Characterizations of discrete distributions by a conditional distribution and a regression function

Summary

The bivariate distribution of (X, Y), whereX andY are non-negative integer-valued random variables, is characterized by the conditional distribution ofY givenX=x and a consistent regression function ofX onY. This is achieved when the conditional distribution is one of the distributions: a) binomial, Poisson, Pascal or b) a right translation of these. In a) the conditional distribution ofY is anx-fold convolution of another random variable independent ofX so thatY is a generalized distribution. A main feature of these characterizations is that their proof does not depent on the specific form of the regression function. It is also indicated how these results can be used for good-ness-of-fit purposes.