Bounds for tail probabilities of martingales using skewness and kurtosis

Let $M_n=\text{fsu}X1n$ be a sum of independent random variables such that $X_k\le 1$, $\text{E}{X}_k=0$ and $\text{E}{X}_k^2={\text{s}}_k^2$ for all $k$. Hoeffding 1963, Theorem 3, proved that $$\mathrm{\P }{M}_n\ge nt\le H^n(t,p),H(t,p)=\text{bgl}(1+qt\mathrm{/}p\text{bgr}{)}^{p+qt}\text{bgl}(1-t\text{bgr}{)}^{q-qt}$$ with $$q=\text{ffrac}11+{\text{s}}^2,p=1-q,{\text{s}}^2=\text{ffrac}{\text{s}}_1^2+\mathrm{.}\mathrm{.}\mathrm{.}+{\text{s}}_n^{2}n,0<t<1.$$ Bentkus 2004 improved Hoeffding’s inequalities using binomial tails as upper bounds. Let ${\text{ga}}_k=\text{E}{X}_k^3\mathrm{/}{\text{s}}_k^3$ and ${\text{vk}}_k=\text{E}{X}_k^4\mathrm{/}{\text{s}}_k^4$ stand for the skewness and kurtosis of $X_k$. In this paper we prove (improved) counterparts of the Hoeffding inequality replacing ${\text{s}}^2$ by certain functions of $\text{fs}\text{ga}1n$ respectively $\text{fs}\text{vk}1n$. 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 $e^2\mathrm{/}2$ the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control so far are known.