Verify that the Khintchine SLLN [Theorem 6.3.13] holds when the random variables Xi have a common distribution F, define on R p , for some p ≥ 1. Hence, or otherwise, show that the case of...

Verify that the Khintchine SLLN [Theorem 6.3.13] holds when the random variables Xi have a common distribution F, define on R^p, for some p ≥ 1. Hence, or otherwise, show that the case of vector-valued or matrix-valued i.i.d. random variables is covered by this extension.

Let (Xi, Yi), i ≥ 1, be i.i.d. random variables with a bivariate distribution function having finit moments up to the second order. Let r_n
be the sample correlation coefficien (for a sample of size n) and ρ be the population counterpart. Show that