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If you have a sequence of random variables $(W_n)$ which converges in distribution to N, and another sequence of random variables $(X_n)$ which converges in distribution to $B$:

i) Will $(W_nX_n)$ converge in distribution to N multiplied by B?

ii) Similarly, will $(W_n + X_n)$ converge in distribution to $N + B$?

I know this is probably a very basic question, but I can't find proofs/discussions properties of convergence in my text books :(

If it's not too complicated, could you let me know whether i and ii happen to be true (or false) for the other basic types of convergence (almost-sure, L^2, pointwise) ?

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  • $\begingroup$ Yes, that's what I meant to say. Ooops. $\endgroup$ – Wanda1989 Dec 6 '13 at 19:33
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If $X_n\to X$ in distribution and $Y_n\to Y$ in distribution, we may not have $X_nY_n\to XY$ in distribution. For example, assume that $X$ takes the values $-1$ and $1$ with probability $1/2$, $X_n=X$ and $Y_n=X$ for $n$ even and $Y_n=-X$ for $n$ odd. We have that $X_n\to X$ in distribution and $Y_n\to X$ in distribution but $X_nY_n$ is equal to $1$ for $n$ even and $-1$ for $n$ odd.

A similar counter-example works for sums.

Note that the results hold when the convergence in distribution is replace by the convergence in probability or almost sure.

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