An Extension of a Classical Martingale Inequality

  • Burgess DavisEmail author
  • Renming Song
Part of the Selected Works in Probability and Statistics book series (SWPS)


A sequence f = (f1, f2, …) of integrable functions on a probability space is a martingale if, for each positive integer n, the difference fn+1- fnis orthogonal to every real bounded continuous function of f1,…, fn. The following beautiful result, due to Ville [11], is now classical: If α and β are positive numbers with α < β and f is a nonnegative martingale with f1= α, then
$$P({\rm {sup} _n}\, {f_n} \geq \beta) \leq \frac{\alpha }{\beta }$$
Thus, since α/β < 1, a gambler with an initial fortune α who plays a sequence of fair games in such a way that he has no chance of going into debt cannot be assured of increasing his fortune to β.


Banach Space Probability Space Maximal Function Left Limit Predictable Sequence 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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