Hölder convergences of multivariate empirical characteristic functions^{∗}

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

Statistique et Probabilités F.R.E. CNRS 2222 Bât. M2, U.F.R. de Mathématiques F-59655 Villeneuve d’Ascq Cedex France

Abstract

We study the asymptotic behavior of the multivariate empirical characteristic function in various Hölder topologies: strong laws of large numbers and central limit theorem for the empirical characteristic process. A rather precise connection is established between the topological framework and the integrability of the marginal distribution of the sample. Some application to test statistics are proposed.

Keywords: Almost sure randomization, central limit theorem, Hölder space, Marcinkiewicz-Zygmund strong laws of large numbers, multivariate empirical characteristic process, strong consistency, test statistic. MSC2000 classifications: 60F05, 60F17, 62E20, 62G10, 62G30.

∗Preprint. Research supported by a cooperation agreement CNRS/LITHUANIA (4714).

1 Introduction

Let X, X_{1}, X_{2}, … ∈ R^{d} be i.i.d. d-dimensional random vectors with common distribution function F (x), x = (x_{1}, …, x_{d}) ∈ R^{d} and characteristic function

𝔠(t) = ∫_{R^{d}} e^{i⟨t,x⟩} dF (x), t = (t_{1}, …, t_{d}) ∈ R^{d},

where ⟨t, x⟩ = ∑_{k=1}^{d} t_{k}x_{k} is the usual inner product. Throughout, |x|_{2} = \sqrt{⟨x, x⟩} and |x| = max(|x_{1}|, …, |x_{d}|) will denote correspondingly the Euclidean and maximum norms on R^{d}.

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

F_{n}(x) = n^{−1} ∑_{k=1}^{n} 1_{{X_{k}≤x}},

where 1_{A} is the indicator function of the set A and inequality x ≤ y for vectors x = (x_{1}, …, x_{d}), y = (y_{1}, …, y_{d}) ∈ R^{d} means that x_{j} ≤ y_{j} for each j = 1, …, d. The empirical characteristic function corresponding to F_{n} is then

𝔠_{n}(t) = ∫_{R^{d}} e^{i⟨t,x⟩} dF_{n}(x) = n^{−1} ∑_{k=1}^{n} exp(i⟨t, X_{k}⟩)

and an empirical characteristic process is

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

We interpret the empirical characteristic function as random element in a certain Hölder function space and establish conditions for a.s. consistency (in Hölder topology) and for the corresponding central limit theorem to hold. As a corollary to our main result we derive convergence in distribution for the statistics e.g. (more examples are provided in Section 5):

V_{n,α} = sup_{s≠t,s,t∈[0,1]^{d}} \frac{|Y_{n}(t) − Y_{n}(s)|}{|t − s|^{α}};

G_{n,ψ} = ∫_{R^{d}} ∫_{R^{d}} |Y_{n}(t) − Y_{n}(s)|^{2}ψ(t, s) dt ds, where ψ is a certain function and α ∈ (0, 1). These statistics can in turn be applied to test the hypothesis that the unknown distribution of X belongs to a family of distributions described by characteristic functions, e.g. p-stable laws.

A rich motivation for investigating the asymptotic properties of the empirical characteristic function 𝔠_{n} in a certain functional framework is found in e.g. Feuerverger and Mureika [7]. The asymptotic behavior of the univariate empirical characteristic process was investigated in Kent [9], Feuerverger and Mureika [7], Feuerverger and McDunnough [5, 6]. Csörgő [2] and Marcus [11] established necessary and sufficient conditions for weak convergence of empirical characteristic process in the space C[−1/2, 1/2] of continuous functions. Recently, the authors [12] considered the stronger Hölder framework ℋ_{α}[0, 1] (0 < α < 1) of functions x such that |x(t) − x(s)| ≤ C|t − s|^{α} and established conditions for weak convergence of empirical characteristic process. The asymptotic behavior of the multivariate empirical characteristic process was investigated in Csörgő [3] in the framework of the space of continuous functions on a compact set.

The present contribution extends the initial results of [12] to the multivariate setting and to a richer ladder of Hölder spaces. It also completes this study by investigating the a.s. consistency (in the sense of Marcinkiewicz-Zygmund SLLN) and clarifying the basic assumptions of [12] in establishing precise relations between the Hölderian asymptotic behavior of 𝔠_{n} and the integrability of X. Section 2 contains some useful background, mainly on Hölder spaces ℋ_{ρ}(𝒯). Section 3 and Section 4 are devoted respectively to SLLN and CLT results. Some statistical applications are suggested in Section 5.

2 Preliminaries

2.1 Hölder spaces

Throughout 𝒯 := [0, 1]^{d} is endowed with the norm |t|. Denote by ℋ_{ρ}(𝒯) the set of continuous complex valued functions x : 𝒯 → C such that ω_{ρ}(x, 1) < ∞, where

ω_{ρ}(x, δ) := sup_{\begin{matrix}t,s∈𝒯, \\ 0<|t−s|<δ \end{matrix}} \frac{|x(t) − x(s)|}{ρ(|s − t|)}

and ρ is a function satisfying conditions (1) to (5), where c_{1}, c_{2} and c_{3} are positive constants:

ρ(0) = 0, ρ(h) > 0, 0 < h ≤ 1;  (1)
ρ is non decreasing on [0, 1];  (2)
ρ(2h) ≤ c_{1}ρ(h), 0 ≤ h ≤ 1/2;  (3)
∫_{0}^{h} \frac{ρ(u)}{u} du ≤ c_{2}ρ(h), 0 < h ≤ 1;  (4)
h ∫_{h}^{1} \frac{ρ(u)}{u^{2}} du ≤ c_{3}ρ(h), 0 < h ≤ 1.  (5)

For instance, elementary computations show that the functions

ρ(h) := h^{α} ln^{β}(\frac{b}{h}), 0 < α < 1, β ∈ R,

satisfy conditions (1) to (5), of course with a suitable choice of the constant b when β ≠ 0, namely b ≥ exp(β/α) if β > 0 and b > exp(−β/(1 − α)) if β < 0.

