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April, 1992 Inequalities for Increments of Stochastic Processes and Moduli of Continuity
Endre Csaki, Miklos Csorgo
Ann. Probab. 20(2): 1031-1052 (April, 1992). DOI: 10.1214/aop/1176989816


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.


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Endre Csaki. Miklos Csorgo. "Inequalities for Increments of Stochastic Processes and Moduli of Continuity." Ann. Probab. 20 (2) 1031 - 1052, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0757.60024
MathSciNet: MR1159584
Digital Object Identifier: 10.1214/aop/1176989816

Primary: 60G07
Secondary: 60F10 , 60F15 , 60G15 , 60G17

Keywords: $\mathscr{B}$-valued stochastic processes , $l^2$-valued Ornstein-Uhlenbeck processes , Gaussian processes , large deviations , large increments , moduli of continuity , regular variation

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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