Deviation probability bounds for fractional martingales and related remarks

TL;DR

This paper establishes exponential bounds for fractional martingales and explores their convergence properties, including a weak law of large numbers, with practical examples illustrating these theoretical results.

Contribution

It introduces Bernstein-type inequalities for fractional martingales and analyzes their convergence behavior under divergence conditions on their variation.

Findings

Proved exponential inequalities for fractional martingales.

Established weak law of large numbers for these processes.

Provided a practical example demonstrating the convergence result.

Abstract

In this paper we prove exponential inequalities (also called Bernstein's inequality) for fractional martingales. As an immediate corollary, we will discuss weak law of large numbers for fractional martingales under divergence assumption on the $\beta-$variation of the fractional martingale. A non trivial example of application of this convergence result is proposed.