## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 6 (1975), 1038-1045.

### Characterizations of Some Stochastic Processes

#### Abstract

In this paper, we extend known characterizations of normal and other distributions. Let $X(t), t \geqq 0$, be a continuous (in probability) homogeneous process, with independent increments. Let $g(s, t)$ and $h(s)$ be continuous functions on $\lbrack a, b \rbrack^2$ and $\lbrack a, b \rbrack, 0 \leqq a < b < \infty$. Define stochastic integrals $Y_1 = \int^b_a h(s)X(ds)$ and $Y_2 = \int^b_a \int^b_a g(s, t)X(ds)X(dt)$. It is known that $Y_1$ exists in the sense of convergence in probability. It is shown here that $Y_2$ exists at least in the sense of convergence in $L_2$, under the additional assumption that $X$ is of second-order. The main results of this paper are to obtain, under additional appropriate assumptions on $g$ and $h$, characterizations of a class of stochastic processes which include the Brownian motion, Poisson, negative binomial and gamma processes, based on the linear regression of $Y_2$ on $Y_1$.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 6 (1975), 1038-1045.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996231

**Digital Object Identifier**

doi:10.1214/aop/1176996231

**Mathematical Reviews number (MathSciNet)**

MR386009

**Zentralblatt MATH identifier**

0336.62008

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E10: Characterization and structure theory

Secondary: 60H05: Stochastic integrals 60G15: Gaussian processes 60K99: None of the above, but in this section

**Keywords**

Characterization stochastic integrals stochastic processes homogeneous independent increments convergence in probability convergence in $L_2$ Brownian motion Poisson process gamma process negative binomial process linear regression

#### Citation

Wang, Y. H. Characterizations of Some Stochastic Processes. Ann. Probab. 3 (1975), no. 6, 1038--1045. doi:10.1214/aop/1176996231. https://projecteuclid.org/euclid.aop/1176996231