An invariance principle for a class of $d$-dimensional polygonal random functions

Abstract:

A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space $(d \geq 1)$ undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if $d \geq 2$, and not being almost surely nonnegative if $d = 1$. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rosén, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.