A unifying view on some problems in probability and statistics

摘要

Abstract Let L be a linear space of real random variables on the measurable space (Ω,A). Conditions for the existence of a probability P on A such that E_p|X| ＜ ∞ and E_P(X) = 0 for all X ∈ L are provided. Such a P may be finitely additive or σ-additive, depending on the problem at hand, and may also be requested to satisfy P ～ P_0 or P (＜＜) P_0 where P_0 is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability P as above. Among them, we mention de Finetti's coherence principle, equivalent martingale measures, equivalent measures with given marginals, stationary and reversible Markov chains, and compatibility of conditional distributions.