A Weak Convergence Theorem with Application to the Robbins-Monro Process

Abstract

In this paper the asymptotic distribution of a sequence of random variables $(X_n)_{n \in \mathbf{N}}$, given by the recursion $$X_{n+1} = X_n(1 - a_n^2g(X_n)) + a_n Y_n,$$ is considered, where $(Y_n)$ is a sequence of independent identically distributed random variables, $g : \mathbb{R} \rightarrow \mathbb{R}$ is a positive continuous function, and $(a_n)$ is a sequence of positive numbers, going to zero. One application to the Robbins-Monro process is discussed, in which the function $g$ will not be constant. Here the asymptotic distribution is no longer normal.