Supremum of the sample mean without first moment assumption.

Question

I have a question concerning the supremum of the sample mean, any help would be greatly appreciated !

Let $(X_n)_{n \geq 1}$ be a sequence of non-negative independent and identically distributed random variables. We only assume that $X_1 < \infty$ a.s. In particular we do $\textbf{not}$ assume $E(X_1) < \infty$. Is it true that the random variable $Z$ definied by :

\begin{equation*} Z = \sup_{n \geq 1} \frac{1}{n}\sum_{i=1}^n X_i \end{equation*} is finite almost surely ?

Answer 1

No, in fact if $E[X_1]=\infty$ then the sample mean will converge to infinity almost surely. In other words, SLLN holds in the case where the first moment is well-defined but infinite. Interesting things happen when the first moment is outright undefined. For instance, it is interesting to run a program like this:

n=1e4;
x=rand(1,n)-1/2;
y=tan(pi*x);
m=cumsum(y)./(1:n)
plot(m)

This samples n elements from a standard Cauchy distribution and then plots the resulting sample means along the way. What will be seen are occasional large jumps which do not dampen out as more samples are taken. (This program is in Matlab syntax but it would be easy enough to translate to anything else.)