[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]

Show that

[[math]] P(S_1 \ge 0,\ S_2 \ge 0,\ \ldots,\ S_{2m} \ge 0) = u_{2m}\ . [[/math]]

Hint: First explain why

[[math]] \begin{eqnarray*} &&P(S_1 \gt 0,\ S_2 \gt 0,\ \ldots,\ S_{2m} \gt 0) \\ && \;\;\;\;\;\;\;\;\;\;\;\;\; = {1\over 2}P(S_1 \ne 0,\ S_2 \ne 0,\ \ldots,\ S_{2m} \ne 0) \ . \end{eqnarray*} [[/math]]

Then use Exercise, together with the observation that if no equalization occurs in the first [math]2m[/math] outcomes, then the path goes through the point [math](1,1)[/math] and remains on or above the horizontal line [math]x = 1[/math].