In this work we address the problem of detecting wether a sampled probability distribution has infinite expectation. 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~$X$ which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis $\mathbf{H_0}$ is: `$X$ 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 statistical 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.

翻译：本文研究了采样概率分布是否具有无穷期望的检测问题。当样本源自复杂数值模拟方法时，此问题尤为重要。例如，在模拟具有奇异麦克恩-弗拉索夫相互作用核的随机粒子系统时便会遇到此类情形。该检测问题本质上是病态的，因此我们针对服从未知稳定分布吸引域的独立同分布随机变量$X$，提出并分析了一种渐近假设检验方法。原假设$\mathbf{H_0}$为：“$X$服从正态分布的吸引域”，备择假设$\mathbf{H_1}$为：“$X$服从指数小于2的稳定分布吸引域”。关键发现是：当拒绝$\mathbf{H_0}$（即接受$\mathbf{H_1}$）时，$X$必不具有有限二阶矩。令人惊奇的是，我们通过随机过程统计学方法推导出检验统计量。具体而言，该假设检验基于受统计方法论启发的统计量——通过离散时间单一路径观测判断半鞅是否具有跳跃性。通过证明布朗桥的离散时间泛函渐近性质，我们为所提出的检验方法提供了理论依据。