We consider several characterizations of distributions by conditional moments. The idea is to assume that some of the conditional moments have the same form as in the normal case, and to look for restrictions on distributions that follow from the assumption in general case.
The research was originated by three sources: by Problem 6 in A. M. Kagan, Yu. V. Linnik and C. R. Rao, Characterization problems in mathematical statistics (1973; Zbl 0271.62002), by the characterization of rotations of invariant distributions in terms of conditional moments, and by the characterization of a Gaussian process by its conditional moments of first two orders. In Section 1 we consider the condition of linearity of regression complemented by restrictions on conditional moments of higher orders. In Section 2 we investigate \(L_ 2\)- differentiable stochastic processes such that conditional moments of first two orders are the same as in the case of Gaussian processes. Section 3 contains our main results. These results were motivated by a symmetric version of Problem 6 in the monograph cited above.