Moderate Deviation Principle for a Stochastic Approximation Process

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 $X_{n+1}=X_n+\frac{b}{n+1}[g(X_n)+U_{n+1}]$, 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.