Some stochastic convergence theorems on vector latices

Rodda, David Francis
Journal Title
Journal ISSN
Volume Title
This thesis presents new generalisations of some classical convergence theorems, thus developing the theory of stochastic processes in vector lattices as set out by Kuo, Labuschagne and Watson [58–60]. This setting has been receiving considerable attention since its introduction, and has already proven to be fruitful by providing generalisations of many classical stochastic process results, which have the classical results as special cases. Such results include, amongst others, generalisations of Brownian motion and Itoˆ integration [44], martingale theory [59,60], Markov processes [88,89], mixingales [68], modes of stochastic convergence [6], and various important inequalities, see for example [7,9,38]. Hence it has been shown how these classical theories depend greatly on the order structure and not on measure theory. In so doing, some new results have been brought even to the classical measure space setting, for example in [65]. This thesis builds on previous work and gives a number of new results. The first main result obtained is the strong sequential completeness of the space L1(T), the natural domain of the conditional expectation operator T. Strong completeness of L∞(T) is also proved. A maximal inequality is generalised which in the classical setting is due to Ha´jek-R´enyi and Chow in [16, Theorem 1], see [30, Proposition (6.1.4)]. The final main result is a measure-free version of Chow’s martingale law of large numbers [16,17] having Kolmogorov’s and L´evy’s laws of large numbers as special cases. This is quite a general result having many areas of application, including Lp(T) martingales for arbitrary p > 1, submartingales, and independent random variables. In the course of proving the above we develop the understanding of functional calculus on Riesz spaces, prove Riesz space versions of Kronecker’s Lemma and Ho¨lder’s inequality for sums, give a submartingale limit law and gain other results.
A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy, 2019
Rodda, David Francis (2019) Some stochastic convergence theorems on vector lattices, University of the Witwatersrand, Johannesburg,