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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!

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    "...such that $X_k \to X$" in what sense? http://en.wikipedia.org/wiki/Convergence_of_random_variables2012-04-22

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It is true that if $X_k$ is multivariate normal random vector and $X_k$ converge in distribution to $X$, then $X$ is also a multivariate normal random vector.

According to Levy continuity theorem convergence in distribution implies point-wise convergence of characteristic functions. Characteristic function $\phi_{X_k}(t) = \exp(- t.A_k.t + B_k.t)$, being characteristic function of a Gaussian random vector. Then

$ \lim_{k \to \infty} \phi_{X_k}(t) = \lim_{k \to \infty} \exp(- t.A_k.t + B_k.t) = \exp(- t.A.t + B.t) $ where $A = \lim_{k\to \infty} A_k$ and $B = \lim_{k \to \infty} B_k$.

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    @MadSi: Like you, I don't understand why the limits $\lim_{k \rightarrow \infty} A_k$ and $\lim_{k \rightarrow \infty} B_k$ must exist. Have you found a satisfactory explanation by now?2014-02-28