Improved Hoeffding’s Lemma and Hoeffding’s Tail Bounds

David Hertz

International Journal of Statistics and Probability, 2021, vol. 10, issue 5, 27

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.

Date: 2021
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