2019/03/10 by Guolie Lan, Lan, Guolie, Ze-Chun Hu +3
Decision Sciences · Mathematics · #60A05 #60F15 #60F25 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Analysis and Transform Methods #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.1903.03917
openalex publication_date 2019/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate the convergence of products of conditional expectation operators. We show that if (Ω,\calF,P) is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub-σ-fields of \calF can be constructed for a large class of random variables in L2(Ω,\calF,P). This settles in the negative a long-open conjecture. On the other hand, we show that if (Ω,\calF,P) is a purely atomic probability space, then products of conditional expectation operators involving any finite set of sub-σ-fields of \calF must converge for all random variables in L1(Ω,\calF,P).