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December, 1976 Purely Atomic Structures Supporting Undominated and Nonuniformly Integrable Martingales
David A. Lane
Ann. Probab. 4(6): 1016-1019 (December, 1976). DOI: 10.1214/aop/1176995946

Abstract

Let $(F_n)_{n=1,2,\cdots}$ be a sequence of sigma-fields on a set $\Omega$, each $F_n$ purely atomic with respect to a measure $P$. Let $C$ denote a nested sequence of sets $C_n$, where $C_n$ is a $P$-atom of $F_n$ for each $n$. Define $S(C) = \Sigma_n(P(C_n - C_{n+1})/P(C_n))$. Then every $L^1$-bounded martingale relative to $(F_n)_{n=1,2,\cdots}$ and $P$ is uniformly integrable if and only if $S$ is finite-valued, and every such martingale is dominated if and only if $S$ is uniformly bounded.

Citation

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David A. Lane. "Purely Atomic Structures Supporting Undominated and Nonuniformly Integrable Martingales." Ann. Probab. 4 (6) 1016 - 1019, December, 1976. https://doi.org/10.1214/aop/1176995946

Information

Published: December, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0369.60050
MathSciNet: MR420835
Digital Object Identifier: 10.1214/aop/1176995946

Subjects:
Primary: 60G45

Keywords: dominated martingale , Purely atomic structure , uniformly integrable martingale

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 6 • December, 1976
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