Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that $$ 0< c_0\leq Y_n, Y \leq c_1. $$ Assume further that $X_nY_n$ convergence in distribution to $Z=\tilde X Y$. Question: can we get $\tilde X$ has the same distribution with $X$?
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No. You can ignore the issues of convergence, and set $X_n=X$ and $Y_n=Y$ for all $n$. Then you get the question: if $XY$ and $\tilde{X}Y$ have the same distribution, does it follow that $X$ and $\tilde{X}$ have the same distribution?
Take $(X, \tilde{X}, Y)$ to be $(0,1,1)$ with probability $1/2$ and $(1/2, 0, 2)$ with probability $1/2$. Then $XY$ and $\tilde{X}Y$ both take values $0$ and $1$ with probability $1/2$ each. But $X$ and $\tilde{X}$ don't have the same distribution.
