On the Almost Sure Convergence of All Conditionings of Positive Random Variables

Abstract

Abstract. We prove that in a non-atomic probability space, for a sequence of positive r.v. (Xn), E(An|2l) -• 0 a.s. for any cr-field 21 of events iff Xn -> 0 a.s. and Esupn>1 |Xn| < 00. We point out these classical theorems on orthogonal series and ergodic means in which the almost sure convergence of all conditionings of a discussed sequence can be obtained. Introduction and main resultsLet 21 be any cr-field of events in any probability space. The operation of conditional expectation E(-|2l) is a positive contraction in the space L\ of integrable random variables. Thus the following theorem can be immediately obtained. Developping such an idea, we obtain the following general THEOREM 1.3. For any non-atomic probability space P) and any sequence (Xi) of positive random variables, the following conditions are equivalent: for any a-fieldThe assumption that (fi, T, P) is non-atomic is essential. Namely,for LO taken from any atom A of 21 with probability P(A) = p, and we have Some consequences of Theorem 1.3 and of (elementary) Theorem 1.1 are collected in Section 3. All these corollaries are obtained by analysing classical proves of specific theorems on almost sure convergence. The proof of Theorem 1.3We shall use the following, rather obvious

On characterization of a.s. convergence of all conditionings for sequences of random variables

“…The same equivalence for sequences of positive random variable was proved in [1], Theorem 1.3. For signed random variables the proof is essentially more complicated.…”

“…The assumption that (Ω, F , P ) is non-atomic is essential, see [1]. The same equivalence for sequences of positive random variable was proved in [1], Theorem 1.3.…”