Exact inequalities for sums of asymmetric random variables, with applications |
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Authors: | Iosif Pinelis |
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Institution: | (1) Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA |
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Abstract: | Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter
. Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in .
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.
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Keywords: | (super)martingales Probability inequalities Generalized moments Self-normalized sums t-statistic |
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