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. We also discuss many numerical experiments which allow us to illustrate satisfying properties of the proposed test.

翻译：在这项工作中,我们解决了探测样本概率分布的随机值为$X的检测问题,这是无限的预期。当样本来自复杂的数字模拟方法的结果时,这个问题就显得特别重要。例如,当一个模拟随机粒子系统,同时使用复杂和单单的麦肯-弗拉索夫互动内核时,就会出现这种情况。如上所述,检测问题是不正确的。因此,我们提议和分析一个独立副本的随机变量~X$的无症状假设测试,该变量应该是属于稳定法律吸引的未知的直径域。无效假设$\mathbf{H_0}$是:`X$是正常法律的吸引力领域',而替代假设是$\mathbf{H_1}:`X$是小于2'的稳定法的吸引力领域。因此,我们的主要观察是,当我们从一个未知的直径直径的直径直径直的直径直的直径直直径直径直径直径直径直径直径直径直的直径直径直径直径直径直径直径直径直径直径直的直的直线路径直径直径直径直径直径直路径直的测量,我们从一个直的直地标直的直的直的直的直路路路路路路路路路路路。我们从一个直的直的直的测量测量测量测量测量,从一个直路由直路由直路由直路路路路路路路路路路路路,我们找到,从一个直到一个直的直的直的直路路路路路由直路由直路的直路的直至直至直的直至直的直的直的直的直的直至直至直的直的直的直的直路路路路路路路路路路路由的直路由的直至直的直的直的直的直的直的直路路路路的直路由的直路由的直路由的直路的直至直至直路的直路的直路由。