论文标题
一个假设检验，用于吸引随机变量的领域
A hypothesis test for the domain of attraction of a random variable
论文作者
论文摘要
在这项工作中，我们解决了检测随机变量$ v $的采样概率分布是否具有无限的第一刻的问题。当样本来自复杂的数值模拟方法时，此问题非常重要。例如，当一个人模拟具有复杂和奇异的McKean-Vlasov相互作用内核时，就会发生这种情况。如前所述，检测问题不足。因此，我们提出和分析了给定随机变量的独立副本的渐近假说检验，该副本应属于稳定定律的吸引力的未知领域。零假设$ \ mathbf {h_0} $是：`$ x = \ sqrt {v} $处于普通法律吸引力的领域，而另一种假设为$ \ mathbf {h_1} $：我们的主要观察结果是，当拒绝$ \ mathbf {h_0} $被拒绝时，〜$ x $不可能有有限的第二刻（因此，接受$ \ mathbf {h_1} $）。 令人惊讶的是，我们发现从随机过程的统计数据中得出测试很有用。更确切地说，我们的假设检验基于一个统计量，该统计数的灵感来自方法论，以确定半障碍在离散时间对单个路径的观察中是否跳跃。 我们通过证明布朗桥离散时间功能的渐近特性来证明我们的测试是合理的。
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.