Now let us remark that for any function ρ fulfilling (1), (2) and (5), there is a positive constant c_{4} such that

ρ(h) ≥ c_{4}h, 0 ≤ h ≤ 1.  (6)

The spaces ℋ_{ρ}^{o}(𝒯) therefore always contain all the Lipschitz functions and in particular the (continuous) piecewise affine functions and the functions t ↦ exp(i⟨t, w⟩), w ∈ R^{d}.

The set ℋ_{ρ}(𝒯) is a Banach space when endowed with the norm

∥x∥_{ρ} := |x(0)| + ω_{ρ}(x, 1).

Obviously an equivalent norm is obtained replacing |x(0)| in the above formula by ∥x∥_{∞} := sup{|x(t)| : t ∈ 𝒯}. Define

ℋ_{ρ}^{o}(𝒯) = {x ∈ ℋ_{ρ}(𝒯); lim_{δ→0} ω_{ρ}(x, δ) = 0}.

Then ℋ_{ρ}^{o}(𝒯) is a closed subspace of ℋ_{ρ}(𝒯). Moreover, the space ℋ_{ρ}^{o}(𝒯) is separable whereas ℋ_{ρ}(𝒯) is not.

We shall use the equivalence of the norm ∥x∥_{ρ} with some sequential norm involving the dyadic second differences of x. To introduce this norm we need some preparation (more details can be found in e.g. [13] or [15]).

Let the set

W_{j} := {k2^{−j}; 0 ≤ k ≤ 2^{j}}^{d}.

Put V_{0} := W_{0} and V_{j} := W_{j} \ W_{j−1} for j ≥ 1. More explicitly, V_{j} is the set of dyadic points v = (k_{1}2^{−j}, …, k_{d}2^{−j}) in W_{j} with at least one k_{i} odd. For a function x : [0, 1]^{d} → C, the coefficients λ_{j,v}(x) are given by:

λ_{0,v}(x) = x(v), v ∈ V_{0};  (7)
λ_{j,v}(x) = x(v) − \frac{1}{2}(x(v^{−}) + x(v^{+})), v ∈ V_{j}, j ≥ 1;  (8)

where v^{−} and v^{+} are defined as follows. Each v ∈ V_{j} admits a unique representation v = (v_{1}, …, v_{d}) with v_{i} = k_{i}/2^{j}, (1 ≤ i ≤ d). The points v^{−} = (v_{1}^{−}, …, v_{d}^{−}) and v^{+} = (v_{1}^{+}, …, v_{d}^{+}) are then defined by

v_{i}^{−} = \{ \begin{matrix}v_{i} − 2^{−j} & if k_{i} is odd; \\ v_{i} & if k_{i} is even; \end{matrix} v_{i}^{+} = \{ \begin{matrix}v_{i} + 2^{−j}, & if k_{i} is odd; \\ v_{i} & if k_{i} is even. \end{matrix}

Since v is in V_{j}, at least one of the k_{i}’s is odd, so v^{−}, v and v^{+} are really three distinct points of 𝒯. Moreover we can write

v^{−} = v − 2^{−j}e(v), v^{+} = v + 2^{−j}e(v) with e(v) := ∑_{k_{i} odd} e_{i}, (9)

so λ_{j,v}(x) is a second difference of the function x directed by the vector e(v). Observe moreover that if v is in V_{j}, then v^{−} and v^{+} are in W_{j−1}.

Now we are in a position to state the equivalence of norms we were looking for. Let us define for any function x : [0, 1]^{d} → C the (possibly infinite) sequential seminorm

∥x∥_{ρ}^{seq} := sup_{j≥0} \frac{1}{ρ(2^{−j})} max_{v∈V_{j}} |λ_{j,v}(x)|.

Observe moreover that when x is continuous, ∥x∥_{ρ}^{seq} = 0 if and only if x = 0. In this case, ∥x∥_{ρ}^{seq} is a (possibly infinite) true norm.

Under the conditions (1) to (5), the norm ∥x∥_{ρ} is equivalent on 𝒞(𝒯), the space of continuous functions on 𝒯, to the sequential norm, i.e. there are positive constants a, b such that for every x ∈ 𝒞(𝒯),

a∥x∥_{ρ}^{seq} ≤ ∥x∥_{ρ} ≤ b∥x∥_{ρ}^{seq},  (10)

with finite values of the norms if and only if x ∈ ℋ_{ρ}(𝒯) (see [13] or [15]).

2.2 Tightness in ℋ_{ρ}^{o}(𝒯)

The following theorem is a straightforward adaptation of [13, Th. 12].

Theorem 1. The sequence (ξ_{n})_{n≥1} of random elements in ℋ_{ρ}^{o}(𝒯) is tight if and only if it satisfies the two following conditions.

(i) For each dyadic t ∈ [0, 1]^{d}, the sequence of random variables (ξ_{n}(t))_{n≥1} is tight.

