Bounds for tail probabilities of martingales using skewness and kurtosis

Details Bounds for tail probabilities of martingales using skewness and kurtosis Bounds for tail probabilities of martingales using skewness and kurtosis

2011-11-28

arxiv.org/abs/1111.6358v1

Mathematics

Probability

Lithuanian Mathematical Journal (2008)

Scientific paper

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

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