Abstract

The purpose of this article is to improve Hoeffding's lemma and consequently Hoeffding's tail bounds. The improvement pertains to left skewed zero mean random variables X\in[a,b], where a<0 and -a>b. The proof of Hoeffding's improved lemma uses Taylor's expansion, the convexity of \exp(sx), s\in \RR, and an unnoticed observation since Hoeffding's publication in 1963 that for -a>b the maximum of  the intermediate function \tau(1-\tau) appearing in Hoeffding's proof is attained at an endpoint rather than at \tau=0.5 as in the case b>-a. Using Hoeffding's  improved lemma we obtain one sided and two sided  tail bounds  for \PP(S_n\ge t) and \PP(|S_n|\ge t), respectively, where S_n=\sum_{i=1}^nX_i and the X_i\in[a_i,b_i],i=1,...,n are independent zero mean  random variables (not necessarily identically distributed). It is interesting to note that we  could  also improve Hoeffding's two sided bound for all \{X_i:  -a_i\ne b_i,i=1,...,n\}. This is  so because here the one sided bound should be  increased by \PP(-S_n\ge t),  wherein the left skewed intervals become right skewed and vice versa.

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