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Suppose a sequence of random variables $X_n$ converges in probability to a random variable $X$. Is it always the case that $\mathrm{Var}(X)=0$?

Silverfish
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Henk
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  • Welcome to the site! Is this for homework or something similar? In that case, see http://stats.stackexchange.com/questions/tagged/self-study and follow the guidelines. – Juho Kokkala Oct 14 '15 at 09:31
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    This is probably a duplicate: http://stats.stackexchange.com/questions/74047/why-dont-asymptotically-consistent-estimators-have-zero-variance-at-infinity or better http://stats.stackexchange.com/questions/53358/counterexample-for-the-sufficient-condition-required-for-consistency/83953#83953 – kjetil b halvorsen Oct 14 '15 at 09:34
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    I disagree this is a dupe as it asks something more fundamental; the misconception here is different. The proposed duplicate thread gives a particular counterexample, in the context of estimators (which the OP isn't specifically asking about) but the flaw in the reasoning in the OP's statement actually deserves addressing here. Other counterexamples could be found, without introducing the added complication of them being estimators. I was going to add @RUser's but they beat me to it - it answers this question but, not mentioning estimators, not the proposed dupe, suggesting it isn't duplicate – Silverfish Oct 14 '15 at 11:28
  • This question sounds like it really wants to be something else of greater import and interest, such as "if $X_n$ converges to a constant, is it always the case that $\text{Var}(X_n)$ converges to zero?" – whuber Oct 14 '15 at 15:12

1 Answers1

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Hint. Think about the simplest example: let $X_n=X$ for every $n$, with $X$ having a non zero variance. Does $X_n$ converge in probability to $X$ ? Then what can be said about the variance of $X$ ?

RUser4512
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