If $f = \{f_t\mid t \in T\}$ is a centered, second-order stochastic process with bounded sample paths, it is then known that $f$ satisfies the central limit theorem in the topology of uniform convergence if and only if the intrinsic metric $\rho^2_f$ (on $T$) induced by $f$ is totally bounded and the normalized sums are eventually uniformly $\rho^2_f$-equicontinuous. We show that a centered, second-order stochastic process satisfies the central limit theorem in the topology of uniform convergence if and only if it has bounded sample paths and there exists totally bounded pseudometric $\rho$ on $T$ so that the normalized sums are eventually uniformly $\rho$-equicontinuous.
Publié le : 1987-01-14
Classification:
Central limit theorem,
eventual boundedness,
eventual uniform equicontinuity,
eventual tightness,
stochastic processes,
60B12,
60F05
@article{1176992262,
author = {Andersen, N. T. and Dobric, V.},
title = {The Central Limit Theorem for Stochastic Processes},
journal = {Ann. Probab.},
volume = {15},
number = {4},
year = {1987},
pages = { 164-177},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992262}
}
Andersen, N. T.; Dobric, V. The Central Limit Theorem for Stochastic Processes. Ann. Probab., Tome 15 (1987) no. 4, pp. 164-177. http://gdmltest.u-ga.fr/item/1176992262/