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 Electronic Communications in Probability > Vol. 15(2010) > Paper 46 open journal systems 


Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

Adam Osekowski, University of Warsaw


Abstract
Let f=(fn)n≥ 0 be a nonnegative submartingale starting from x and let g=(gn)n≥ 0 be a sequence starting from y and satisfying |dgn|≤ |dfn|,  |E(dgn|Fn-1)|≤ E(dfn|Fn-1) for n≥ 1. We determine the best universal constant U(x,y) such that P(supngn≥ 0)≤ ||f||1+U(x,y). As an application, we deduce a sharp weak type (1,1) inequality for the one-sided maximal function of g and determine, for any t in [0,1] and any real β, the number L(x,y,t,β)=inf{||f||1: P(supngn≥ β)≥ t}. The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.


Full text: PDF

Pages: 508-521

Published on: October 26, 2010
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Electronic Communications in Probability. ISSN: 1083-589X