[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]

Let [math]\{X_k\}[/math], [math]1 \leq k \leq n[/math], be a sequence of independent random

variables, all with mean 0 and variance 1, and let [math]S_n[/math], [math]S_n^*[/math], and [math]A_n[/math] be their sum, standardized sum, and average, respectively. Verify directly that [math]S_n^* = S_n/\sqrt{n} = \sqrt{n} A_n[/math].