Integrated empirical processes in $L^{p}$ with applications to estimate probability metrics

Abstract

We discuss the convergence in distribution of the $r$-fold (reverse) integrated empirical process in the space $L^{p}$, for $1\le p\le\infty$. In the case $1\le p<\infty$, we find the necessary and sufficient condition on a positive random variable $X$ so that this process converges weakly in $L^{p}$. This condition defines a Lorentz space and can be also characterized in terms of several integrability conditions related to the process $\{(X-t)^{r}_{+}:t\ge0\}$. For $p=\infty$, we obtain an integrability requirement on $X$ guaranteeing the convergence of the integrated empirical process. In particular, these results imply a limit theorem for the stop-loss distance between the empirical and the true distribution. As an application, we derive the asymptotic distribution of an estimator of the Zolotarev distance between two probability distributions. The connections of the involved processes with equilibrium distributions and stochastic integrals with respect to the Brownian bridge are also briefly explained.