Characterizations of mixtures of discrete distributions by a regression point

Given that the conditional distribution p_s(y|x) of Y, given X = x is an x-fold convolution of a nonnegative integer-valued r.v. ξ for every s= Pξ = 0] > 0, the distribution of X, hence also of Y, is characterized by the regression point m(0) = EX|Y = 0]. An infinite variety of generalized distributions (of Y) can be characterized by arbitrarily varying the distribution of X.