[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 random variables, all with

mean [math]\mu[/math] and variance [math]\sigma^2[/math], and [math]Y_k = X_k^*[/math] be their standardized versions. Let [math]S_n[/math] and [math]T_n[/math] be the sum of the [math]X_k[/math] and [math]Y_k[/math], and [math]S_n^*[/math] and [math]T_n^*[/math] their standardized version. Show that [math]S_n^* = T_n^* = T_n/\sqrt{n}[/math].