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Newsgroups: sci.math
From: Yihong <yihon...@princeton.edu>
Date: Wed, 04 Nov 2009 19:59:45 EST
Local: Thurs, Nov 5 2009 12:59 am
Subject: variance and convergence in distribution
Dear all,
I have a question as follows: we know that convergence in distribution does not imply convergence of moments. In fact by Fatou's lemma and Skorohod's representation, if X_n -> X in distribution, we have liminf E[X_n^2] >= E[X^2] and strict inequality is possible, e.g., consider P{X_n = 0} = 1-1/n and P{X_n = n} = 1/n, X_n -> 0 in distribution. Now, how about variance? My question is that is it possible to construct an example such that X_n -> X in distribution and liminf var X_n < var X < Infinity? Clearly this could happen only when E[X_n^2] is unbounded. Thanks! You must Sign in before you can post messages.
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