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April, 1981 Order Convergence of Martingales in Terms of Countably Additive and Purely Finitely Additive Martingales
Kenneth A. Astbury
Ann. Probab. 9(2): 266-275 (April, 1981). DOI: 10.1214/aop/1176994467

Abstract

Let $(E, \mathscr{B}, \mu)$ be a measure space, let $\theta$ be a directed set with a countable cofinal subset, and let $(\mathscr{B}_\tau)_{\tau \in \theta}$ be an increasing family of sub-$\sigma$-algebras of $\mathscr{B}$. A martingale $(f_\tau)_{\tau \in \theta}$ is said to be of semibounded variation whenever the set $\{\int_B f_\tau d\mu \mid \tau \in \theta, B \in \mathscr{B}_\tau\}$ is bounded either from above or below. Denote conditional expectation by $\mathscr{E}$. We show that if every martingale of the form $\mathscr{E}(f \mid \mathscr{B}_\tau)_{\tau\in\theta}$ for some $\mathscr{B}$-measurable function $f$ with $\int|f| d\mu < \infty$ is order convergent, then every martingale of semibounded variation is order convergent. When the family $(\mathscr{B}_\tau)_{\tau\in\theta}$ satisfies a certain refinement condition, we obtain a sufficient condition for order convergence of martingales of semibounded variation in terms of order convergence of martingales which converge stochastically to 0.

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Kenneth A. Astbury. "Order Convergence of Martingales in Terms of Countably Additive and Purely Finitely Additive Martingales." Ann. Probab. 9 (2) 266 - 275, April, 1981. https://doi.org/10.1214/aop/1176994467

Information

Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60047
MathSciNet: MR606988
Digital Object Identifier: 10.1214/aop/1176994467

Subjects:
Primary: 60G45

Keywords: countably additive martingales , Order convergence , purely finitely additive martingales

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 2 • April, 1981
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