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Let $X$ and $Y$ be (possibly dependent) random variables. I want to show that if the variance of $X$ and $Y$ both go to zero, the variance of the product $XY$ also goes to zero. In other words, for any small $\epsilon$, we can make $Var(XY) < \epsilon$ by making $Var(X)$ and $Var(Y)$ small enough.

I have tried to show this by expanding into expectations but I can’t seem to get the result I want.

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It's not true. Try $X=Y= \delta^{-1/3}$ with probability $\delta$, $0$ otherwise.

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  • $\begingroup$ I'm not sure I understand this, but maybe I need additional conditions. I mean that the variance of X, Y goes to zero with the expected value unchanged, e.g. if X and Y are sample means then by increasing the sample size. $\endgroup$ – Flash Nov 21 '16 at 6:44
  • $\begingroup$ You can adjust these slightly so the expected values are $0$: take $X=Y=-\delta^{2/3}/(1-\delta)$ otherwise, instead of $0$. $\endgroup$ – Robert Israel Nov 21 '16 at 7:18
  • $\begingroup$ Hmm, what if we limit ourselves to discrete distributions and their support must stay the same as the variance goes to zero? $\endgroup$ – Flash Nov 21 '16 at 11:30

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