Products of Conditional Expectation Operators: Convergence and Divergence

Guolie Lan, Ze-Chun Hu and Wei Sun ()
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Guolie Lan: Guangzhou University
Ze-Chun Hu: Sichuan University
Wei Sun: Concordia University

Journal of Theoretical Probability, 2021, vol. 34, issue 2, 1012-1028

Abstract: Abstract In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $$(\Omega ,\mathcal {F},P)$$ ( Ω , F , P ) is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub- $$\sigma $$ σ -fields of $$\mathcal {F}$$ F can be constructed for a large class of random variables in $$L^2(\Omega ,\mathcal {F},P)$$ L 2 ( Ω , F , P ) . This settles in the negative a long-open conjecture. On the other hand, we show that if $$(\Omega ,\mathcal {F},P)$$ ( Ω , F , P ) is a purely atomic probability space, then products of conditional expectation operators involving any finite set of sub- $$\sigma $$ σ -fields of $$\mathcal {F}$$ F must converge for all random variables in $$L^1(\Omega ,\mathcal {F},P)$$ L 1 ( Ω , F , P ) .

Keywords: Product of conditional expectation operators; Amemiya–Ando conjecture; Non-atomic $$\sigma $$ σ -field; Purely atomic $$\sigma $$ σ -field; Linear compatibility; Deeply uncorrelated; 60A05; 60F15; 60F25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-01000-5

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