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

TL;DR

This paper extends classical theorems by Hsu-Robbins and Spitzer to the context of fractional Brownian motion, using advanced probabilistic techniques to analyze correlated increments.

Contribution

It introduces new proofs of Hsu-Robbins and Spitzer's theorems for fractional Brownian motion increments employing Stein's method and Wiener-Itô integrals.

Findings

Established Hsu-Robbins theorem for fractional Brownian motion increments.

Proved Spitzer's theorem in the context of correlated fractional Brownian motion.

Demonstrated the applicability of Stein's method to dependent Gaussian processes.

Abstract

Using recent results on the behavior of multiple Wiener-It\^o 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.