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

¹ CIMFAV – Instituto de Ingeniería Matemática, Universidad de Valparaíso. Chile
² INRIA Saclay, France

^* Corresponding author: hector.olivero@uv.cl

Received: 11 December 2023
Accepted: 5 July 2024

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.