1{|X_{k}| ≤ τ|a_{n,k}|^{2α−1}}. Applying (35) to the variables |a_{n,k}|^{1−2α}|X_{k}|, we obtain with q > p = p(α),

∑_{k=1}^{k_{n}} σ_{n,k}^{−2αq}E|X_{n,k,τ} |^{q} = ∑_{k=1}^{k_{n}} E(|a_{n,k}|^{1−2α}|X_{k}|)^{q}1{|a_{n,k}|^{(1−2α)}|X_{k}| ≤ τ}

≤ ∫_{0}^{τ} qt^{q−1} ∑_{k=1}^{k_{n}} P(|X_{k}| ≥ t|a_{n,k}|^{2α−1}) dt

≤ \frac{Mq}{q − p}τ^{q−p}.

As q > p, Condition (10) is satisfied and this completes the proof of the convergence (15).

Let us now show the necessity of (13) for this same convergence. By continuous mapping theorem, the convergence in distribution of Ξ_{a,n} to W in H_{α}^{o} gives for every δ > 0 :

w_{α}(Ξ_{a,n}, δ) ⟶_{n→∞}^{D} w_{α}(W, δ).  (36)

Due to (12), we have max_{1≤k≤n} a_{n,k}^{2} < δ for n ≥ n_{0}(δ), which enables us to bound the increments of Ξ_{a,n} between two consecutive vertices by the modulus of Hölderian regularity w_{α}(Ξ_{a,n}, δ). More precisely, we get

P\{ max_{1≤k≤n} |a_{n,k}X_{k}|a_{n,k}^{−2α} > u\} = P\{ max_{0<k≤n} \frac{|S_{n}(k) − S_{n}(k − 1)|}{a_{n,k}^{2α}} > u\}
≤ P\{ sup_{0<|t−s|<δ} \frac{|Ξ_{a,n}(t) − Ξ_{a,n}(s)|}{|t − s|^{α}} > u\}
= P{w_{α}(Ξ_{a,n}, δ) > u}.

By the convergence (36)

lim sup_{n→∞} P(w_{α}(Ξ_{n}, δ) > u) ≤ P(w_{α}(W, δ) > u).

It follows that for every δ > 0,

lim sup_{n→∞} P{ max_{1≤k≤n} |a_{n,k}|^{1−2α}|X_{k}| > u} ≤ P(w_{α}(W, δ) > u).

As P(w_{α}(W, δ) > u) goes to 0 when δ goes to 0 and accounting the independence of the X_{k}’s we deduce that

lim_{n→∞} ∑_{k=1}^{k_{n}} P(|X_{k}| > u|a_{n,k}|^{2α−1}) = 0,

which gives (13) since the X_{k}’s have the same distribution.

3.5 Proof of Corollary 5

To establish the sufficiency of (18) for the convergence in distribution of Ξ_{a,n} to W in H_{α}^{o}, we use Theorem 4. As a matter of fact, we check (13) under the asumption (12) which is more general than (17), by writing

∑_{k=1}^{k_{n}} P{|X_{1}| ≥ ε|a_{n,k}|^{2α−1}} = ∑_{k=1}^{k_{n}} a_{n,k}^{2}a_{n,k}^{−2}P{|X_{1}| ≥ ε(|a_{n,k}|^{−2})^{1/2−α}}

≤ ∑_{k=1}^{k_{n}} a_{n,k}^{2} sup_{t≥|a_{n,k}|^{−2}} tP{|X_{1}| ≥ εt^{1/p(α)}}

≤ sup_{t≥d_{n}} tP{|X_{1}| ≥ εt^{1/p(α)}} ∑_{k=1}^{k_{n}} a_{n,k}^{2},

where d_{n} is defined by (34). This upper bound goes to zero due to (11), (12) and (18). As ε is arbitrary, (13) is satisfied. The fulfilment of (14) is immediate due to Condition (17), indeed

sup_{t>0} t^{p(α)} ∑_{k=1}^{k_{n}} P(|X_{1}| ≥ t|a_{n,k}|^{2α−1}) ≤ sup_{t>0} t^{p(α)}k_{n}P(|X_{1}| ≥ tβ^{α−1/2}k_{n}^{1/p(α)})
= β^{(1/2−α)α} sup_{u>0} u^{p(α)}P(|X_{1}| ≥ u).

