I couldn't find the answer in literature so any idea would be helpful.
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$\begingroup$ This should actually have been posted to math.stackexchange.com instead as it is not research level. I've voted to migrate it there. $\endgroup$– Nate EldredgeCommented Apr 25, 2020 at 17:48
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$\begingroup$ (I forgot which site I was on - I wouldn't have answered the question if I had noticed it was on MathOverflow.) $\endgroup$– Nate EldredgeCommented Apr 25, 2020 at 17:49
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1 Answer
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Not in general. The standard counterexample is to let $U \sim U(0,1)$, $X_n = n 1_{\{U \le 1/n\}}$, and $X=0$.
There are several basic theorems giving sufficient conditions for this to hold, e.g. monotone convergence, dominated convergence, uniform integrability. They can be found in any graduate-level probability textbook.
answered Apr 25, 2020 at 15:30
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$\begingroup$ Thank you for the notice. What I am working on concretely are variables in form of
e^tXiY, where t is a real number, and Xi, Yj are independent sequences of random variables having binomial distribution. How to figure out would the condition hold in this case? $\endgroup$ Commented Apr 25, 2020 at 16:11 -
$\begingroup$ Maybe look in the literature for results on sufficient conditions, as the answer suggests... $\endgroup$– R.P.Commented Apr 25, 2020 at 17:01