Stable central limit theorems for discrete-time lag martingale difference arrays

Abstract

Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove a stable central limit theorem (CLT) via a Bernstein blocking scheme and an application of the classical martingale CLT. We generalize our limit theorem to vector-valued processes via the Cram\'er-Wold device and develop a simple form for the limiting variance. We demonstrate the application of these results to a problem in dynamic causal inference and present a simulation study supporting their validity.