(ii) For each positive ε,

lim_{J→∞} sup_{n≥1} P(sup_{j≥J} \frac{1}{ρ(2^{−j})} max_{v∈V_{j}} |λ_{j,v}(ξ_{n})| > ε) = 0.  (11)

2.3 More on the Hölder moduli

We collected in this subsection some useful complements about the functions ρ. The proof being elementary will be omitted.

Proposition 2. Suppose that ρ satisfies (1), (2) and there is some δ in (0, 1/2] such that

∀h ∈ [0, δ], ρ(2h) ≤ 2ρ(h).  (12)

Then for every h ∈ [0, δ/2] and every r ∈ [0, 1],

ρ(rh) ≥ \frac{r}{2}ρ(h).  (13)

Proposition 3. Conditions (1)–(5) are satisfied by the non decreasing functions ρ of the form

ρ(h) = h^{α}L(h), 0 ≤ h ≤ 1, 0 < α < 1,

where the function L is continuous and positive on (0, 1] and verifies one of the following extra assumptions.

Case 1: L is non increasing on (0, 1], has right limit ∞ at 0 and for some 0 < ε < α, 0 < δ < 1, the function h ↦ h^{ε}L(h) is non decreasing on

(0, δ].

Case 2: L is non decreasing on (0, 1], L(0) = 0 and for some 0 < ε < 1 − α and 0 < δ < 1, the function h ↦ h^{−ε}L(h) is non increasing on (0, δ].

In both cases, we may choose c_{1} = 2 in (3) when the range of h is restricted to [0, δ/2] instead of [0, 1/2].

To exhibit a concrete class of functions L satisfying the conditions of Proposition 3, let us define inductively the sequence of functions ℓ_{k} on (u_{k}, ∞) by ℓ_{1}(u) = ln u, u_{1} = 1 and for k ≥ 2

ℓ_{k}(u) = ln(ℓ_{k−1}(u)), u > u_{k}, ℓ_{k−1}(u_{k}) = 1.

Corollary 4. For any 0 < α < 1 and any finite sequence β_{1}, …, β_{m} of real exponents (with at least one non null), there exists a corresponding sequence of positive constants b_{1}, …, b_{m} such that the function ρ(h) = h^{α}L(h) with

L(h) := ∏_{k=1}^{m} ℓ_{k}^{β_{k}}(\frac{b_{k}}{h} ), 0 < h ≤ 1,

fulfills the conditions (1)–(5).

2.4 Miscellaneous

For the proof of the following lemma see e.g. [10].

Lemma 5. Let (η_{k}) be positive random variables on some probability space and let ψ be some Young function. Then

E max_{1≤k≤n} η_{k} ≤ ψ^{−1}(n) max_{1≤k≤n} ||η_{k}||_{ψ}.  (14)

Here ||η||_{ψ} = inf{c > 0 : E ψ(η/c) ≤ 1} is the ψ-Orlicz norm of the positive random variable η. We apply this lemma with the function

ψ(t) = ψ_{p}(t) = exp(t^{p}) − 1, t ∈ R.

3 Hölderian consistency

From the Glivenko-Cantelli and continuity theorems follows (see Theorem 2.1 in Csörgő (1981)) uniform a.s. consistency of e.ch.f. on each bounded set S ⊂ R^{d}:

sup_{t∈S} |𝔠_{n}(t) − 𝔠(t)| ⟶_{n→∞}^{a.s.} 0.

In this section we investigate consistency of empirical characteristic function 𝔠_{n} with respect to Hölder norm.

Theorem 6. Assume that ρ satisfies Conditions (1), (2), (6) and (12). Then the convergence

sup_{\begin{matrix}t,s∈𝒯, \\ s≠t \end{matrix}} \frac{|𝔠_{n}(t) − 𝔠(t)|}{ρ(|t − s|)} ⟶_{n→∞}^{a.s.} 0  (15)

holds if and only if

E ρ^{∗}(|X|) < ∞,  (16)

where

ρ^{∗}(h) := \frac{1}{ρ(min(1; 1/h))}, 0 < h < ∞.  (17)

Corollary 7. Let ρ be of the form (see Corollary 4)

ρ(h) = h^{α} ∏_{k=1}^{m} ℓ_{k}^{β_{k}}(\frac{b_{k}}{h} ),

with 0 < α < 1 and β_{1}, …, β_{m} ∈ R. Then the convergence (15) holds if and only if

E {|X|^{α} ∏_{k=1}^{m} ℓ_{k}^{−β_{k}}(b_{k} max(|X|; 1))}< ∞.

Corollary 8. For each bounded set S ⊂ R^{d} the convergence

sup_{\begin{matrix}t,s∈S, \\ s≠t \end{matrix}} \frac{|𝔠_{n}(t) − 𝔠(t)|}{|t − s|^{α}} ⟶_{n→∞}^{a.s.} 0,  (18)

holds if and only if E |X_{1}|^{α} < ∞.

