Sample Variance in Free Probability

Wiktor Ejsmont, Franz Lehner

Let $X_1,X_2,\dots ,X_n$ denote i.i.d.~centered standard normal random variables, then the law of the sample variance $Q_n={\sum }_{i=1}^n(X_i-\bar{X}{)}^2$ is the ${\chi }^2$-distribution w…