An Analogue of the Lévy-cramér Theorem for Rayleigh Distributions

In the present paper we prove that every k-dimensional Cartesian product of Kingman convolutions can be embedded into a k-dimensional symmetric convolution (k=1, 2, . . . ) and obtain an analogue of the Cramér-Lévy theorem for multi-dimensional Rayleigh distributions. A new class of multidimensional Rayleigh distributions and associated higher-dimensional Bessel processes are introduced and studied. This class of processes inherits the wellknown characteristics of Brownian motions: They are independent stationary ”increments” processes with continuous sample paths.