Conditional moments

Abstract

In this chapter we shall use assumptions that mimic the behavior of conditional moments that would have followed from independence. Strictly speaking, corresponding characterization results do not generalize the theorems that assume independence, since weakening of independence is compensated by the assumption that the moments of appropriate order exist. However, besides just complementing the results of the previous chapters, the theory also has its own merits. Reference [37] points out the importance of description of probability distributions in terms of conditional distributions in statistical physics. From the mathematical point of view, the main advantage of conditional moments is that they are “less rigid” than the distribution assumptions. In particular, conditional moments lead to characterizations of some non-Gaussian distributions, see Problems 7.8 and 7.9.