Optimal two-value zero-mean disintegration of zero-mean random variables

Abstract

For any continuous zero-mean random variable $X$, a reciprocating function $r$ is constructed, based only on the distribution of $X$, such that the conditional distribution of $X$ given the (at-most-)two-point set $\{X,r(X)\}$ is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations - of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) - go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.