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Products of Conditional Expectation Operators: Convergence and Divergence

Author

Listed:
  • Guolie Lan

    (Guangzhou University)

  • Ze-Chun Hu

    (Sichuan University)

  • Wei Sun

    (Concordia University)

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 ) .

Suggested Citation

  • Guolie Lan & Ze-Chun Hu & Wei Sun, 2021. "Products of Conditional Expectation Operators: Convergence and Divergence," Journal of Theoretical Probability, Springer, vol. 34(2), pages 1012-1028, June.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:2:d:10.1007_s10959-020-01000-5
    DOI: 10.1007/s10959-020-01000-5
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