Bounds for tail probabilities of martingales using skewness and kurtosis

TL;DR

This paper develops new bounds for tail probabilities of martingales by incorporating skewness and kurtosis, extending classical Hoeffding inequalities and providing the first such bounds that include these higher moments.

Contribution

It introduces the first inequalities that incorporate skewness and kurtosis into tail probability bounds for martingales, extending Hoeffding's classical results.

Findings

Bounds are improved by replacing variance with functions of skewness and kurtosis.

Results extend to martingale differences with combined knowledge of moments.

Bounds are nearly optimal, differing by a factor bounded by e^2/2.

Abstract

Let $M_n= \fsu X1n$ be a sum of independent random variables such that $ X_k\leq 1$, $\E X_k =0$ and $\E X_k^2=\s_k^2$ for all $k$. Hoeffding 1963, Theorem 3, proved that $$\P{M_n \geq nt}\leq H^n(t,p),\quad H(t,p)= \bgl(1+qt/p\bgr)^{p +qt} \bgl({1-t}\bgr)^{q -qt}$$ with $$q=\ffrac 1{1+\s^2},\quad p=1-q, \quad \s^2 =\ffrac {\s_1^2+...+\s_n^2}n,\quad 0<t<1.$$ Bentkus 2004 improved Hoeffding's inequalities using binomial tails as upper bounds. Let $\ga_k =\E X_k^3/\s_k^3$ and $ \vk_k= \E X_k^4/\s_k^4$ stand for the skewness and kurtosis of $X_k$. In this paper we prove (improved) counterparts of the Hoeffding inequality replacing $\s^2$ by certain functions of $\fs \ga 1n$ respectively $\fs \vk 1n$. 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.