  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.
