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\(L^p\)-continuity of conditional expectations. (English) Zbl 0917.60034

A necessary and sufficient condition on a sequence \(({\mathcal A}_n)_{n\in\mathbb{N}}\) of \(\sigma\)-subalgebras is given such that the conditional expectations \(E(f\mid {\mathcal A}_n)\) converge in \(L^p\)-norm \((1\leq p<\infty)\). This result generalizes the convergence theorem of the \(L^p\)-martingales, the Fetter and Boylan equi-convergence theorems. Some examples and counterexamples are also given.
Reviewer: F.Weisz (Budapest)

MSC:

60F25 \(L^p\)-limit theorems
60G48 Generalizations of martingales
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References:

[1] Alonso, A., A counterexample on the continuity of conditional expectations, J. Math. Anal. Appl., 129, 1-5 (1988) · Zbl 0641.60053
[2] Boylan, E., Equiconvergence of martingales, Ann. Math. Statist., 42, 552-559 (1971) · Zbl 0218.60049
[3] F. Brambila, A. Alonso, Continuity of conditional expectations; F. Brambila, A. Alonso, Continuity of conditional expectations · Zbl 0917.60034
[4] Fetter, H., On the continuity of conditional expectations, J. Math. Anal. Appl., 61, 227-231 (1977) · Zbl 0415.60003
[5] Neveu, J., Mathematical Foundations of the Calculus of Probability (1965), Holden-Day: Holden-Day San Francisco · Zbl 0137.11301
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