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.
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