## The Annals of Probability

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

Lawrence E. Myers

#### 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

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
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