Let $\{\Gamma(t), t \in \mathbb{R}\}$ be a Banach space $\mathscr{B}$-valued stochastic process. Let $P$ be the probability measure generated by $\Gamma(\cdot)$. Assume that $\Gamma(\cdot)$ is $P$-almost surely continuous with respect to the norm $\| \|$ of $\mathscr{B}$ and that there exists a positive nondecreasing function $\sigma(a), a > 0$, such that $P\{\|\Gamma(t + a) - \Gamma(t)\| \geq x\sigma(a)\} \leq K \exp(-\gamma x^\beta)$ with some $K, \gamma, \beta > 0$. Then, assuming also that $\sigma(\cdot)$ is a regularly varying function at zero, or at infinity, with a positive exponent, we prove large deviation results for increments like $\sup_{0\leq t\leq T-a}\sup_{0\leq s\leq a}\|\Gamma(t + s) - \Gamma(t)\|$, which we then use to establish moduli of continuity and large increment estimates for $\Gamma(\cdot)$. One of the many applications is to prove moduli of continuity estimates for $l^2$-valued Ornstein-Uhlenbeck processes.
Publié le : 1992-04-14
Classification:
$\mathscr{B}$-valued stochastic processes,
large deviations,
moduli of continuity,
large increments,
regular variation,
Gaussian processes,
$l^2$-valued Ornstein-Uhlenbeck processes,
60G07,
60F10,
60G17,
60G15,
60F15
@article{1176989816,
author = {Csaki, Endre and Csorgo, Miklos},
title = {Inequalities for Increments of Stochastic Processes and Moduli of Continuity},
journal = {Ann. Probab.},
volume = {20},
number = {4},
year = {1992},
pages = { 1031-1052},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989816}
}
Csaki, Endre; Csorgo, Miklos. Inequalities for Increments of Stochastic Processes and Moduli of Continuity. Ann. Probab., Tome 20 (1992) no. 4, pp. 1031-1052. http://gdmltest.u-ga.fr/item/1176989816/