## The Annals of Probability

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

### Large Deviations for Processes with Independent Increments

James Lynch and Jayaram Sethuraman

#### 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