The H\'ajek-R\'enyi-Chow maximal inequality and a strong law of large numbers in Riesz spaces

TL;DR

This paper extends a maximal inequality to Riesz spaces with conditional expectations, leading to a convergence theorem and a strong law of large numbers within this abstract mathematical framework.

Contribution

It introduces a Riesz space version of the H"ajek-R"enyi-Chow inequality and derives related convergence results, advancing the theory in this area.

Findings

Established a Riesz space variant of Clarkson's inequality for 1≤p≤2

Proved a submartingale convergence theorem in Riesz spaces

Derived a strong law of large numbers in Riesz spaces

Abstract

In this paper we generalize the H\'ajek-R\'enyi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for $1\le p\le 2$.