Bounds for tail probabilities of martingales using skewness and kurtosis

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.