Proof of Theorem 6. Write

𝔠_{n}(t) − 𝔠(t) = n^{−1} ∑_{k=1}^{n}(ξ_{k}(t) − 𝔠(t)),

where ξ_{k}(t) := exp(i⟨t, X_{k}⟩), so that ξ_{k} = (ξ_{k}(t), t ∈ [0, 1]^{d}), k = 1, …, n are independent copies of the same random element ξ_{1} in ℋ_{ρ}^{o}(𝒯). Due to (6), ℋ_{ρ}^{o}(𝒯) contains all the continuous piecewise affine functions and is easily seen to be separable (use affine interpolation at dyadic points for instance). By the strong law of large numbers in ℋ_{ρ}^{o}(𝒯),

∥𝔠_{n} − 𝔠∥_{ρ} ⟶_{n→∞}^{a.s.} 0  (19)

if and only if E ∥ξ∥_{ρ} < ∞ (and then 𝔠 may be viewed as the Bochner integral of ξ in the space ℋ_{ρ}^{o}(𝒯)). Observing that

ξ(t) − ξ(s) = e^{i⟨t,X⟩} − e^{i⟨s,X⟩} = 2 exp(i\frac{⟨t + s, X⟩}{2}) sin(\frac{⟨t − s, X⟩}{2}),

the convergence (19) is clearly equivalent to

M := E sup_{\begin{matrix}t,s∈𝒯, \\ s≠t \end{matrix}} \frac{| sin(\frac{⟨t−s,X⟩}{2})|}{ρ(|t − s|)} < ∞.  (20)

Choosing s = (1/2, …, 1/2) at the center of 𝒯 and t − s = u|X|_{2}^{−2}X and using the fact that ρ is non decreasing on [0, 1], we get the lower bound

M ≥ E sup_{0<u≤1/2} \frac{| sin(u/2)|1_{\{|X|≥1\}}}{ρ(u|X||X|_{2}^{−2})} ≥ sin(1/4)E \frac{1_{\{|X|≥1\}}}{ρ(1/|X|)}.

Now (16) clearly follows from (20).

Conversely, by the estimates | sin θ| ≤ min(|θ|; 1) and |t|_{2} ≤ \sqrt{d}|t|,

Q := sup_{\begin{matrix}t,s∈𝒯, \\ s≠t \end{matrix}} \frac{| sin(⟨t − s, X⟩/2)|}{ρ(|t − s|)} ≤ sup_{\begin{matrix}t,s∈𝒯, \\ s≠t \end{matrix}} \frac{min(|⟨t − s, X⟩/2|; 1)}{ρ(d^{−1/2}|t − s|_{2})}.  (21)

Since the denominator of this upper bound depends only on the Euclidean norm, we are free to choose t − s = 2u|X|_{2}^{−2}X, with u > 0, in order to maximize the scalar product. To avoid a range depending on X for u, we enlarge it in (0, ∞). Then it is convenient to extend ρ by putting

ρ̃(h) := ρ(min(h; 1)), 0 ≤ h < ∞.

Noting also that ρ̃ is non decreasing on (0, ∞), we get

Q ≤ sup_{u>0} \frac{min(u; 1)}{ρ̃(2d^{−1/2}u/|X|_{2})} ≤ sup_{u>0} \frac{min(u; 1)}{ρ̃(2d^{−1}u/|X|)}.  (22)

On the event {X < 2/δ}, with δ given by (12), we have (recalling (6) and noting that ρ̃(1) = ρ(1) ≥ c_{4}):

Q ≤ sup_{u>0} \frac{min(u; 1)}{ρ̃(uδ/d)} ≤ sup_{u>0} \frac{min(u; 1)}{c_{4} min(uδ/d; 1)} < ∞.

By non decreasingness of ρ̃ and Proposition 2, it is clear that

ρ̃(rh) ≥ min(r/2; 1)ρ̃(h), 0 ≤ h ≤ \frac{δ}{2}, 0 ≤ r < ∞.

Therefore on the event {X ≥ 2/δ},

Q ≤ sup_{u>0} \frac{min(u; 1)}{min(uδ/d; 1)} \frac{1}{ρ̃(1/|X|)}.

So (20) follows from (16) and the proof is complete.

It is worth to note that none of Conditions (4) and (5) was used in Theorem 6. This is due to the fact that we did not use the sequential norm. We now turn to a Marcinkiewicz-Zygmund type result for which this sequential norm will show to be useful and we shall assume from now on that ρ satisfies (1)–(5). Our main assumptions will be expressed in term of the quantities

τ_{p}(u) := ∑_{l=1}^{d} E | sin(\frac{u}{2}⟨X, e_{l}⟩)|^{2p}, u ∈ R, 1 ≤ p ≤ 2,  (23)

whose relevance is motivated as follows.

Recalling (8) and (9), we can write the coefficients λ_{j,v}(ξ) as

λ_{j,v}(ξ) = exp(i⟨v, X⟩)[1 − cos(2^{−j}⟨e(v), X⟩)],

so that

|λ_{j,v}(ξ)| = 2 sin^{2}(2^{−j−1}⟨e(v), X⟩).  (24)

Now the coefficients τ_{p} arise from the following estimate.

Lemma 9. For every x ∈ R^{d}, v ∈ V_{j}, u ∈ R, p ≥ 1,

| sin(u⟨e(v), x⟩)|^{2p} ≤ d^{2p−1} ∑_{l=1}^{d} | sin(u⟨x, e_{l}⟩)|^{2p}.

Proof. By induction it is easily seen that

sin(x_{1} + ⋯ + x_{d}) = ∑_{l=1}^{d} a_{l} sin x_{l},

where each a_{l} is function of the (x_{k}, k ≠ l) and |a_{l}| ≤ 1. The result follows from this remark and the convexity of r ↦ r^{2p}.

