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I am getting confused with the following set of notes http://www.stat.umn.edu/geyer/8112/notes/metric.pdf.

On page 10, the author states Prokhorovs theorem as follows: Let $X_1,X_2,\dots$ be a sequence of random elements of a Polish space. If the sequence is tight, then every subsequence contains a weakly convergent subsubsequence. Conversely, if the sequence weakly converges, then it is tight.

In the next section, he mentions the subsequence principle which says that $X_n$ converges weakly to $X$ if and only if for every subsequence of $X_n$, we can extract a further subsequence which weakly converges to $X$.

However, does this then not mean that if $(X_n)$ are random elements in a Polish space, then tightness is equivalent to weak convergence?

This seems fishy to me. Could somebody please expand on this?

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  • $\begingroup$ Take two sequences that converge to different things. Interlace them to get one sequence. Tight but not weakly convergent. $\endgroup$ – GEdgar May 23 at 19:53
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    $\begingroup$ Notice that in Prokhorovs Theorem, the weak limit of the sub-subsequences need not be the same. In this sense tightness is more like boundedness in the Heine-Borel Theorem in $\mathbb{R} ^n$. $\endgroup$ – Jose27 May 23 at 19:54
  • $\begingroup$ Thanks @Jose27, that makes things far more clear! $\endgroup$ – Artur May 23 at 19:59
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    $\begingroup$ @Jose27 This seems like an answer rather than a comment. Why not post it as one? $\endgroup$ – Rhys Steele May 23 at 20:17

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