Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent.

Let $Z_n$ = $X_n$ + $Y_n$

I want to study the law convergence of the sample mean of $Z_n$. That is:

$$ \overline{Z_n} = \frac{\sum X_i + \sum Y_i}{n} $$

So, first of all there is an hint: I cannot use the Law of large numbers. Why is that?

Anyway, I am really out of tools to attack this problem. Does someone have a hint?

EDIT: Managed to prove that sample mean of Cauchy is still Cauchy! Still not sure about the solution.


Hint: what does $\sum X_i / n$ converge to? As you note, the sum $\sum Y_i / n$ is equal in distribution to a standard Cauchy.

  • $\begingroup$ I guess it converges to $0$? Not sure how to prove it though. Followup question: can I say something about $Z_n$ belonging to some $L^p?$ $\endgroup$ – qcc101 Jan 15 at 17:08
  • $\begingroup$ @qcc101, yes it converges to $0$. The $X_n$'s are very nice random variables (they're bounded, even), so what theorem can we use to show that their average converges to 0? You can't say anything about the $\overline{Z}_n$'s being in a nice $L^p$ space since Cauchy random variables aren't. $\endgroup$ – Marcus M Jan 15 at 17:10
  • $\begingroup$ I think I would try to use law of large numbers and then Slutsky to conclude, is that right? $\endgroup$ – qcc101 Jan 15 at 17:15
  • $\begingroup$ @qcc101, yep that'll do it! $\endgroup$ – Marcus M Jan 15 at 17:17
  • $\begingroup$ Nice, thank you for the follow-ups it was exactly how I wanted to solve it. $\endgroup$ – qcc101 Jan 15 at 17:19

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