B0469
Title: The central limit theorem for empirical processes in $L^p(\mu)$ with applications
Authors: Javier Carcamo - Universidad Autonoma de Madrid (Spain) [presenting]
Abstract: The empirical process associated with a random variable $X$ with distribution function $F$ is shown to converge in distribution to the $F$-Brownian bridge in $L^p(\mu)$ if and only if it is pregaussian. The result is valid for all $1\le p <\infty$ and all $\sigma$-finite Borel measure $\mu$. Additionally, we determine several easy to check requirements equivalent to the empirical process being pregaussian. In this way, we determine necessary and sufficient conditions to obtain a Donsker theorem for the empirical process in $L^p(\mu)$. When $\mu$ is the Lebesgue measure, the convergence of the empirical process amounts to the membership of $X$ to the Lorentz space $\mathcal{L}^{2/p,1}$. A characterization of the finiteness of the second strong moment of the empirical process in $L^p(\mu)$ is also provided. As an application, we obtain the asymptotic distribution of the plug-in estimators of the $L^p$-distance between distribution functions. The case $p=1$ corresponds to the so-called $L^1$-Wasserstein metric.
