[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]

Given a random walk [math]W[/math] of length [math]m[/math], with summands

[[math]] \{X_1, X_2, \ldots,X_m\}\ , [[/math]]

define the reversed random walk to be the walk [math]W^*[/math] with summands

[[math]] \{X_m, X_{m-1}, \ldots, X_1\}\ . [[/math]]

• Show that the [math]k[/math]th partial sum [math]S^*_k[/math] satisfies the equation
  [[math]] S^*_k = S_m - S_{n-k}\ , [[/math]]
  where [math]S_k[/math] is the [math]k[/math]th partial sum for the random walk [math]W[/math].
• Explain the geometric relationship between the graphs of a random walk and its reversal. (It is not in general true that one graph is obtained from the other by reflecting in a vertical line.)
• Use parts (a) and (b) to prove Theorem~.