Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {Ut: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {Utn([mu]1 * [mu]2*...*[mu]n)*[delta]xn}, where {[mu]n}[subset, double equals]Q, {xn}[subset, double equals]X and measures Utn[mu]j (j=1, 2,..., n; N=1, 2,...) form an infinitesimal triangular array. We define classes Lm() as follows: L0()=((X); ), Lm()=(Lm-1(); ) for m=1, 2,... and L[infinity]()=[down curve]m=0[infinity]Lm(). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from Lm(), m=0, 1, 2,..., [infinity], are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.