Iterates of a Product of Conditional Expectation Operators

Let (Ω,F , μ) be a probability space and let T = P1P2 · · ·Pd be a finite product of conditional expectations with respect to the sub σ-algebras F1,F2, . . . ,Fd. Since conditional expectations are contractions of all Lp(μ) spaces, p ∈ [1,∞], so is T . When d = 2, Burkholder and Chow [2] proved that for every f ∈ L2(μ) the iterates T f converge a.s. (and thus also in L2-norm) to the conditional expectation with respect to F1 ∩ F2. The L2-norm convergence had been proved by von-Neumann [5, Lemma 22]. The main property of T when d = 2 is that T n = (P1P2P1) P2 with P1P2P1 self-adjoint in L2, so from the work of Stein [9] it follows that the a.e. convergence of {T f} holds also for any f ∈ Lp(μ), p > 1 (one needs to show only for p < 2). Rota’s work [7] yields a different proof, which in fact proves the a.e. convergence of {T f} when f ∈ L log L (see [1]). Ornstein [6] showed that for f ∈ L1(μ) almost everywhere convergence need not hold (although L1-norm convergence does). For arbitrary d, the L2-norm convergence of T f , f ∈ L2(μ), was proved by Halperin [4] (and the limit is the conditional expectation with respect to F1 ∩ F2 ∩ · · · ∩ Fd). Zaharopol [12] proved that the iterates T f converge in Lp-norm for f ∈ Lp(μ), p ≥ 1 (for p ≤ 2 this follows from [4]). Delyon and Delyon [3] proved that T f converges a.e. for any f ∈ L2(μ). In this note we show that for every f ∈ Lp(μ), p > 1, the sequence {T f} converges μ-a.e., with