I have a question regarding sequences of multivariate normal vectors:
Let $ (X_k)_{k \geq 1} $ be a sequence of random vectors of fixed length $ n \geq 1 $ with multivariate normal distribution on a probability space $ (\Omega, \mathcal{F}, P) $ , such that $ X_k \rightarrow X, \ P-a.s.,$ as $ k \rightarrow + \infty $.
Is is true then that $ X $ has again multivariate normal distribution? And would you happen to know where I can find a proof of this?
Thanks a lot for your help & have a nice week!