A CENTRAL-LIMIT-THEOREM FOR ISOTROPIC RANDOM-WALKS ON N-SPHERES FOR N-]INFINITY

Let (Y-1)(1 greater than or equal to 0) be an isotropic random walk on the n-sphere S-n subset of R(n+1) starting at x(0) is an element of S -n. Then the random variables X(i) := cos angle(Y-1, x(0)) form a Mark ov chain on [-1, 1] whose transition probabilities are closely related to ultraspherical convolutions on [-1, 1]. We prove that root nX(1) i s normally distributed for n, l --> infinity provided that the spheric al distances of the jumps of Y-1 tend to O. The expectation of this di stribution depends on relations between n, l and the spherical distanc es of the random jumps. (C) 1995 Academic Press, Inc.