Talk:Prokhorov's theorem

Correct theorem statement ?
The first statement of the theorem seems to be wrong. I recall that for a general metric space $$S$$ (even separable), tightness of a family of measures implies pre-compactness in $$\mathcal{P}(S)$$, but not the other way round (although I don't have an example at hand). Also, since $$\mathcal{P}(S)$$ is metrizable, the concepts of sequential compactness and compactness are the same in $$\mathcal{P}(S)$$.

Compare with the Theorem 14.3 in Kallenberg's "Foundations of Modern Probability": For any sequence of random elements in a metric space $$S$$, tightness implies relative compactness in distribution, and the two conditions are equivalent when $$S$$ is separable and complete. — Preceding unsigned comment added by 79.180.115.167 (talk) 08:46, 28 April 2014 (UTC)