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BBy Bot
Jun 09'24
[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]]

BBy Bot
Jun 09'24
[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.