To show the necessity of (18) for the convergence in distribution in H_{α}^{o} of Ξ_{a,n} to W, we simply note that according to Theorem 4, this convergence implies Condition (13). This later together with the lower bound

∑_{k=1}^{k_{n}} a_{n,k}^{2}a_{n,k}^{−2}P(|X_{1}| ≥ ε|a_{n,k}|^{2α−1}) ≥ \frac{k_{n}}{γ} P(|X_{1}| ≥ εβ^{α−1/2}k_{n}^{1/p(α)}) ∑_{k=1}^{k_{n}} a_{n,k}^{2},

valid for every ε > 0, give

lim_{k_{n}→∞} k_{n}P(|X_{1}| ≥ k_{n}^{1/p(α)}) = 0.

Due to (16) and the increasingness of k_{n}, we easily deduce (18).

4 Application to bootstrap

We consider now an application of the Hölderian invariance principles for triangular arrays to the Hölderian convergence in distribution of partial sums processes built on bootstraped samples. We obtain necessary and sufficient conditions for the convergence almost sure or in probability which are stated in the subsection 4.1 and proved in the following subsections.

4.1 Hölderian FCLT under bootstrap

Let X_{1}, …, X_{n} be random variables defined on the same probability space (Ω, F, P), independent, with null expectation and having the same distribution function F. We write F̂_{n} for the corresponding empirical distribution function. Let X_{1}^{∗}, …, X_{n}^{∗} be a bootstraped sample of F̂_{n}, i.e. conditionally to X_{1}, …, X_{n}, the X_{1}^{∗}, …, X_{n}^{∗} are independent random variables with the same distribution function F̂_{n}. We denote by E^{∗} the conditional expectation given X_{1}, …, X_{n}.

Put S_{n}^{∗}(0) := 0,

S_{n}^{∗}(t) := σ_{n}^{−1} ∑_{j≤t}(X_{j}^{∗} − \overline{X}_{n}), 0 < t ≤ n,

with the classical notations

\overline{X}_{n} := n^{−1}(X_{1} + ⋯ + X_{n}) and σ_{n}^{2} := n^{−1} ∑_{k=1}^{n}(X_{k} − \overline{X}_{n})^{2}.

Consider the polygonal random line ξ_{n}^{∗}(t), t ∈ [0, 1], with vertices the points (k/n, n^{−1/2}S_{n}^{∗}(k)), k = 0, 1, …, n. This polygonal line being doubly random, it is suitable to precise the convergence mode under consideration when studying its asymptotic behavior. Denote by d some metric inducing the convergence in distribution in H_{α}^{o}. The sequence ξ_{n}^{∗} is said to converge in D-probability to W in H_{α}^{o} if d(ξ_{n}^{∗}, W) converges in probability to 0 (with an obvious abuse of notation since d(ξ_{n}^{∗}, W) denotes the distance between the distribution of ξ_{n}^{∗} and that of W, both distributions being viewed as measures on the Borel σ-field of H_{α}^{o}). This convergence will be denoted by

ξ_{n}^{∗} ⟶_{n→∞}^{D−Pr} W in H_{α}^{o}.

Similarly, if d(ξ_{n}^{∗}, W) converges P-almost surely on Ω to 0, we say that ξ_{n}^{∗} converges D-almost surely to W in H_{α}^{o}, which we denote by

ξ_{n}^{∗} ⟶_{n→∞}^{D−a.s.} W in H_{α}^{o}.

Theorem 6. The moment condition

E|X_{1}|^{p(α)} < ∞  (37)

is necessary and sufficient for the convergence

ξ_{n}^{∗} ⟶_{n→∞}^{D−a.s.} W in H_{α}^{o}.  (38)

Theorem 7. The condition

lim_{t→∞} t^{p(α)}P(|X_{1}| > t) = 0  (39)

is necessary and sufficient for the convergence

ξ_{n}^{∗} ⟶_{n→∞}^{D−Pr} W in H_{α}^{o}.  (40)

4.2 Proof of Theorem 6

To establish the sufficiency of (37), we use Theorem 2, conditionally to X_{1}, …, X_{n}, putting

X_{n,k} := n^{−1/2}σ_{n}^{−1}(X_{k}^{∗} − \overline{X}_{n}), k = 1, …, n.

