The Annals of Probability

Characterizations of Some Stochastic Processes

Y. H. Wang

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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

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

First available in Project Euclid: 19 April 2007

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Primary: 62E10: Characterization and structure theory
Secondary: 60H05: Stochastic integrals 60G15: Gaussian processes 60K99: None of the above, but in this section

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


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

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