Bounds for tail probabilities of martingales using skewness and kurtosis

摘要

M n = X 1 + ? + X n be a sum of independent random variables such that X k ? 1, and EX k 2 = σ k 2 for all k. Hoeffding [15, Theorem 3] proved that with. Bentkus [5] improved Hoeffding’s inequalities using binomial tails as upper bounds. Let and stand for the skewness and kurtosis of X k . In this paper we prove (improved) counterparts of the Hoeffding inequality replacing σ 2 by certain functions of γ 1, ..., γ n (respectively ?1, ..., ?1). Our bounds extend to a general setting where X k are martingale differences, and they can combine the knowledge of skewness and/or kurtosis and/or variances of X k . Up to factors bounded by e 2/2 the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control are known so far.