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June, 1980 Martingale Transform and Random Abel-Dini Series
Louis H. Y. Chen
Ann. Probab. 8(3): 475-482 (June, 1980). DOI: 10.1214/aop/1176994722


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.


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Louis H. Y. Chen. "Martingale Transform and Random Abel-Dini Series." Ann. Probab. 8 (3) 475 - 482, June, 1980.


Published: June, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0434.60030
MathSciNet: MR573288
Digital Object Identifier: 10.1214/aop/1176994722

Primary: 60F15
Secondary: 60G45

Keywords: Burkholder's strong law , conditional strong law , martingale transform , random Abel-Dini series

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 3 • June, 1980
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