Theorem 10. Let 1 < p < 2 and q = p/(p − 1). If

∑_{j=1}^{∞} \frac{τ_{p}^{1/p}(2^{−j})}{ρ(2^{−j})} j^{1/q} < ∞,  (25)

then

n^{1−1/p}(𝔠_{n} − 𝔠) ⟶_{n→∞}^{a.s.} 0 in ℋ_{ρ}^{o}(𝒯).  (26)

Corollary 11. If

∑_{j=1}^{∞} 2^{αj}j^{1/q}τ_{p}^{1/p}(2^{−j}) < ∞,  (27)

then, for each bounded set S ⊂ R^{d}

n^{1−1/p} sup_{\begin{matrix}t,s∈S, \\ s≠t \end{matrix}} \frac{|𝔠_{n}(t) − 𝔠(t)|}{|t − s|^{α}} ⟶_{n→∞}^{a.s.} 0.  (28)

Proof of Theorem 10. By Marcinkiewicz-Zygmund strong law of large numbers in Banach spaces (see e.g. Ledoux Talagrand [10, Th. 7.9.]), the convergence (26) holds if and only if both following conditions are satisfied.

E ∥ξ∥_{ρ}^{p} < ∞;  (29)

n^{1−1/p}(𝔠_{n} − 𝔠) ⟶_{n→∞}^{Pr} 0 in ℋ_{ρ}^{o}(𝒯).  (30)

It seems worth to recall here that (29) would be sufficient for the Marcinkiewicz-Zygmund SLLN if the space ℋ_{ρ}^{o}(𝒯) had the type p. But precisely, ℋ_{ρ}^{o}(𝒯) contains c_{0} and then has only the trivial type 1. It is known that in such spaces, even strong integrability conditions do not imply a.s. convergence in general (see e.g. [10, Ex. 7.11]).

In all the proof we work with the equivalent sequential norm ∥x∥_{ρ}^{seq}. First to check (29), we have

E (∥ξ∥_{ρ}^{seq})^{p} = E sup_{j≥0} \frac{1}{ρ(2^{−j})^{p}} max_{v∈V_{j}} |λ_{j,v}(ξ)|^{p}.

Lemma 9 and (24) provide a bound uniform on V_{j}, which leads after integration, to

E (∥ξ∥_{ρ}^{seq})^{p} ≤ \frac{1}{ρ(1)^{p}} + d^{2p−1} ∑_{j≥1} \frac{τ_{p}(2^{−j})}{ρ(2^{−j})^{p}} .  (31)

Since p > 1, the convergence of this last series follows clearly from (25).

To prove (30), let us introduce some additional notations. Define the processes

S_{n} = n^{1−1/p}(𝔠_{n} − 𝔠) = n^{−1/p} ∑_{k=1}^{n}(ξ_{k} − E ξ_{k}).

Consider for A > 0, the processes

ξ_{k,A}(t) = 1_{{|X_{k}|≤A}} exp(i⟨t, X_{k}⟩), t ∈ 𝒯

and

S_{n,A} := n^{−1/p} ∑_{k=1}^{n}(ξ_{k,A} − E ξ_{k,A}).

To obtain (30), it clearly suffices to prove, that

sup_{n≥1} E ∥S_{n} − S_{n,A}∥_{ρ}^{seq} ⟶_{A→∞} 0  (32)

and that for each A large enough,

E ∥S_{n,A}∥_{ρ}^{seq} ⟶_{n→∞} 0.  (33)

With ξ′_{k} = ξ_{k} − E ξ_{k} − (ξ_{k,A} − E ξ_{k,A}), k = 1, …, n, we can write

S_{n} − S_{n,A} = n^{−1/p} ∑_{k=1}^{n} ξ′_{k}.

Now let (ε_{k})_{k≥1} be a Rademacher sequence independent of (X_{k})_{k≥1}. For notational convenience, we may consider that the sequences (X_{k})_{k≥1}, (ε_{k})_{k≥1} and (ε_{k}ξ_{k})_{k≥1} are defined respectively on the probability spaces (Ω_{X}, F_{X}, P_{X}), (Ω_{ε}, F_{ε}, P_{ε}) and (Ω, F, P) which is the product of the two preceding ones. We shall denote respectively by E_{X}, E_{ε} partial integration with respect to P_{X} or P_{ε}. By symmetrization (see e.g. [10, Lemma 6.3]),

E ∥S_{n} − S_{n,A}∥_{ρ}^{seq} ≤ 2E ∥n^{−1/p} ∑_{k=1}^{n} ε_{k}ξ′_{k}∥_{ρ}^{seq},  (34)

whence

E ∥S_{n} − S_{n,A}∥_{ρ}^{seq} ≤ 2n^{−1/p} ∑_{j≥0} \frac{1}{ρ(2^{−j})}E max_{v∈V_{j}}|∑_{k=1}^{n} ε_{k}λ_{j,v}(ξ′_{k})|.  (35)

Now using Lemma 5 together with a classical estimate for the ψ_{q}-Orlicz norms of Rademacher sums [10, Lemma 4.3] and partial integration with respect to P_{ε}, we obtain with an absolute constant C_{q}

E max_{v∈V_{j}}|∑_{k=1}^{n} ε_{k}λ_{j,v}(ξ′_{k})| ≤ C_{q}j^{1/q}E_{X} max_{v∈V_{j}} ( ∑_{k=1}^{n} |λ_{j,v}(ξ′_{k})|^{p})^{1/p}.  (36)

