Bounds for tail probabilities of martingales using skewness and kurtosis

Abstract

Let M _n = X ₁ + ⋯ + X _n be a sum of independent random variables such that X _k ⩽ 1, $\mathbb{E}X_k = 0$ and EX ² _k = σ ² _k for all k. Hoeffding [15, Theorem 3] proved that

$$\mathbb{P}\{ M_n \geqslant nt\} \leqslant H^n (t,p),H(t,p) = (1 + {{qt} \mathord{\left/ {\vphantom {{qt} p}} \right. \kern-\nulldelimiterspace} p})^{ - p - qt} (1 - t)^{qt - q} $$

with

$$q = \frac{1}{{1 + \sigma ^2 }},p = 1 - q,\sigma ^2 = \frac{{\sigma _1^2 + \cdots + \sigma _n^2 }}{n},0 < t < 1.$$

. Bentkus [5] improved Hoeffding’s inequalities using binomial tails as upper bounds. Let $\gamma _k = \mathbb{E}{{X_k^3 } \mathord{\left/ {\vphantom {{X_k^3 } {\sigma _k^3 }}} \right. \kern-\nulldelimiterspace} {\sigma _k^3 }}$ and $\kappa _k = \mathbb{E}{{X_k^4 } \mathord{\left/ {\vphantom {{X_k^4 } {\sigma _k^4 }}} \right. \kern-\nulldelimiterspace} {\sigma _k^4 }}$ 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 ϰ₁, ..., ϰ₁). 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.