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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?

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    $\begingroup$ As a general thing, it might be rather tricky; how might one distinguish (as an example) a random variable which has a Pareto distribution with $\alpha=1$ from one with $\alpha=1+\varepsilon$? $\endgroup$
    – Glen_b
    Aug 25, 2013 at 11:01
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    $\begingroup$ See stats.stackexchange.com/questions/2504/test-for-finite-variance/… for the same question about testing for a finite variance. $\endgroup$
    – whuber ♦
    Aug 24, 2019 at 16:47

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