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Guest
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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!
YH |
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