Math 6010 • Linear Models

Let \(\mathbf {X}_n:=(X_{1,n},\ldots ,X_{d,n})\) be a random variable in \(\mathbf {R}^d\) for every \(n\geq 1\). Let \(\mathbf {Y}:=(Y_1,\ldots ,Y_d)\) be another random variable in \(\mathbf {R}^d\). Prove that \(\mathbf {X}_n\Rightarrow \mathbf {Y}\) if and only if the 1-dimensional random variable \(\sum _{i=1}^{d}\lambda _iX_{i,n}\) converges weakly to \(\sum _{i=1}^{d}\lambda _iY_i\) for every \(\lambda :=(\lambda _1,\ldots ,\lambda _d)\in \mathbf {R}^d\).