To establish the relation between weak convergence and characteristic functions, the book I'm studying suggest without proof the following theorem:

For any tight sequence of probability measures, there exists a weakly convergent subsequence.

For the proof the author refers to "a rather difficult theorem", the Helly selection principle.

Can anyone suggest me a more simple and direct proof, at least for unidimensional case?

  • $\begingroup$ the skeleton of the proof appear obvious and common... associate a sequence of distribution functions to probability measures.. use the fact that distributions functions are bounded and the Bolzano-Weierstrass to construct a pseudo-distribution-function over rationals, extend this to reals, then show this function is a true distribution function using the thightness... but after that we need to show that distributions functions converges to the constructed distribution function in any continuity point... and this last part appear difficult $\endgroup$
    – unlikely
    Jul 8, 2011 at 12:47

1 Answer 1


I would refer you to to Billingsley's book Convergence of Probability Measures. This is an excellent introduction to the area of convergence in distribution.


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