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Continuity of conditional expectation in Orlicz spaces

A. Hosseini, Y. Estaremi

TL;DR

This work addresses the continuity of conditional expectation in Orlicz spaces by extending $L^{p}$-space convergence results to the broader setting of Orlicz spaces $L^{\varphi}$ under the $\Delta_2$ condition. It introduces and analyzes three notions of convergence for sequences of $\sigma$-subalgebras: $\mu$-convergence, $\perp$-convergence, and their combination $\mu\perp$-convergence, establishing precise links to the $L^{\varphi}$-convergence of conditional expectations $E(f|\mathcal{A}_n)$. The main contributions are necessary and sufficient conditions for $L^{\varphi}$-convergence of conditional expectations under each convergence notion, including equivalences that connect algebraic convergence with analytic convergence in $L^{\varphi}$, and a structured analysis of the associated limit algebras $\underline{\mathcal{A}},\mathcal{A}_{\mu},\mathcal{A}_{\perp},\overline{\mathcal{A}}$. The results broaden martingale-convergence-type theorems to Orlicz spaces and provide a framework for analyzing conditional expectations in nonlinear growth contexts, with illustrative examples using dyadic partitions.

Abstract

The continuity of conditional expectation on Orlicz spaces is investigated. Indeed, we provide some necessary and sufficient conditions on a sequence $\{\mathcal{A}_n\}_{n\in\mathbb{N}}$ of $σ$-subalgebras for $L^{\varphi}$-convergence of the related conditional expectations. Our results generalize similar results in $L^p$-spaces.

Continuity of conditional expectation in Orlicz spaces

TL;DR

This work addresses the continuity of conditional expectation in Orlicz spaces by extending -space convergence results to the broader setting of Orlicz spaces under the condition. It introduces and analyzes three notions of convergence for sequences of -subalgebras: -convergence, -convergence, and their combination -convergence, establishing precise links to the -convergence of conditional expectations . The main contributions are necessary and sufficient conditions for -convergence of conditional expectations under each convergence notion, including equivalences that connect algebraic convergence with analytic convergence in , and a structured analysis of the associated limit algebras . The results broaden martingale-convergence-type theorems to Orlicz spaces and provide a framework for analyzing conditional expectations in nonlinear growth contexts, with illustrative examples using dyadic partitions.

Abstract

The continuity of conditional expectation on Orlicz spaces is investigated. Indeed, we provide some necessary and sufficient conditions on a sequence of -subalgebras for -convergence of the related conditional expectations. Our results generalize similar results in -spaces.
Paper Structure (4 sections, 16 theorems, 91 equations)

This paper contains 4 sections, 16 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. $\mu$-Convergence
  3. $\perp$-Convergence
  4. $\mu\perp$-Convergence

Key Result

Lemma 2.2

7 Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $\{\mathcal{A}_n\}_{n \in \mathbb{N}}$ be a sequence of $\sigma$-subalgebras of $\mathcal{A}$ and $\mathfrak{D}$ be a $\sigma$-subalgebra of $\mathcal{A}$. Then $\mathcal{A}_n \xrightarrow{\mu} \mathfrak{D}$ if and only if $\mathfrak{D} \subs

Theorems & Definitions (33)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 23 more