Martingale Transform and Random Abel-Dini Series

Abstract

Let $X_1, X_2, \cdots$ be identically distributed random variables defined on a probability space $(\Omega, \mathscr{F}, P)$ such that $E|X_1| < \infty$ and let $\mathscr{F}_0 \subset \mathscr{F}_1 \subset \cdots$ be nondecreasing sub-$\sigma$-algebras of $\mathscr{F}$ such that $X_n$ is $\mathscr{F}_n$-measurable for $n \geqslant 1$. Define $S_n = X_1 + \cdots + X_n$ and $\xi_n = E(X_n \mid \mathscr{F}_{n-1})$. The convergence and divergence of the series $\Sigma^\infty_{n=1} \operatorname{sgn}(S_{n+k})|S_{n+k}|^{-\alpha}(X_n - \xi_n)$, where $\alpha$ is a real number and $k$ a nonnegative integer, is considered and related to that of martingale transforms. This paper answers a question raised by Kai Lai Chung.