Exact inequalities for sums of asymmetric random variables, with applications

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Let BS1,⋯,BSn be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter p ∈ (0, 1). Let m*(p):=(1 + p + 2 p 2)/(2√{p - p2 + 4 p2) if 0 < p ≤ 1/2 and m*(p):= 1 if 1/2 ≤ p < 1. Let m ≥ m *(p). Let f be such a function that f and f″ are nondecreasing and convex. Then it is proved that for all nonnegative numbers c1,⋯,cn one has the inequality equation presented where equation presented. The lower bound m*(p) on m is exact for each p ∈ (0,1). Moreover, Ef(c1BS1+⋯ +cnBSn is Schur-concave in c12m,⋯,cn2m. A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given. © 2007 Springer-Verlag.

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Probability Theory and Related Fields