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I understand that in general it is NOT true that $X_n \overset{D} \to X$ and $Y_n \overset{D} \to Y$ imply $X_nY_n \overset{D} \to XY$. However, if we take $Y_n = X_n$, we have that $X_nX_n \to X^2$, by the continuous mapping theorem. Hence, it seems that if the other sequence is the same as the first, we have the product of two random variables converging to their product limits. In general what are the rules for this to hold, aside from $Y$ being a constant? It appears that $Y$ being a constant is a sufficient but NOT necessary condition, as $X_nX_n \to X^2$. Thanks.

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  • $\begingroup$ If $Y_n=f_n(X_n)$ for each $n$, with $f_n$ measurable, one can guess the result shall hold. $\endgroup$ – Did Mar 29 '17 at 6:46
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    $\begingroup$ Or if $(X_n,Y_n) \stackrel{d}{\to} (X,Y)$. $\endgroup$ – saz Mar 29 '17 at 6:50

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