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 $Y_1$.