LIMIT-THEOREMS FOR COMPACT 2-POINT HOMOGENEOUS SPACES OF LARGE DIMENSIONS

Let K be the field R, C, or H of real dimension nu. For each dimension d greater than or equal to , we study isotropic random walks (Y-l)(l greater than or equal to 0) on the projective space P-d(K) with natura l metric D where the random walk starts at some x(0)(d) is an element of P-d(K) with jumps at each step of a size depending on d Then the ra ndom variables X(l)(d) : = cos D(Y-l(d), x(0)(d)) form a Markov chain on [-1, 1] whose transition probabilities are related to Jacobi convol utions on [-1, 1]. We prove that, for d --> infinity, the random varia bles (vd/2)(X(l(d))(d) + 1) tend in distribution to a noncentral chi(2 )-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive anothe r limit theorem for P-d(K) as well as the d-spheres S-d for d --> infi nity.