Limit Theorems for Random Evolutions with Explicit Error Estimates*

We think of x (t, y) as the position of a particle at time t when its velocity is v (t). The process x (t, y) is the simplest example of a random evolution: one-dimensional motion at a constant but random velocity determined by the state of the Markov chain associated with v(t). We denote by P(y,~i){" }, Y real, v~sA, the probability laws of the joint process (x (t, y), v (t)), where v (0)= v/. E(y, v,) will denote integration with respect to P(y, v~). The purpose of this paper is to prove the following two theorems, which correspond respectively to the weak law of large numbers and the central limit theorem for x (t).