# How to test that a distribution has infinite mean?

I observe a sample from a distribution that I expect to be the hitting time

$$\tau = \inf\{t>0| X(t)>a\}$$

where $X(t)$ is a Lévy process with $X(0)=0$ and $a$ is some constant. $X$ is not a Brownian motion and the experimental fit to the Lévy distribution is poor.

However, I do not need to know the exact formula for the law of $\tau$. For my needs I only need to know that the expectation of $\tau$ is infinite (as in the case of $\tau$ for a Brownian motion). Is it possible to formulate and test this as a statistical hypothesis?

• Perhaps you could take increasing subsets $A_n$ of your sample, test on $A_n$ whether the mean is greater than $a_n$, where $a_n \uparrow \infty$. If the likelihood of these hypotheses is "stable" as $n$ increases, maybe that would support the mean being infinite. – quasi Aug 26 '13 at 20:55
• +1 but I would suggest moving that to one of mathematical websites: MSE or MO. Perhaps, cross-validated also could be useful – Ilya Sep 25 '13 at 19:05
• @quasi, surely the results would vary based on what sequence $a_n$ you chose? – jwg Sep 1 '14 at 10:49
• For the Brownian motion case, I think an application of Dynkin's formula should do the job (en.wikipedia.org/wiki/Dynkin's_formula). If the gernator of your Levy process is of a handy form then maybe Dynkin helps in your case too. Maybe check out the Brownian case first (see also the section in Oksendal's book about this amazon.com/…). – Ric Sep 1 '14 at 15:50