2020/12/07 by Hertz, David
#FOS: Electrical engineering #FOS: Mathematics #Probability (math.PR) #Signal Processing (eess.SP) #electronic engineering #information engineering
paper · doi:10.48550/arxiv.2012.03535
The purpose of this letter is to improve Hoeffding's lemma and consequently Hoeffding's tail bounds. The improvement pertains to left skewed zero mean random variables X∈[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∈ \bf R and an unnoticed observation since Hoeffding's publication in 1963 that for -a>b the maximum of the intermediate function τ(1-τ) appearing in Hoeffding's proof is attained at an endpoint rather than at τ=0.5 as in the case b>-a. Using Hoeffding's improved lemma we obtain one sided and two sided tail bounds for P(Sn≥ t) and P(|Sn|≥ t), respectively, where Sn=∑i=1nXi and the Xi∈[ai,bi],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 \Xi: ai≠ bi,i=1,...,n\. This is so because here the one sided bound should be increased by P(-Sn≥ t), wherein the left skewed intervals become right skewed and vice versa.