SCHWEITZER 2009 Problem 12

Let $Z_1, Z_2\dots, Z_n$ be $d$ -dimensional independent random (column) vectors with standard normal distribution, $n - 1 > d$ . Furthermore let $$\overline Z = \frac {1}{n}\sum_{i = 1}^n Z_i, S_n = \frac {1}{n - 1}\sum_{i = 1}^n(Z_i - \overline Z)(Z_i - \overline Z)^\top$$ be the sample mean and corrected empirical covariance matrix. Consider the standardized samples $Y_i = S_n^{ - 1/2}(Z_i - \overline Z)$ , $i = 1,2,\dots,n$ . Show that $$\frac {E|Y_1 - Y_2|}{E|Z_1 - Z_2|} > 1,$$ and that the ratio does not depend on $d$ , only on $n$ .