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Martingale Transforms

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

Abstract

Let f= (f 1, f 2,…) be a martingale on a probability space (Ω,α,P]) Let \({d_1} = {f_1},{d_2} = {f_2} - {f_1},...\)so that \({f_n} = \sum\nolimits_{k = 1}^n {{d_k},n \geq 1.}\)It is convenient to say that g=(gi,gi…) is a transform of fif \({g_n} = \sum\nolimits_{k = 1}^n {v_k}{d_k},\)where vn is a real v n is a real αn-1-measurable function, n ≧ 1, and α0⊂ α1⊂… ⊂ α are σ-fields such that {f n, αn, n ≧ 1} is a martingale. Note that gneed not be a martingale. It is easy to see that gis a martingale if and only if E∣gn∣ is finite for all n. This condition is satisfied, for example, if each v nis bounded. Transforms of real (but not of extended real) submartingales may be defined similarly.

Keywords

Probability Space Positive Real Number Maximal Function Difference Sequence Interpolation Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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