# Relationship between tightness and weak convergence.

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?

• Take two sequences that converge to different things. Interlace them to get one sequence. Tight but not weakly convergent. – GEdgar May 23 at 19:53
• 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 $\mathbb{R} ^n$. – Jose27 May 23 at 19:54
• Thanks @Jose27, that makes things far more clear! – Artur May 23 at 19:59
• @Jose27 This seems like an answer rather than a comment. Why not post it as one? – Rhys Steele May 23 at 20:17