A counter example to central limit theorem in Hilbert spaces under a strong mixing condition

Davide Giraudo,Dalibor Volny

doi:10.1214/ecp.v19-3249

Abstract

We show that in a separable infinite dimensional Hilbert space, uniform integrability of the square of the norm of normalized partial sums of a strictly stationary sequence, together with a strong mixing condition, does not guarantee the central limit theorem.

Highlights

Introduction and notationsLet (Ω, F, μ) be a probability space and (S, d) a separable metric space
We show that in a separable infinite dimensional Hilbert space, uniform integrability of the square of the norm of normalized partial sums of a strictly stationary sequence, together with a strong mixing condition, does not guarantee the central limit theorem
We say that the sequence of random variables (Xn)n∈Z from Ω to S is strictly stationary if for all integer d and all integer k, the d- uple (X1, . . . , Xd) has the same law as (Xk+1, . . . , Xk+d)

Summary

Introduction and notations

Let (Ω, F , μ) be a probability space and (S, d) a separable metric space. We say that the sequence of random variables (Xn)n∈Z from Ω to S is strictly stationary if for all integer d and all integer k, the d- uple (X1, . . . , Xd) has the same law as (Xk+1, . . . , Xk+d). There exists a probability space (Ω, F , μ) such that given 0 < q < 1, we can construct a strictly stationary sequence X = (f ◦T k) = (Xk)k∈N defined on Ω, taking its values in H, such that: a) E [f ] = 0, E [ f pH] is finite for each p; b) the limit limN→∞ σN (f ) is infinite; c) the process (Xk)k∈N is β-mixing, more precisely, βX(j) = O. Tone has established in [20] a central limit theorem for strictly stationary random fields with values in H under ρ -mixing conditions These coefficients are defined by ρ (n) := sup |E [ f , g H] − E [f ] , E [g] H| , f L2(H) g L2(H). From this inequality, they deduce a central limit theorem for a stationary sequence (Xj)j∈Z of H-valued zero-mean random variables satisfying α−1(u)Q2X0.

Construction of f

Preliminary results

Proof of Theorem A

Proof of Theorem A’