Martingale Transforms

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


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.


Probability Space Positive Real Number Maximal Function Difference Sequence Interpolation Theorem 
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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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