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

Question

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.

Answer 1

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.