A Remark on Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand Inequalities

Abstract

Let X ₁,..., X_n be independent random variables such that ℙ{X_j≤ 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality ℙ{M _n≥x}≤ 3.7 ℙ{S_n≥ x}, where S _n=ε₁+...+ ε_n is a sum of centered independent identically distributed Bernoulli random variables such that E S ²_n =ME M ²_n and ℙ{ε_k=1}=E S ²_n /(n+E S ²_n ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x∈∝ at which the survival function x↦ℙ{S _n≥x} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate ℙ{S _n≥x} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.