[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]

For a random walk of length [math]2m[/math], define [math]\epsilon_k[/math] to equal 1 if

[math]S_k \gt 0[/math], or if [math]S_{k-1} = 1[/math] and [math]S_k = 0[/math]. Define [math]\epsilon_k[/math] to equal -1 in all other cases. Thus, [math]\epsilon_k[/math] gives the side of the [math]t[/math]-axis that the random walk is on during the time interval [math][k-1, k][/math]. A “law of large numbers” for the sequence [math]\{\epsilon_k\}[/math] would say that for any [math]\delta \gt 0[/math], we would have

[[math]] P\biggl(-\delta \lt {{\epsilon_1 + \epsilon_2 + \cdots + \epsilon_n}\over{n}} \lt \delta \biggr) \rightarrow 1 [[/math]]

as [math]n \rightarrow \infty[/math]. Even though the [math]\epsilon[/math]'s are not independent, the above assertion certainly appears reasonable. Using Theorem, show that if [math]-1 \le x \le 1[/math], then

[[math]] \lim_{n \rightarrow \infty} P\biggl({{\epsilon_1 + \epsilon_2 + \cdots + \epsilon_n}\over{n}} \lt x\biggr) = {2\over{\pi}} \arcsin\sqrt{{{1 + x}\over{2}}}\ . [[/math]]