Iterates of Conditional Expectation Operators

Abstract

Let {T_n} be a sequence of conditional expectation operators in $L_1=L_1(W, \ F, \ P)$ where (W, F, P) is a probability space. Let $S_n=T_n \ldots T_2 T_1$. It is known [l, p. 331] that if {T _n } is monotone decreasing, that is, if the range of $T_{n+1}$ is a subset of the range of $T_n$ for all n, then for each x in L ₁ the sequence {S _n x} converges almost everywhere. Here, the pointwise convergence behavior of {S_nx} is studied under other conditions. For example, if $T_{2n-1}={T_1} \ {\rm and} \ {T_{2n}}={T_2}$ for all n, does {S_nx} converge almost everywhere? This question was first posed by J. L. Doob. It is proved here that if x is in L₂, then this is indeed the case, and, furthermore, ${\rm sup}_{n} \ |S_{n}x| \ {\rm is \ in} \ L_2$. Several of the preliminary results needed, especially Theorems 1 and 2, seem to be of some interest in their own right. The linear spaces mentioned in this paper may be either real or complex. All of our results hold with either interpretation.

Presented to the Society, August 31, 1960; received by the editors July 6, 1960.