Moderate Deviation Principle for a Stochastic Approximation Process

TL;DR

This paper establishes a moderate deviation principle for a specific stochastic approximation process, providing new insights into its probabilistic behavior and auxiliary inequalities.

Contribution

It introduces the first moderate deviation principle for a class of stochastic approximation processes with bounded martingale differences.

Findings

Proves the moderate deviation principle for the process $(X_n)$

Derives exponential inequalities for the process

Establishes moderate deviation for weighted sums of martingale differences

Abstract

In this paper, we investigate a stochastic approximation procedure $\left(X_n\right)_{n\ge 0}$ taking values in $R$. The process is adapted to a filtration $(F_n)_{n\ge 0}$ and satisfies the recursion $X_{n+1}=X_n+\frac{b}{n+1}\big[g(X_n)+U_{n+1}\big]$, where $b>0$, $g:R \to R$ is a function and $\left(U_n\right)_{n\ge 1}$ is a sequence of bounded martingale differences adapted to the filtration $(F_n)_{n\ge 1}$. We establish the moderate deviation principle for the stochastic process $(X_n)_{n\ge 0}$. As auxiliary results, we also obtain the exponential inequality for $(X_n)_{n\ge 0}$ and the moderate deviation principle for weighted sums of bounded martingale differences.