Proves E[max_j | (1/n) sum_i ε_ij |] ≥ min{255/256, (1/sqrt(2 log 2)) sqrt(log(2p)/n)} with equality for (n,p)=(2,1) and (2,8).
A full proof of universal inequalities for the distribution function of the binomial law
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abstract
We present a new form and a short full proof of explicit two-sided estimates for the distribution function F_{n,p}(x) of the binomial law from the paper published by D.Alfers and H.Dinges in 1984. These inequalities are universal (valid for all binomial distribution and all values of argument) and exact (namely, the upper bound for F_{n,p}(k) is the lower bound for F_{n,p}(k+1)). By means of such estimates it is possible to bound any quantile of the binomial law by 2 subsequent integers.
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Notes on constants for maxima of Rademacher averages
Proves E[max_j | (1/n) sum_i ε_ij |] ≥ min{255/256, (1/sqrt(2 log 2)) sqrt(log(2p)/n)} with equality for (n,p)=(2,1) and (2,8).