MATH 280A-B-C Lecture Notes - Lecture 9: Maximal Function, Stopping Time, Xm Satellite Radio
4 Dec 2017
School
Department
Course
Professor
Document Summary
Everything that follows takes place on a probability space ( , f, p ) equipped with a ltration. {fn : n = 0, 1, 2, . }, with fn f for all n: submartingale maximal inequality. Let {xn} be a non-negative submartingale (for ex- ample, xn = |mn| if {mn} is a martingale, or xn = s+ n if {sn} is a submartingale), and de ne. P [x n t] t 1e[xn; x n b] t 1e[xn], For the proof of this maximal inequality we require the following simple lemma, a hint of better things: lemma. If {yn} is a submartingale and t is a stopping time bounded above by a positive integer n , then. E[yn ; a {t = n}] . E[yt ; a {t = n}] = e[yt ; a], where the inequality follows from the submartingale property of y because a {t = n} fn: proof of the maximal inequality.
