Martingale-like sequences in Banach lattices

TL;DR

This paper explores martingale-like sequences within Banach lattices, establishing their structure as Banach spaces and lattices, and providing conditions for their order properties and completeness.

Contribution

It introduces the concept of martingale-like sequences in Banach lattices and characterizes their lattice and Banach space structures under various conditions.

Findings

Bounded $X$-martingales form a Banach space under the supremum norm.

Under certain conditions, these collections are Banach lattices with coordinate-wise order.

Necessary and sufficient conditions are given for $ ext{E}$-martingales to form a vector lattice.

Abstract

Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437--456]. In these frameworks, a collection of bounded $X$-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\mathcal{E}$-martingales is a normed lattice but not necessarily a Banach space under the supremum norm.