Non-convergence of Cauchy Random Variables

Question

Suppose $X_1,X_2,\ldots$ is a sequence of Cauchy random variables with density $$f(x)=\frac{1}{\pi(1+x^2)}, \hspace{3mm}x\in \mathbb{R}$$ and let $S_n=X_1+\ldots+X_n$.

It's easy to show that $\frac{S_n}{n}$ converges in distribution using characteristic functions (in fact, $S_n/n$ has the same distribution as $X$ for every $n$).

On the other hand, $S_n/n$ does not converge in probability. Does this follow from the fact that these random variables are not integrable (if they were, we could apply WLLN)? Or do we need to use the definition of convergence in probability directly?

Answer 1

Define $Z_n:=\frac{S_{2n}}{2n}-\frac{S_n}n$. Then $$A_n:=\frac{S_{2n}}{2n}+Z_n$$ is independent of $B_n:=\frac{S_n}n$. We assume that $B_n\to Y$ in probability. Then $A_n-B_n\to 0$ in probability, and denoting $\varphi$ the characteristic function of $Y$, we should have $\varphi(t)\varphi(-t)=1$ for each real number $t$. Since $Y$ is necessarily a Cauchy random variable, we get a contradiction.

There is a necessary and sufficient condition for the weak law of large numbers for independent sequences, namely, if $(X_n,n\geqslant 1)$ is an i.i.d. sequence and $\varphi$ is the characteristic function of $X_1$, the NSC is $\varphi'(0)$ exists.

Here, we can't use the law of large numbers because we know in advance that the limit would not be constant, but Cauchy.