Bounds for tail probabilities of martingales using skewness and kurtosis

Abstract

Abstract. Let M n = X 1 + · · · + X n be a sum of independent random variables such that X k ≤ 1, E X k = 0 and E X 2 k = σ 2 k for all k. Hoeffding 1963, Theorem 3, proved thatBentkus 2004 improved Hoeffding's inequalities using binomial tails as upper bounds. Letk 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 κ 1 , . . . , κ n . 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 so far are known.

An Engineered Empirical Bernstein Bound

“…Our first probability bound is a Chernoff bound on the sample mean called Bennett's inequality. This bound is not new and was derived by [18] and [7] and has subsequently been a subject of discussion and many further developments [8,24,29]; we provide a proof in Appendix A.2.…”

“…For each arm k, there is a unique α k and β k parameters of its beta distribution over rewards, and for each realization of the problem, these α k and β k are drawn uniformly from between 0 and 3. For these problems, we used the different confidence bound approaches, and measured their performance in terms of regret, defined in (8). As noted above, regret is a measure of the performance of bandit algorithms identified by the expected loss of selecting an arm against choosing only the ideal arm.…”

On the Bennett–Hoeffding inequality

“…The right-tail inequality (1.9) can be proved quite similarly; alternatively, it follows from general results of [38]. Various generalizations and improvements of inequality (1.9) as well as related results were given by Pinelis [38,39,41,43,44,46] and Bentkus [2,3,4,5] (with co-authors). For Rademacher η i 's, a version of (1.9) with a better constant factor c, which is about 1% off the best possible one, was given in [47]; related inequalities were obtained in [8,48].…”