Products of conditional expectation operators: convergence and divergence. (English) Zbl 1469.60010
Summary: In this paper, we investigate the convergence of products of conditional expectation operators. We show that if \((\Omega,\mathcal{F},P)\) is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving \(3\) or \(4\) sub-\(\sigma\)-fields of \(\mathcal{F}\) can be constructed for a large class of random variables in \(L^2(\Omega,\mathcal{F},P)\). This settles in the negative a long-open conjecture. On the other hand, we show that if \((\Omega ,\mathcal{F},P)\) is a purely atomic probability space, then products of conditional expectation operators involving any finite set of sub-\(\sigma\)-fields of \(\mathcal{F}\) must converge for all random variables in \(L^1(\Omega,\mathcal{F},P)\).
MSC:
| 60A05 | Axioms; other general questions in probability |
| 60F15 | Strong limit theorems |
| 60F25 | \(L^p\)-limit theorems |
Keywords:
product of conditional expectation operators; Amemiya-Ando conjecture; non-atomic \(\sigma\)-field; purely atomic \(\sigma\)-field; linear compatibility; deeply uncorrelatedReferences:
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