Here σ_{n,k}^{2} = E^{∗}X_{n,k}^{∗2} = n^{−1}, which reduces the proof of (38) to that of the convergence

n^{qα−q/2}σ_{n}^{−q}nE^{∗}|X_{1}^{∗} − \overline{X}_{n}|^{q} ⟶_{n→∞}^{a.s.} 0,  (41)

for some q > p(α) > 2. As EX_{1}^{2} is finite by (37), the strong law of large numbers gives

\overline{X}_{n} ⟶_{n→∞}^{a.s.} 0 and σ_{n}^{2} ⟶_{n→∞}^{a.s.} 1.

So (41) reduces to

n^{qα−q/2}nE^{∗}|X_{1}^{∗}|^{q} = n^{−q/p(α)} ∑_{k=1}^{n} |X_{k}|^{q} ⟶_{n→∞}^{a.s.} 0.  (42)

According to Marcinkiewicz-Zygmund strong law of large numbers, if (Y_{k})_{k≥1} is an i.i.d. sequence such that E|Y_{1}|^{β} < ∞ for some β ∈]0, 1[, then n^{−1/β}(Y_{1} + ⋯ + Y_{n}) converges almost surely to 0. Choosing Y_{k} = |X_{k}|^{q} and β = p(α)/q makes clearly (42) follow from (37).

Assuming now (38) satisfied, let us show the necessity of (37). From (38), the sequence of processes (ξ_{n}^{∗})_{n≥1} is P-almost surely tight in H_{α}^{o}. It follows [8, Th. 1.4] that

a.s., ∀ε > 0, lim_{δ→0} sup_{n≥1} P^{∗}{w_{α}(ξ_{n}^{∗}, δ) > ε} = 0,

where P^{∗} is the probability conditional to the sequence (X_{n}). In particular

a.s., w_{α}(ξ_{n}^{∗}, 1/n) ⟶_{n→∞}^{P^{∗}} 0.

Considering the weighted increments of ξ_{n}^{∗} between two consecutive vertices we deduce from the above convergence that

a.s., max_{1≤k≤n} \frac{|ξ_{n}^{∗}(k/n) − ξ_{n}^{∗}((k − 1)/n)|}{(1/n)^{α}} ⟶_{n→∞}^{P^{∗}} 0,

which may be recast with p = p(α) as

a.s., max_{1≤k≤n} \frac{|X_{k}^{∗} − \overline{X}_{n}|}{n^{1/p}σ̂_{n}} ⟶_{n→∞}^{P^{∗}} 0,

or more explicitly

∀ε > 0, P( max_{1≤k≤n} \frac{|X_{k}^{∗} − \overline{X}_{n}|}{n^{1/p}σ̂_{n}} > ε | X_{1}, …, X_{n}) ⟶_{n→∞}^{a.s.} 0.  (43)

As the X_{k}^{∗} are conditionally to (X_{1}, …, X_{n}), independent identically distributed,

(43) is equivalent to

∀ε > 0, nP(|X_{1}^{∗} − \overline{X}_{n}| > εn^{1/p}σ̂_{n} | X_{1}, …, X_{n}) ⟶_{n→∞}^{a.s.} 0.  (44)

Now observe that

P(|X_{1}^{∗} − \overline{X}_{n}| > εn^{1/p}σ̂_{n} | X_{1}, …, X_{n}) = \frac{1}{n} ∑_{k=1}^{n} 1\{|X_{k} − \overline{X}_{n}| > εn^{1/p}σ̂_{n}\},

whence going back to (44),

∀ε > 0, ∑_{k=1}^{n} 1\{|X_{k} − \overline{X}_{n}| > εn^{1/p}σ̂_{n}\} ⟶_{n→∞}^{a.s.} 0.

