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A Martingale Convergence Theorem in Vector Lattices

Published online by Cambridge University Press:  20 November 2018

Ralph DeMarr
Ralph DeMarr*
Affiliation:
University of Washington
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The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 424 - 432
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Birkhoff, G., Lattice theory (Providence, R.I., 1948).Google Scholar
2. Bochner, S., Partial ordering in the theory of martingales, Ann. of Math., 62 (1955), 162–169.Google Scholar
3. Doob, J. L., Stochastic processes (New York, 1953).Google Scholar
4. Kantorovich, L. V., Pinsker, A. G., and Vulikh, B. Z., Functional analysis in semi-ordered spaces (Russian) (Moscow-Leningrad, 1950).Google Scholar
5. Loève, M., Probability theory (Princeton, 1955).Google Scholar
6. Vulikh, B. Z., Introduction to the theory of semi-ordered spaces (Russian) (Moscow, 1961).Google Scholar