My question is about basic probability.

We have two sequences of random variables, $ \{ X_n \}$ and $\{ Y_n \}$, such that each converge in distribution - i.e. there exist random variables $X$ and $Y$ such that:

$X_n \rightarrow X$ as $n \rightarrow \infty$ in law, and $Y_n \rightarrow Y$ as $n \rightarrow \infty$ in law.

What can we say about the sequence $X_n + Y_n$? I would like to say that it converges in law to $X +Y$ - but is this true?

Many apologies if this question has already been asked and answered. Thank you so much!!


p.s. Feel free to add some assumptions like independence of the $X_n$ and $Y_n$ in the process of answering this question.


In general, this is not true. If one of the two sequences converges in probability (which is the case of $X$ or $Y$ are constant), the result holds by Slutsky's theorem.

  • $\begingroup$ Thanks, especially for the link! :) $\endgroup$ – John Allan May 12 '14 at 0:03

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