Central Limit Theorem for product of dependent random variables

JunTao Duan, Ionel Popescu, Fan Zhou

Published 2021-06-28Version 1

Given $\{X_k\}$ is a martingale difference sequence. And given another $\{Y_k\}$ which has dependency within the sequence. Assume $\{X_k\}$ is independent with $\{Y_k\}$, we study the properties of the sums of product of two sequences $\sum_{k=1}^{n} X_k Y_k$. We obtain product-CLT, a modification of classical central limit theorem, which can be useful in the study of random projections. We also obtain the rate of convergence which is similar to the Berry-Essen theorem in the classical CLT.