if $X_n$ converges to $X$ and $Y_n$ converges to $Y$ in distribution, what about $X_n + Y_n $ would that converge to $X+Y$ in distribution ? any ideas how i could prove or disprove this
Linearity of convergence in distribution of random variables
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probability-theory
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3It should be true if they are independent, see http://math.stackexchange.com/questions/591708/sum-of-two-independent-random-variables-converges-in-distribution – 2015-11-02