Jensen’s and martingale inequalities in Riesz spaces. (English) Zbl 1307.47095
The role of positivity in probability theory and stochastic processes was investigated in several papers by C. C. A. Labuschagne, B. A. Watson, and Wen-Chi Kuo, V. Troitsky and the author of the present paper. The ideas on stochastic processes in vector lattices were also applied by S. F. Cullender and C. C. A. Labuschagne to get information about martingale convergence in Bochner spaces. Stochastic variables are elements of function spaces and a natural correspondence exists between stochastic variables and elements of a general vector lattice, events (i.e., sets) and order projections in the vector lattice, \(\sigma\)-algebras of sets and Boolean algebras of order projections. The role of a probability measure is played by a conditional expectation operator, i.e., a positive order continuous projection mapping the vector lattice onto a Dedekind complete Riesz subspace. Thus, the theory of probability and stochastic processes relies on many abstract notions involving positivity. The present paper gives a proof of Jensen’s inequality for conditional expectations. Halmos’s optimal skipping theorem, a new approach to the upcrossing theorem, and the upcrossing inequality, martingale inequalities, and Doob’s inequality are also given.
Reviewer: Şafak Alpay (Ankara)
MSC:
| 47N30 | Applications of operator theory in probability theory and statistics |
| 60E15 | Inequalities; stochastic orderings |
| 60G44 | Martingales with continuous parameter |
| 46A40 | Ordered topological linear spaces, vector lattices |
| 47B65 | Positive linear operators and order-bounded operators |
| 47A60 | Functional calculus for linear operators |
Keywords:
vector lattice; stochastic process; Jensen’s inequality; Doob’s \(L^p\)-inequality; Doob’s maximal inequality; upcrossing theorem; downcrossing theoremReferences:
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