## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 1 (1974), 32-39.

### A Random Walk with Nearly Uniform $N$-Step Motion

#### Abstract

Let $N$ be a strictly positive integer. Motivated by a certain discrete evasion game, we search for a $\{0, 1\}$-valued discrete time stochastic process whose conditional-on-the-past distributions of the sum of the next $N$ terms are as close to uniform as possible. A process is found for which none of the sums ever occurs with conditional probability more than $2e/(N + 1)$. The process is characterized by invariance under interchange of 0 and 1, and its waiting times between successive transitions, which are independently, identically, and uniformly distributed over $\{1,2, \cdots, N + 1\}$.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 1 (1974), 32-39.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996749

**Digital Object Identifier**

doi:10.1214/aop/1176996749

**Mathematical Reviews number (MathSciNet)**

MR368137

**Zentralblatt MATH identifier**

0277.60057

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G17: Sample path properties

Secondary: 60K99: None of the above, but in this section 60C05: Combinatorial probability

**Keywords**

Random walk $N$-step motion $m$-dependent process

#### Citation

Myers, Lawrence E. A Random Walk with Nearly Uniform $N$-Step Motion. Ann. Probab. 2 (1974), no. 1, 32--39. doi:10.1214/aop/1176996749. https://projecteuclid.org/euclid.aop/1176996749