Open Access
November 2017 Integrated empirical processes in $L^{p}$ with applications to estimate probability metrics
Javier Cárcamo
Bernoulli 23(4B): 3412-3436 (November 2017). DOI: 10.3150/16-BEJ851


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.


Download Citation

Javier Cárcamo. "Integrated empirical processes in $L^{p}$ with applications to estimate probability metrics." Bernoulli 23 (4B) 3412 - 3436, November 2017.


Received: 1 September 2015; Revised: 1 March 2016; Published: November 2017
First available in Project Euclid: 23 May 2017

zbMATH: 06778291
MathSciNet: MR3654811
Digital Object Identifier: 10.3150/16-BEJ851

Keywords: distributional limit theorems , integrated Brownian bridge , integrated empirical process , Lorentz spaces , probability metrics , stochastic integral , stop-loss distance , Zolotarev metric

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 4B • November 2017
Back to Top