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Context: I'm trying to understand a proof of Wigner's theorem in An Introduction to Random Matrices by Anderson et al. and I'm having trouble proving an intermediate fact in the proof that the book glosses over.

The general form of my question is as follows. Let $\{X_n\}$ and $\{Y_n\}$ be sequences of non-negative random variables such that $Y_n\rightarrow 0$ in probability as $n\rightarrow\infty$. I would like to show that $$\lim_{n\rightarrow\infty} P(X_n + Y_n > \epsilon) = \lim_{n\rightarrow\infty} P(X_n > \epsilon)$$ for any $\epsilon > 0$. (Note that $X_n$ and $Y_n$ are not assumed to be independent.) This is actually a slightly more general result than what I need, but it seems like it should be true.

I've tried to write the probability on the LHS as an integral so I can use a dominated convergence argument, but I'm having trouble writing the joint law down in a way which is useful for getting a handle on the integral.

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  • $\begingroup$ What does it mean for a random variable to tend to zero in probability? $\endgroup$
    – snulty
    Commented Jun 19, 2018 at 21:25
  • $\begingroup$ You should add what you have done so far in addition to your question. That way, people can help you figure out where you went wrong. $\endgroup$
    – an4s
    Commented Jun 19, 2018 at 21:33
  • $\begingroup$ The limit converges for all $\epsilon > 0$ possibly except at countably many points. $\endgroup$ Commented Jun 20, 2018 at 6:19

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After some more thought, it turns out this statement is false. We can take $X_n = 1$, $Y_n = 1/n$, both with probability $1$, and $\epsilon = 1$. Then $P(X_n + Y_n > 1) = 1$ for all $n$ but $P(X_n > 1) = 0$ for all $n$, so the two limits do not match.

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