The Annals of Probability

Large Deviations for Processes with Independent Increments

James Lynch and Jayaram Sethuraman

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Abstract

Let $\mathscr{X}$ be a topological space and $\mathscr{F}$ denote the Borel $\sigma$-field in $\mathscr{X}$. A family of probability measures $\{P_\lambda\}$ is said to obey the large deviation principle (LDP) with rate function $I(\cdot)$ if $P_\lambda(A)$ can be suitably approximated by $\exp\{-\lambda \inf_{x\in A}I(x)\}$ for appropriate sets $A$ in $\mathscr{F}$. Here the LDP is studied for probability measures induced by stochastic processes with stationary and independent increments which have no Gaussian component. It is assumed that the moment generating function of the increments exists and thus the sample paths of such stochastic processes lie in the space of functions of bounded variation. The LDP for such processes is obtained under the weak$^\ast$-topology. This covers a case which was ruled out in the earlier work of Varadhan (1966). As applications, the large deviation principle for the Poisson, Gamma and Dirichlet processes are obtained.

Article information

Source
Ann. Probab., Volume 15, Number 2 (1987), 610-627.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992161

Digital Object Identifier
doi:10.1214/aop/1176992161

Mathematical Reviews number (MathSciNet)
MR885133

Zentralblatt MATH identifier
0624.60045

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations
Secondary: 60J30 60E07: Infinitely divisible distributions; stable distributions

Keywords
Large deviations rate function stationary and independent increments Dirichlet process

Citation

Lynch, James; Sethuraman, Jayaram. Large Deviations for Processes with Independent Increments. Ann. Probab. 15 (1987), no. 2, 610--627. doi:10.1214/aop/1176992161. https://projecteuclid.org/euclid.aop/1176992161


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