Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

Adam Osekowski
2010 Electronic Communications in Probability  
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|F n−1 )| ≤ E(dfn|F n−1 ) for n ≥ 1. We determine the best universal constant U (x, y) such that As an application, we deduce a sharp weak type (1, 1) inequality for the onesided maximal function of g and determine, for any t ∈ [0, 1] and β ∈ R, the number L(x, y, t) = inf{||f || 1 : P(g * ≥ β) ≥ t}. The results extend some earlier work of
more » ... lier work of Burkholder and Choi in the martingale setting. 2000 Mathematics Subject Classification. Primary: 60G42. Secondary: 60G44.
doi:10.1214/ecp.v15-1582 fatcat:jw7xbvdzqfghjdg5euuu3scq5u