A CLT for empirical processes involving time-dependent data

Abstract

For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\mathrm{Pr}(X_t\le y):t\in E,y\in \mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in \mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.

Published in at http://dx.doi.org/10.1214/11-AOP711 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Weak Convergence of the Scaled Median of Independent Brownian Motions

Cited by in corpus (2)

• Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions
• A CLT for weighted time-dependent uniform empirical processes