A Martingale Convergence Theorem in Vector Lattices

Abstract

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

Martingales in Banach Lattices

“…DeMarr in [8] introduced a martingale in a vector lattice F as double sequences (x n , E n ) where x n is an element of F, E n is a positive linear projection, E n E m = E n∧m and E n x m = x n whenever n m. DeMarr then generalized the almost everywhere part of Doob's Convergence Theorem. Namely, he proved that under certain special conditions, a martingale in a vector lattice is order convergent.…”

Continuous stochastic processes in Riesz spaces: the Doob–Meyer decomposition

“…Martingales were studied by DeMarr [3] as early as 1966, followed by Stoica [17] and [18], and Troitsky [19]. The general theory was considered by Kuo, Labuschagne and Watson (see [6][7][8][9][10][11][12]), who studied countable processes in this setting.…”

Doob’s optional sampling theorem in Riesz spaces

“…For more background on what has been done in this field we refer the reader to DeMarr [3] who studied martingales in Banach lattices as early as 1966, followed by Stoica [18][19][20][21], and Troitsky [22]. The general theory was considered by Kuo, Labuschagne and Watson (see [8][9][10][11][12][13][14]), who studied countable processes in this setting.…”