Here we have to pay some attention to the control of λ_{j,v}(ξ′_{k}). Write η_{k} = ξ_{k}1_{{|X_{k}|>A}} and note that ξ′_{k} = η_{k} − E η_{k}. We clearly have λ_{j,v}(η_{k}) = λ_{j,v}(ξ_{k})1_{{|X_{k}|>A}}. This remark together with (24) and Lemma 9 gives for j ≥ 1,

|λ_{j,v}(η_{k})|^{p} ≤ 2^{p}d^{2p−1} ∑_{l=1}^{d} | sin(2^{−j−1}⟨X_{k}, e_{l}⟩)|^{2p}1_{{|X_{k}|>A}} =: M_{j,k}. (37)

This upper bound M_{j,k} being uniform with respect to v ∈ V_{j}, we obtain for j ≥ 1,

E_{X} max_{v∈V_{j}} ( ∑_{k=1}^{n} |λ_{j,v}(ξ′_{k})|^{p})^{1/p} ≤ E_{X}( ∑_{k=1}^{n} M_{j,k} + E_{X}M_{j,k})^{1/p}
≤ ( ∑_{k=1}^{n} 2E_{X}M_{j,k})^{1/p}
= 2^{1/p}n^{1/p}E_{X}(M_{j,1})^{1/p}.  (38)

This bound holds also in the special case j = 0, noting that |λ_{0,v}(ξ_{k})| = 1 and defining M_{0,k} := 1_{{|X_{k}|>A}}.

Going back to (36), (35) and (34), we get finally with a constant C = C(d, p),

E ∥S_{n} − S_{n,A}∥_{ρ}^{seq} ≤ C ∑_{j≥0} \frac{1}{ρ(2^{−j})}j^{1/q}E_{X}(M_{j,1})^{1/p}.  (39)

This upper bound is uniform in n. Its convergence to zero (when A goes to ∞) follows from our assumption (25) by dominated convergence. So (32) is established.

To prove the convergence (33) for the process S_{n,A}, we use the same symmetrization technique, but with the ψ_{2}-Orlicz norm instead of the ψ_{q} one. So putting ξ̃_{k,A} = ξ_{k,A} − E ξ_{k,A}, we get

E ∥S_{n,A}∥_{ρ}^{seq} ≤ 2E ∥∑_{k=1}^{n} ε_{k}ξ̃_{k,A}∥_{ρ}^{seq}
≤ 2 ∑_{j≥0} \frac{1}{ρ(2^{−j})}E max_{v∈V_{j}}|n^{−1/p} ∑_{k=1}^{n} ε_{k}λ_{j,v}(ξ̃_{k,A})|
≤ C′n^{−1/p} ∑_{j≥0} \frac{\sqrt{j}}{ρ(2^{−j})}E_{X} max_{v∈V_{j}}(∑_{k=1}^{n} |λ_{j,v}(ξ̃_{k,A})|^{2})^{1/2}.

Similarly to (37) and (38), we have

E_{X} max_{v∈V_{j}}(∑_{k=1}^{n} |λ_{j,v}(ξ̃_{k,A})|^{2})^{1/2} ≤ 2^{1/2}n^{1/2}E_{X}(M′_{j,1})^{1/2},  (40)

where for j ≥ 1

M′_{j,k} := 4d^{3} ∑_{l=1}^{d} sin^{4}(2^{−j−1}⟨X_{k}, e_{l}⟩)1_{{|X_{k}|≤A}},

and for j = 0,

M′_{0,k} := 1_{{|X_{k}|≤A}}.

Now for j ≥ 1,

(M′_{j,1})^{1/2} ≤ d^{1/2}A^{2}2^{−2j}

and this bound is valid also in the case j = 0 if we take A ≥ 1. All this leads to

E ∥S_{n,A}∥_{ρ}^{seq} ≤ C′′A^{2}n^{1/2−1/p} ∑_{j≥0} \frac{\sqrt{j}2^{−2j}}{ρ(2^{−j})} .  (41)

By (6), the convergence of the series in (41) is obvious, so (33) is established and the proof complete.

4 Weak Hölder convergence

In this section we consider the empirical characteristic process

Y_{n}(t) = \sqrt{n}(𝔠_{n}(t) − 𝔠(t)), t ∈ [0, 1]^{d}.

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

ξ_{k}(t) = exp(i⟨t, X_{k}⟩) − 𝔠(t), k = 1, …, n.

Hence Y_{n} is a random element in ℋ_{ρ}^{o}(𝒯) if and only if the function 𝔠(t) is in ℋ_{ρ}^{o}(𝒯). By the multidimensional CLT, the finite dimensional distributions of (Y_{n}) converge to those of a complex Gaussian process Y with zero mean 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 of a stochastic integral (see e.g. [3])

Y (t) = ∫_{R^{d}} exp(i⟨t, x⟩) dB_{F} (x),

where B_{F} (x) is a d-variate Brownian bridge process associated with the distribution function F, i.e., B_{F} is a d-variate Gaussian process with the following properties:

E B_{F} (x) = 0, E B_{F} (x)B_{F} (y) = F (x ∨ y) − F (x)F (y),

lim_{x_{j}→−∞} B_{F} (x_{1}, …, x_{d}) = 0, j = 1, …, d;

lim_{x_{j}→∞,1≤j≤d} B_{F} (x_{1}, …, x_{d}) = 0,

where for x, y ∈ R^{d} we write x ∨ y = (min{x_{1}, y_{1}}, …, min{x_{d}, y_{d}}). An other representation of the process Y is given in S. Csörgő [3].

