Sum of sequences of random variables converging in distribution

Question

Even if $\{X_n\}, \{Y_n\}$ converges in distribution to $X,Y$, I know that $X_n +Y_n$ need not converge to $X+Y$ . Can it happen that $\{X_n+Y_n\}$ doesn't converge in distribution to anything at all ?

Answer 1

Yes, it can happen. Take $X_1$ Bernoulli ($\frac12$) distributed: $\mathbb P(X_1=0)=\mathbb P(X_1=1)=\frac12$, and $X_{2k+1}=X_1$, $X_{2k}=1-X_1$ for $k\geq 1$, $Y_n=X_1$ for all $n\geq 1$. Then both sequences converge in distribution to Bernoulli ($\frac12$) distribution, and sum $X_n+Y_n$ does not converges since $X_{2k}+Y_{2k}=1$, $X_{2k+1}+Y_{2k+1}=2X_1$.