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When examining a sequence of stochastic processes $(\textbf{X}_n)$, $n\geq1$ convergence of marginals, i.e. $\mathbf{X}_n(t)\to\mathbf{X}(t)$ (in distribution) is often not too hard to establish for any fixed $t\in [0,\infty)$.

On the other hand convergence of finite dimensional distributions (FDD-convergence) is rather hard. Are there any "standard" techniques, or at least nice tricks one can try? I don't need an exhaustive list, maybe just ideas/starting points/references/example proofs,...

My impressions so far: If one has some kind of Markov property one tries to prove those results by induction. I also saw some argument with test functions somewhere, but I cannot recall where and how it worked.

Thanks!

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A technique is to use the Cramer Wold theorem. This reduces the proof of the convergence in distribution of the random vector $\left(\mathbf X_n(t_i)\right)_{i=1}^d$ to that of linear (deterministic) combinations of this vector. Thus we work with real valued random variables instead of vectors.

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  • $\begingroup$ Thanks for the input! A very good point, since actually I have never seen this argument in the context of stochastic processes so far. Although imho it is a standard argument when proving that some random vector is e.g. normally distributed. $\endgroup$
    – Mr. Barrrington
    Jun 1, 2015 at 12:32

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