Martingale-like sequences in Banach lattices

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 coordinatewise order. Moreover, a necessary and sufficient condition is presented for the collection of E -martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded E -martingales is a normed lattice but not necessarily a Banach space under the supremum norm.