Random variables $x_i$ with $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$

Question

I am looking for a sequence $(x_n)_{n\in\mathbb N}$ of random variables such that the sequence hasn't any expected value and $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$.

I thought about using a Cauchy distribution $f(x)=\frac1\pi\cdot\frac{1}{1+x^2}$ since there doesn't exist an expected value.

But how exactly? Can you just define $x_n:=\frac12 y_n$ with $y_n$ Cauchy-distributed? And how can you show the limit above?

Answer 1

Choose any non integrable random variable $x_1$ and let $x_{i+1}=-\frac1{2^i}x_1$ for every $i\geqslant1$. Then $\frac1k\sum\limits_{i=1}^kx_i=\frac2{k2^k}x_1\to0$ almost surely although no $\frac1k\sum\limits_{i=1}^kx_i$ is integrable.