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Suppose that $|X_n - Y_n|$ converges in probability to 0, and that $X_n$ converges in distribution to X. Show that $Y_n$ converges in distribution to X.

Thanks in advance.

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    $\begingroup$ thanks for editing your post, can you include what you tried as well? $\endgroup$ Apr 11 at 4:53
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Goal: show $P(Y_n \le t) \to P(X \le t)$ [for $t$ at which $F_X(t):=P(X \le t)$ is continuous].

Hints:

  • $P(Y_n \le t) = P(Y_n \le t, |X_n - Y_n| > \epsilon) + P(Y_n \le t, |X_n - Y_n| \le \epsilon)$.
  • $P(Y_n \le t, |X_n - Y_n| > \epsilon) \le P(|X_n - Y_n| > \epsilon)$. What does the right-hand side converge to?
  • $P(Y_n \le t, |X_n - Y_n| \le \epsilon) \le P(X_n \le t + \epsilon)$. What does the right-hand side converge to?
  • Use the assumption that $F_X(t)$ is continuous at $t$.
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