The Annals of Mathematical Statistics

Convergence in Distribution, Convergence in Probability and Almost Sure Convergence of Discrete Martingales

David Gilat

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Abstract

Examples are provided of Markovian martingales that: (i) converge in distribution but fail to converge in probability; (ii) converge in probability but fail to converge almost surely. This stands in sharp contrast to the behavior of series with independent increments, and settles, in the negative, a question raised by Loeve in 1964. Subsequently, it is proved that a discrete, real-valued Markov-chain with stationary transition probabilities, which is at the same time a martingale, converges almost surely if it converges in distribution, provided the limiting measure has a mean. This fact does not extend to non-discrete processes.

Article information

Source
Ann. Math. Statist., Volume 43, Number 4 (1972), 1374-1379.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692494

Digital Object Identifier
doi:10.1214/aoms/1177692494

Mathematical Reviews number (MathSciNet)
MR324769

Zentralblatt MATH identifier
0243.60031

JSTOR
links.jstor.org

Citation

Gilat, David. Convergence in Distribution, Convergence in Probability and Almost Sure Convergence of Discrete Martingales. Ann. Math. Statist. 43 (1972), no. 4, 1374--1379. doi:10.1214/aoms/1177692494. https://projecteuclid.org/euclid.aoms/1177692494


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