This chapter aims to enhance Hoeffding’s lemma and, as a result, Hoeffding’s tail limits. To begin, we’ll offer Hoeffding’s lemma with a proof that differs from the original, and then show and prove the better Hoeffding’s lemma. The enhancement is for left skewed zero mean random variables X[a,b], with a0 and -a>b. The proof of Hoeffding’s improved lemma employs Taylor’s expansion, the convexity of exp(sx),sR, and an unnoticed observation made since Hoeffding’s publication in 1963 that the maximum of the intermediate function (1-) appearing in Hoeffding’s proof is attained at an endpoint rather than at =0.5 as in the case b>-a. We get one-sided and two-sided tail limits for P(Snt) and P(Snt) using Hoeffding’s improved lemma. P(Snt) and P(|S n |t), where S n= (i=1)n X i and X i[a i,b i],i=1,…,n are independent zero mean random variables, respectively (not necessarily identically distributed). For any X i:-a ib i,i=1,…,n, we might additionally enhance Hoeffding’s two-sided bound. This is because P(-Snt) should raise the one-sided bound, forcing left-skewed intervals to become right-skewed and vice versa.
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