Hsu-Robbins and Spitzer's theorems for the variations of fractional Brownian motion

Ciprian Tudor

doi:10.1214/ecp.v14-1481

Abstract

Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.

Highlights

A famous result by Hsu and Robbins [7] says that if X1, X2, . . . is a sequence of independent identically distributed random variables with zero mean and finite variance and Sn := X1 +. . .+Xn, P |Sn| > ǫn < ∞n≥1 for every ǫ > 0
The purpose of this paper is to prove Hsu-Robbins and Spitzer’s theorems for sequences of correlated random variables, related to the increments of fractional Brownian motion, in the spirit of
We note that the techniques generally used in the literature to prove the Hsu-Robbins and Spitzer’s results are strongly related to the independence of the random variables X1, X2, . . . . In our case the variables are correlated

Summary

Introduction

A famous result by Hsu and Robbins [7] says that if X1, X2, . . . is a sequence of independent identically distributed random variables with zero mean and finite variance and Sn := X1 +. . .+Xn, . The purpose of this paper is to prove Hsu-Robbins and Spitzer’s theorems for sequences of correlated random variables, related to the increments of fractional Brownian motion, in the spirit of [5] or [12]. We note that the techniques generally used in the literature to prove the Hsu-Robbins and Spitzer’s results are strongly related to the independence of the random variables X1, X2, . We need to estimate the difference between the tail probabilities of Zn(1), Zn(2) and the tail probabilities of their limits To this end, we will use the estimates obtained in [2], [10] via Malliavin calculus and we are able to prove that this difference converges to zero in all cases. Throughout the paper we will denote by c a generic strictly positive constant which may vary from line to line (and even on the same line)

Preliminaries

Spitzer’s theorem

Hsu-Robbins theorem for the variations of fractional Brownian motion