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

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 H0 is: ‘√V is in the domain of attraction of the Normal law’ and the alternative hypothesis is H1: ‘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 H0 is rejected (and therefore H1 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.