Read e-book online Amarts and Set Function Processes PDF

By A. Gut, K. D. Schmidt

ISBN-10: 3540128670

ISBN-13: 9783540128670

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3 to hold in general (if D =N) . 2 together with the function ~(x) ~ [ x l P , and ~Y(P) p > 1 . Then, ~ - n "~nP ) ~" n e N {,~,cx(P)~n "'~P)}nn EN is an (p > I) Ll-bounded ~m~rt Ll-bounded semiamart but not an amart. 3 with "{~(Xn)}n 6 N is uniformly Ll-bounded'' imply each other (combine 1/2 < p < i) both conditions had to be investigated. Now, let D=-N. Suppose that the assumption that {~0(Xn),~n}nE_N weakened (cf. 6 together with the function ~0(x) -- Ix[p , p > i. Then, ^'X p) } ~ n ~ nE-N an amart.

Let D = N , assume, in addition, be a function such that ~: R ~ R is continuous and lim ~(x) and lim ~(x) X exist and are finite. X X ~ X ~ --~ Then, {~(Xn),~n} n C D is an Ll-bounded amart. 2. The cases obviously included. 2) are necessary and sufficient for an {~(Xn)'~n}nCN to be Ll-bounded amart. Proof. We first assume that X > O, n ~(O) = 0 and lim ~,x~ = O. s. as Ini '" ® . s. 5) {~(XT)}T q T N is a semiamart. is uniformly integrable. We first consider the Ll-boundedness of By assumption, x-l-l~(x)l < E if 0 < x < M.

2. e. as n,m ~ with n > m. l. is a game fairer with time. Further, Blake (1978) and Edgar and Sucheston (1977 a) show that e~ePyumuPt is a mc~ti~gule in the limit. l. converges almost surely. -theory just as has been done for amarts. We have, for example, noticed above that some kind of convergence theorems remain valid. However, it turns out that there areseveral other basic and useful properties which hold for martingales and amarts which do not hold for martingales in the limit (and games which become fairer with time), such as the maximal ineuqality, the Riesz decomposition theorem.

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Amarts and Set Function Processes by A. Gut, K. D. Schmidt

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