Limiting distributions of randomly accelerated motions

Abstract

In this paper, the process {X(t); t > 0}, representing the position of a uniformly accelerated particle (with Poisson-paced) changes of its acceleration, is studied. It is shown that the distribution of X (t) (suitably normalized), conditionally on the number n of changes of acceleration, tends in distribution to a normal variate as n goes to infinity. The asymptotic normality of the unconditional distribution of X(t) for large values of t is also shown. The study of these limiting distributions is motivated by the difficulty of evaluating exactly the conditional and unconditional probability laws of X(t). In fact, the results obtained in this paper permit us to give useful approximations of the probability distributions of the position of the particle.