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February, 1976 Convergence of Conditional $p$-Means Given a $\sigma$-Lattice
D. Landers, L. Rogge
Ann. Probab. 4(1): 147-150 (February, 1976). DOI: 10.1214/aop/1176996194

Abstract

Let $\mathbf{P}_n \mid \mathscr{A}, n \in \mathbb{N}$, be a sequence of probability measures converging in total variation to the probability measure $\mathbf{P} \mid \mathscr{A}$ and $\mathscr{C}_n \subset \mathscr{A}, n \in \mathbb{N}$, be a sequence of $\sigma$-lattices converging increasing or decreasing to the $\sigma$-lattice $\mathscr{C}$. Then for every uniformly bounded sequence $f_n, n \in \mathbb{N}$, converging to $f$ in $\mathbf{P}$-measure we show in this paper that the conditional $p$-mean $\mathbf{P}_n^\mathscr{C}k f_j$ converge to $\mathbf{P}^\mathscr{C}f$ in $\mathbf{P}$-measure if $n, k, j$ tends to infinity. The methods used in this paper are completely different from those used to prove the corresponding result for $\sigma$-fields instead of $\sigma$-lattices.

Citation

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D. Landers. L. Rogge. "Convergence of Conditional $p$-Means Given a $\sigma$-Lattice." Ann. Probab. 4 (1) 147 - 150, February, 1976. https://doi.org/10.1214/aop/1176996194

Information

Published: February, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0348.60008
MathSciNet: MR391206
Digital Object Identifier: 10.1214/aop/1176996194

Subjects:
Primary: 60B10

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 1 • February, 1976
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