Proving the weak ℋ_{ρ}^{o}(𝒯)-convergence of Y_{n} to Y is exactly proving that the random element ξ_{1} satisfies the CLT in this space (in the sense of [10, Chap. 10]). As ℋ_{ρ}^{o}(𝒯) contains a subspace isomorphic to c_{0}, the problem is not easy. In particular, we should not expect to obtain the CLT for ξ_{1} from its square integrability in the norm of ℋ_{ρ}(𝒯). Our assumption (42) to prove this CLT may be roughly interpreted as square integrability in a norm a bit stronger than ∥ξ_{1}∥_{ρ}. The relation between (42) and the integrability properties of X is discussed in Propositions 14 and 15 below.

Theorem 12. If

∑_{j=1}^{∞} \frac{\sqrt{j}}{ρ(2^{−j})}τ_{2}^{1/2}(2^{−j}) < ∞,  (42)

then (Y_{n}) converges in distribution to Y in the space ℋ_{ρ}^{o}(𝒯).

Proof. To prove the theorem we shall use randomization arguments which are based on the following well known result.

Lemma 13 ([10, Th. 10.14]). Let Z be a mean zero Borel random variable with values in a separable Banach space 𝔛 and ε a Rademacher variable independent of Z. Denote by (Z_{k}) (resp. (ε_{k})) independent copies of Z (resp. ε). These two sequences are understood to be independent and constructed on different probability spaces. The following are equivalent:

a) E ∥Z∥^{2} < ∞ and n^{−1/2} ∑_{k=1}^{n} Z_{k} converges in distribution in 𝔛;

b) for almost every ω of the probability space supporting the Z_{k}’s, the sequence (n^{−1/2} ∑_{k≤n} ε_{k}Z_{k}(ω)) converges in distribution in 𝔛.

In either case, the limit is distributed as G(Z), the Gaussian distribution with the same covariance structure as Z.

We use the part b) ⇒ a) of this lemma with 𝔛 = ℋ_{ρ}^{o}(𝒯) and Z_{k} = ξ_{k}, where ξ_{k}(t) = exp(i⟨t, X_{k}⟩) − 𝔠(t), t ∈ 𝒯, k = 1, …, n. So we consider the randomized field

ζ_{n}^{ω}(t) := n^{−1/2} ∑_{k=1}^{n} ε_{k}ξ_{k}(ω, t), t ∈ 𝒯,

defined on the probability space Ω_{ε} supporting the Rademacher sequence (ε_{k}).

Using the part a) ⇒ b) of Lemma 13 with 𝔛 = (R^{d})^{m}, together with the CLT in R^{d}, we have for almost every ω the convergence of the finite dimensional distributions of ζ̃_{n}^{ω} to those of Y. So to obtain for a.e. ω the ℋ_{ρ}^{o}(𝒯)-weak convergence of ζ̃_{n}^{ω} to Y, it remains to prove that for a.e. ω, for every ϵ > 0,

lim_{J→∞} sup_{n≥1} P_{ε}( sup_{j>J} \frac{1}{ρ(2^{−j})} max_{v∈V_{j}} |λ_{j,v}(ζ_{n}^{ω})| > ϵ) = 0.  (43)

Since

λ_{j,v}(ζ_{n}^{ω}) = 2n^{−1/2} ∑_{k=1}^{n} ε_{k}[e^{i⟨v,X_{k}(ω)⟩} sin^{2}⟨2^{−j−1}e(v), X_{k}(ω)⟩ −
− E e^{i⟨v,X_{k}⟩} sin^{2}⟨2^{−j−1}e(v), X_{k}⟩],

we see by the well known tail behavior of Rademacher sums and Lemma 9, that

∥λ_{j,v}(ζ_{n}^{ω})∥_{ϕ_{2}} ≤ ad^{3}[n^{−1} ∑_{k=1}^{n} ∑_{l=1}^{d} sin^{4}⟨2^{−j−1}X_{k}(ω), e_{l}⟩ + τ_{2}(2^{−j})]^{1/2},

with an absolute constant a. By the strong law of large numbers in the probability space supporting (X_{k}), this is bounded for a.e. ω by 2ad^{3}τ_{2}^{1/2}(2^{−j}) for n ≥ N(ω). So by Lemma 5, we have for a.e. ω, for each n ≥ N(ω),

E_{ε} max_{v∈V_{j}} |λ_{j,v}(ζ_{n}^{ω})| ≤ c_{d}\sqrt{jτ_{2}(2^{−j})},

where the constant c_{d} depends on d only. In view of Hypothesis (42), this gives (43) and completes the proof.

We remark that (42) is simply the limit case p = 2 in Assumption (25) in Theorem 10. Both conditions express some integrability property of X. This is illustrated by the following two propositions. Actually it is enough to consider the case d = 1.

Proposition 14. For a random variable X ∈ R, put

τ_{p}(h) = E | sin(hX/2)|^{2p}

and suppose that for some 1 < p ≤ 2,

∑_{j=1}^{∞} \frac{j^{1/q}}{ρ(2^{−j})}τ_{p}^{1/p}(2^{−j}) < ∞,  (44)

where q = p/(p − 1). Then with ρ^{∗} as in (17),

E ρ^{∗}(|X|)^{p}|ln |X||^{p−1}1_{{|X|>1}} < ∞.  (45)

Proof. First note that (44) yields

∑_{j=1}^{∞} \frac{j^{p/q}}{ρ(2^{−j})^{p}} τ_{p}(2^{−j}) < ∞.  (46)

Next, the elementary minoration

| sin(t)| ≥ \frac{1}{\sqrt{2}}1_{\{π/4<|t|≤π/2)\}},

gives for each j ≥ 1,

τ_{p}(2^{−j}) ≥ 2^{−p}P(2^{j−1}π < |X| ≤ 2^{j}π).

