Martingale difference arrays and stochastic integrals

Abstract

Consider MDAs (Xni) and (Yni), and stopping times τn(t), 0≦t≦1. Denote {Mathematical expression} and let φ{symbol}: ℝ→ℝ be a function. If the common distribution converges and if St, Tt denote the corresponding limiting processes then we give conditions such that the martingale transforms {Mathematical expression} converge weakly to the stochastic integral {Mathematical expression} This result has important consequences for functional central limit theorems: (1) If the MDAs are connected by a difference equation of the form {Mathematical expression}, then weak convergence of Tn(t) implies that of Sn(t), and the limit satisfies the stochastic differential equation {Mathematical expression}. This observation leads to functional limit theorems for diffusion approximations. E.g. we obtain easily a result of Lindvall, [4], on the diffusion approximation of branching processes. (2) If the MDA (Xni) arises from a likelihood ratio martingale then the limit satisfies {Mathematical expression} which leads to the representation of the limiting likelihood ratios as exponential martingale: {Mathematical expression} This approximation by an exponential martingale has been proved previously by Swensen, [9], using a Taylor expansion of the log-likelihood ratio. (3) As a consequence we obtain a general functional central limit theorem: If {Mathematical expression} converges weakly to ([S, S]t), then {Mathematical expression} converges weakly to (St), provided that the distribution of (St) is uniquely determined by that of ([S, S]t). This assertion embraces previous central limit theorems, dealing with cases where the increasing process ([S, S]t) is deterministic. © 1986 Springer-Verlag.