A Weak Convergence Theorem for Functionals of Sums of Martingale Differences

For each n , let ( S nk ), 1 ≦ k ≦ k n , be a mean zero square — integrable martingale adapted to increasing s̀‐fields (b nk ), 0 ≦ k ≦ k n , and let ( b nk ), 0 ≦ k ≦ k n , be a system of random variables such that b n 0 = 0 < b n 1 <…< b   nk   n= 1 and such that b nk is b n , k −1 measurable for each k . We present sufficient conditions under which \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 0}^{k_n - 1} {f_n (b_{ni,\;} S_{ni})\;(S_{n,i + 1} \; - \;S_{ni})\; \to \int\limits_0^1 {f(t,\;W(t))\;d{\rm W(t)}} } $\end{document} as n → ∞, where { W ( t ) : 0 ≦ t ≦ 1} is a standard W IENER process.