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

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.
• 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 $L^p?$ – qcc101 Jan 15 at 17:08
• @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. – Marcus M Jan 15 at 17:10