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 $\{1,2, \cdots, N + 1\}$.