By comparison between series and integral, it follows then from (46) that

E \frac{|ln |X||^{p−1}}{ρ(π/|X|)^{p}} 1_{\{|X|>π\}} < ∞.

By (3), ρ(π/|X|) ≤ ρ(4/|X|) ≤ c_{1}^{2}ρ(1/|X|), so (45) holds.

Remark. Consider for simplicity the case ρ(h) = h^{α}. Then (45) becomes

E |X|^{pα}|ln |X||^{p−1} < ∞,

where we dropped 1_{{|X|>1}} in view of the boundedness of |x|^{pα}| ln |x||^{p−1} on [0, 1]. On the other hand, if E |X|^{pγ} < ∞ for some 1 > γ > α, then (44) readily follows from the elementary inequality | sin t| ≤ |t|^{γ/2}.

Proposition 15. Suppose that the random variable X ∈ R and consider the process ξ(t) = exp(itX), t ∈ [0, 1]. If for some 1 < p < 2, ξ satisfies the Marcinkiewicz-Zygmund SLLN (26), then E ρ^{∗}(|X|)^{p} < ∞. If ξ satisfies the central limit theorem in ℋ_{ρ}^{o}(𝒯)[0, 1], then E ρ^{∗}(|X|)^{p} < ∞ for any 0 < p < 2.

Proof. If ξ satisfies (26), then E ∥ξ∥_{ρ}^{p} is finite for the corresponding p, while if ξ satisfies the CLT in ℋ_{ρ}^{o}(𝒯) the same finiteness holds for any 0 < p < 2, see e.g. [10, Th. 7.9, Lem. 10.1].

For |X(ω)| ≥ π, choosing t = t(ω) = π/|X(ω)|, we obtain

∥ξ∥_{ρ} ≥ \frac{|ξ(t) − ξ(0)|}{ρ(t)} = \frac{2| sin(tX/2)|}{ρ(t)} = \frac{2}{ρ(|π/X|)} ≥ \frac{2}{c_{1}^{2}ρ(1/|X|)}.

The result follows.

5 Some suggestions for possible applications

First we outline application of our results to the problem of testing for symmetry of the distribution function. Feuerverger and Mureika [7] proposed for that the statistic (in the univariate case):

S_{n}(q) = ∫_{R^{d}} |Im 𝔠_{n}(t)|^{2}q(t) dt,

where q : R^{d} → R is a nonnegative integrable function. This is suggested by the fact, that a characteristic function of a random variable is real if and only if the corresponding distribution is symmetric. We have the following result.

Proposition 16. Let the random vector X be symmetric and Condition (42) be satisfied. Then for any function q : R^{d} → R such that

∫_{R^{d}} ρ(|x|)q(x) dx < ∞,  (47)

the statistic nS_{n}(q) converges in distribution to the random variable S(q) = ∫_{R^{d}} |Im Y (t)|^{2}q(t) dt.

Proof. The result follows easily by Theorem 12 and Theorem 4.2 of Billingsley [1].

Consider univariate case and apply the above result to the function q_{α}(x) = q(x) = |x|^{−1−α}, where α ≥ 0. Evidently, the result of Feuerverger and Mureika (1977) is not applicable to the statistic S_{n}(q_{α}). By simple calculations the statistic S_{n}(q) reduces to cS_{n}(α), where

S_{n}(α) = n^{−2} ∑_{k,j=1}^{n} [|X_{k} + X_{j}|^{α} − |X_{k} − X_{j}|^{α}]

and c = ∫_{−∞}^{∞} sin^{2}(t/2)|t|^{−1−α} dt. Of course, this is a V -statistic. Asymptotic of V -statistics is well investigated. As a corollary to the proposition above, we have the following.

Corollary 17. If the random variable X ∈ R is symmetric, then the statistic S_{n}(α) converges in distribution provided the condition

∑_{j=1}^{∞} \frac{\sqrt{j}}{ρ(2^{−j})}(E sin^{4}(\frac{X}{2^{j+1}} ))^{1/2} < ∞

is satisfied, where

∫_{−∞}^{∞} ρ(t)|t|^{−1−α} dt < ∞.

The use of the empirical characteristic function for goodness-of-fit tests becomes very popular especially after the works of Csórgő [2] and Marcus [11] (see e.g. [8], [4] and references therein). The classes of statistics used for the tests consist mainly of certain weighted integrals of the squared modulus of the difference between the empirical characteristic function and the unknown characteristic function of the population. To that big amount of statistics we suggest the one more class

D_{n}(g) = ∫_{R^{d}} ∫_{R^{d}} |Y_{n}(t) − Y_{n}(s)|^{2}g(t, s) dt ds,

where g is a certain measurable function.

Proposition 18. Assume the condition (42) is satisfied. Then for any function q : R^{d} × R^{d} → R such that

∫_{R^{d}} ∫_{R^{d}} ρ(|t − s|)g(s, t) ds dt < ∞,  (48)

the statistic nD_{n}(g) converges in distribution to the random variable D(g) = ∫_{R^{d}} ∫_{R^{d}} |Y (t) − Y (s)|^{2}g(t, s) dt ds.

Proof. The result follows easily by Theorem 12 and [1, Th. 4.2].

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