This work tackles the problem of detecting whether a sampled probability distribution has infinite expected values. This issue is especially important when the samples are the result of complex numerical simulation techniques. For example, such situations arise when simulating stochastic particle systems with complex and singular McKean-Vlasov interaction kernels. As mentioned above, the detection problem is inadequate. Therefore, we propose and analyze an asymptotic hypothesis test for independent copies of a given random variable ~X$, which are considered to belong to the unknown gravitational region of the stability law. The null hypothesis $\mathbf{H_0}$ is: `$X$ is in the region of ordinary law attraction’, the alternative is $\mathbf{H_1}$: `$X$ is in the region of attraction A stable law with some exponent less than 2′. Our key observation is that if $\mathbf{H_0}$ is rejected (and thus $\mathbf{H_1}$ is accepted), ~$X$ can have a finite second instant. It means you can’t.
Surprisingly, it turns out to be useful to derive tests from the statistics of random processes. More precisely, our hypothesis test is based on statistical methodology-inspired statistics to determine whether a semimartingale jumps from her one-pass observations in discrete time.
We justify the test by proving the asymptotic property of the discrete-time functional of the Brownian bridge. We also describe a number of numerical experiments that allow us to explain the satisfactory properties of the proposed test.
