Martingale Invariance Principles

Abstract

Let $\{(S_{nj}, \mathscr{F}_{nj}), 1 \leqq j \leqq k_n\}$ be a square-integrable martingale for each $n = 1,2,3,\cdots$. Define $X_{nj} = S_{nj} - S_{n,j-1} (S_{n0} = 0), U^2_{nj} = \sum^j_{i=1} X^2_{ni}, U^2_n = U^2_{nk_n}$, and for each $z \in \lbrack 0, 1\rbrack$ let $\xi_n(z) = U^{-1}_n \sum^{k_n}_{j=1} X_{nj} I(U^{-2}_n U^2_{nj} \leqq z)$ and $\eta_n(z) = \sum^{k_n}_{j=1} X_{nj} I(U^{-2}_n U^2_{nj} \leqq z); \xi_n$ and $\eta_n$ are random elements of $D\lbrack 0, 1\rbrack$. Sufficient conditions are given for $\xi_n$ to converge in distribution to Brownian motion and for $\eta_n$ to converge to a mixture of Brownian motion distributions. We give several applications and examples.