数学物理学报(英文版) ›› 2002, Vol. 22 ›› Issue (3): 329-334.
梅韬, 刘培德
MEI Tao, LIU Pei-De
摘要:
Let 1,2 be nonnegative nondecreasing functions, and 1 be concave. The authors prove the equivalence of the following two conditions:
(i) E1(Mf) cE2(Z0+A1) for every nonnegative submartingale f = (fn)n0 with it’s Doob’s Decomposition: f = Z + A, where Z is a martingale in L1 and A is a nonnegative incrasing and predictable process. (ii) There exists positive constants c, t0 such that R 1t 1(s) s2 ds c2(t) t , 8t > t0.
When 1 = 2 the condition (ii) above is equivalent to the classical condition p < 1. As a consequence, for a concave function , p < 1 if and only if E1(Mf) cE2(Z0+A1) for every nonnegative submartingale f.
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