Leading All The Way

Abstract

Xavier and Yushi run a ``random race'' as follows. A continuous probability distribution $\mu$ on the real line is chosen. The runners begin at zero. At time $i$ Xavier draws ${\mathbf{X}}_i$ from $\mu$ and advances that distance, while Yushi advances by an independent drawing ${\mathbf{Y}}_i$. After $n$ such moves, Xavier wins a valuable prize provided he not only wins the race but leads after every step; that is, ${\sum }_{i=1}^k{\mathbf{X}}_i>{\sum }_{i=1}^k{\mathbf{Y}}_i$ for all $k=1,2,\dots ,n$. What distribution is best for Xavier, and what then is his probability of getting the prize?

9 pages