This sum of indicator functions cannot converge to zero unless it is null for every n bigger than some (random) n_{0}. This remark gives immediatly that

max_{1≤k≤n} \frac{|X_{k} − \overline{X}_{n}|}{n^{1/p}σ̂_{n}} ⟶_{n→∞}^{a.s.} 0.  (45)

Let us check now that

\frac{|\overline{X}_{n}|}{n^{1/p}σ̂_{n}} ⟶_{n→∞}^{a.s.} 0.  (46)

Observing that σ_{n}^{2} = n^{−1} ∑_{i=1}^{n} X_{i}^{2} − \overline{X}_{n}^{2}, we can bound the denominator from below by n^{1/p−1/2}(X_{1}^{2} + ⋯ + X_{n}^{2})^{1/2}. In the numerator, Cauchy-Schwarz inequality gives |X_{1} + ⋯ + X_{n}| ≤ n^{1/2}(X_{1}^{2} + ⋯ + X_{n}^{2})^{1/2}. So (46) follows from the upper bound

\frac{|\overline{X}_{n}|}{n^{1/p}σ̂_{n}} ≤ \frac{1}{n^{1/p}} .

From (45) and (46) we immediatly obtain

\frac{max_{1≤k≤n} |X_{k}|^{2}}{n^{2/p−1}(X_{1}^{2} + ⋯ + X_{n}^{2})} ⟶_{n→∞}^{a.s.} 0.  (47)

Using the following Hölder inequality

∑_{k=1}^{n} X_{k}^{2} ≤ n^{1−2/p}( ∑_{k=1}^{n} |X_{k}|^{p})^{2/p}

we finally deduce from (47) the convergence

\frac{max_{1≤k≤n} |X_{k}|^{p}}{|X_{1}|^{p} + ⋯ + |X_{n}|^{p}} ⟶_{n→∞}^{a.s.} 0.

By Lemma 4.1 in Maller and Resnik [7] we can now conclude that E|X_{1}|^{p} is finite.

4.3 Proof of Theorem 7

Like in the proof of Theorem 6, we establish the sufficiency of (39) by checking that

n^{qα−q/2}σ_{n}^{−q}nE^{∗}|X_{1}^{∗} − \overline{X}_{n}|^{q} ⟶_{n→∞}^{P} 0,  (48)

for some exponent q > p = p(α). Since p(α) > 2, Condition (39) implies the finiteness of EX_{1}^{2}. The weak law of large numbers gives us then the convergences in probability of \overline{X}_{n} to 0 and of σ_{n}^{2} to 1, which reduce the proof of (48) to that of

n^{qα−q/2}nE^{∗}|X_{1}^{∗}|^{q} = n^{−q/p} ∑_{k=1}^{n} |X_{k}|^{q} ⟶_{n→∞}^{P} 0.

This last convergence follows from (39) by Marcinkiewicz-Zygmund weak law of large numbers.

Assume now (40) satisfied, that is

d(ξ_{n}^{∗}, W) ⟶_{n→∞}^{P} 0,

for some metric d inducing the convergence in distribution in H_{α}^{o}. Then from every subsequence of indexes (n′) ⊂ (n) we can extract a subsequence (n′′) ⊂ (n′) such that

d(ξ_{n′′}^{∗}, W) ⟶_{n′′→∞}^{a.s.} 0.

In view of the proof of Theorem 7, the a.s. convergence (47) is then satisfied for the subsequence indexed by (n′′). By arbitrariness of the initial subsequence (n′) it follows

\frac{max_{1≤k≤n} |X_{k}|^{2}}{n^{2/p−1}(X_{1}^{2} + ⋯ + X_{n}^{2})} ⟶_{n→∞}^{P} 0.  (49)

As p > 2, we immediatly deduce from (49) that

\frac{max_{1≤k≤n} |X_{k}|^{2}}{(X_{1}^{2} + ⋯ + X_{n}^{2})} ⟶_{n→∞}^{P} 0,

which due to Lemme 4.1 in Maller and Resnik [7], gives us the finiteness of E|X_{1}|^{2}. From this together with (49), we deduce

\frac{max_{1≤k≤n} |X_{k}|}{n^{1/p}} ⟶_{n→∞}^{P} 0,

which is equivalent to (39). The proof is complete.

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