[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

A lead change in a random walk occurs at time

[math]2k[/math] if [math]S_{2k-1}[/math] and [math]S_{2k+1}[/math] are of opposite sign.

• Give a rigorous argument which proves that among all walks of length [math]2m[/math] that have an equalization at time [math]2k[/math], exactly half have a lead change at time [math]2k[/math].
• Deduce that the total number of lead changes among all walks of length [math]2m[/math] equals
  [[math]] {1\over 2}(g_{2m} - u_{2m})\ . [[/math]]
• Find an asymptotic expression for the average number of lead changes in a random walk of length [math]2m[/math].