## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 16, 12 pp.

### Self-averaging sequences which fail to converge

Eric Cator and Henk Don

#### Abstract

We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

#### Article information

**Source**

Electron. Commun. Probab., Volume 22 (2017), paper no. 16, 12 pp.

**Dates**

Received: 3 October 2016

Accepted: 30 January 2017

First available in Project Euclid: 18 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1487386904

**Digital Object Identifier**

doi:10.1214/17-ECP48

**Mathematical Reviews number (MathSciNet)**

MR3615667

**Zentralblatt MATH identifier**

1362.60034

**Subjects**

Primary: 60F99: None of the above, but in this section

**Keywords**

self-averaging recursion non-convergence shooting problem

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Cator, Eric; Don, Henk. Self-averaging sequences which fail to converge. Electron. Commun. Probab. 22 (2017), paper no. 16, 12 pp. doi:10.1214/17-ECP48. https://projecteuclid.org/euclid.ecp/1487386904