Consider two sequences of random variables $X_n, Y_n$ and suppose $X_n\to X$ in distribution. Does the following hold: $\lim_{n\to\infty} E[|X_n-Y_n|]=0 \implies Y_n\to X$ in distribution?

  • $\begingroup$ What do you mean by convergence in distribution? maybe you meant in probability? $\endgroup$ – Mercy King Jun 22 '12 at 12:11
  • $\begingroup$ @Mercy en.wikipedia.org/wiki/… $\endgroup$ – user940 Jun 22 '12 at 12:26

Let $Z_n = Y_n - X_n$. From the assumption, we have $Y_n = X_n + Z_n$, where $X_n \to X$ in distribution and $Z_n \to 0$ in $L^1$.

Then $Z_n \to 0$ in probability, and therefore by Slutsky's theorem we have the desired conclusion.


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