## The Annals of Mathematical Statistics

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

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

#### 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