Prove Part 2 of Theorem 3.6. That is, let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of \(d\) dimensional random vectors and let \(\mathbf{X}\) be another \(d\)-dimensional random vector. and prove that \(\mathbf{X}_{n} \xrightarrow{\text { a.c. }} \mathbf{X}\) as \(n ightarrow \infty\) if and only if \(X_{k n} \xrightarrow{\text { a.c. }} X_{k}\) as \(n ightarrow \infty\) for all \(k \in\{1, \ldots, d\}\).

Transcribed Image Text:

Theorem 3.6. Let {X} be a sequence of d-dimensional random vectors and let X be another d-dimensional random vector where X' = (X1,..., Xd) and X = (X1,n..., Xd,n) for all n N. 1. XnX as n if and only if Xk,n Xk as n for all k {1,..., d}. 2. Xn X as n if and only if Xk,nc., Xk as n for all k = {1,..., d}.