Moderate Deviation Principle for a Stochastic Approximation Process

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.