A Continuum of Collision Process Limit Theorems

Abstract

Let $\{x_i(t): i = \cdots -1, 0, 1, \cdots\}$ be a collection of one-dimensional symmetric stable processes of order $\gamma \in (0, 1\rbrack$ with the property that the starting positions $\cdots < x_{-1}(0) < x_0(0) = 0 < x_1(0) < \cdots$ form a Poisson system with rate one. By generalizing the order preserving property of elastic collision, these can be used to define a set of collision processes $\{y_i(t)\}$. It is shown in this paper that for large values of $A$, the finite dimensional distributions of $y_0(At)/A^{1/2\gamma}$ approach the Gaussian distribution with mean zero and covariance $r(t, s) = c(t^{1/\gamma} + s^{1/\gamma} - |t - s|^{1/\gamma})$.