A hypothesis test for the domain of attraction of a random variable

TL;DR

This paper develops an asymptotic hypothesis test to determine whether a distribution's domain of attraction indicates an infinite first moment, especially relevant for complex simulations, using statistics inspired by jump detection in semimartingales.

Contribution

It introduces a novel hypothesis test based on Brownian bridge functionals to detect infinite moments in distributions, addressing an ill-posed problem in complex stochastic systems.

Findings

Test effectively distinguishes between finite and infinite first moments.

Asymptotic properties of the test are rigorously proven.

Applicable to complex numerical simulations with singular kernels.

Abstract

In this work we address the problem of detecting whether a sampled probability distribution of a random variable $V$ has infinite first moment. This issue is notably important when the sample results from complex numerical simulation methods. For example, such a situation occurs when one simulates stochastic particle systems with complex and singular McKean-Vlasov interaction kernels. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis $\mathbf{H_0}$ is: `$X=\sqrt{V}$ is in the domain of attraction of the Normal law' and the alternative hypothesis is $\mathbf{H_1}$: `$X$ is in the domain of attraction of a stable law with index smaller than 2'. Our key observation is that~$X$ cannot have a finite second moment when $\mathbf{H_0}$ is rejected (and therefore $\mathbf{H_1}$ is accepted). Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times. We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges.