## The Annals of Probability

### Convergence of Conditional $p$-Means Given a $\sigma$-Lattice

#### 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.

#### Article information

Source
Ann. Probab., Volume 4, Number 1 (1976), 147-150.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996194

Digital Object Identifier
doi:10.1214/aop/1176996194

Mathematical Reviews number (MathSciNet)
MR391206

Zentralblatt MATH identifier
0348.60008

JSTOR
Landers, D.; Rogge, L. Convergence of Conditional $p$-Means Given a $\sigma$-Lattice. Ann. Probab. 4 (1976), no. 1, 147--150. doi:10.1214/aop/1176996194. https://projecteuclid.org/euclid.aop/1176996194