A central limit theorem for some generalized martingale arrays

Abstract

Let $(X_{n,j})$ and $(Y_{n,j})$ be two arrays of real random variables and $f:\mathbb{R}\to \mathbb{R}$ a Borel function. Define $D_n={\sum }_{j}f\left({\sum }_{i=1}^{j-1}{Y}_{n,i}\right)X_{n,j}$ and $D=Z\sqrt{{\int }_0^1{f}^2(B_{G(t)})dF(t)}$ where B is a standard Brownian motion, Z a standard normal random variable independent of B, and F and G are distribution functions. Conditions for $D_n\to D$, in distribution or stably, are given. Among other things, such conditions apply to certain sequences of stochastic integrals, when the quadratic variations of the integrand processes converge in distribution but not in probability. An upper bound for the Wasserstein distance between the probability distributions of $D_n$ and D is obtained as well.