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An extension of a classical martingale inequality. (English) Zbl 0594.60020

Probability theory and harmonic analysis, Pap. Mini-Conf., Cleveland/Ohio 1983, Pure Appl. Math., Marcel Dekker 98, 21-30 (1986).
[For the entire collection see Zbl 0577.00016.]
Let \(g=(g_ 1,g_ 2,...)\) be a transform of a real martingale \(f=(f_ 1,f_ 2,...)\) by a predictable sequence \(v=(v_ 1,v_ 2,...)\), i.e. \(g_ n=\sum^{n}_{k=1}v_ k(f_ k-f_{k-1}).\) Let each \(v_ k\) take its values in [0,1]. It is proved that \[ \lambda P(\sup_{n}| g_ n| \geq \lambda)\leq \sup_{n}\| f_ n\|_ 1,\quad \lambda >0. \] Inequalities of this type are also proved for Banach- space-valued martingales and stochastic integrals.
Reviewer: K.Kubilius

MSC:

60E15 Inequalities; stochastic orderings
60B05 Probability measures on topological spaces
60G42 Martingales with discrete parameter
60H05 Stochastic integrals

Citations:

Zbl 0577.00016