Sharp tail distribution estimates for the supremum of a class of sums of i.i.d. random variables

We take a class of functions FF with polynomially increasing covering numbers on a measurable space (X,X)(X,X) together with a sequence of i.i.d. XX-valued random variables ξ1,…,ξnξ1,…,ξn, and give a good estimate on the tail behaviour of supf∈F∑j=1nf(ξj) if the relations supx∈X|f(x)|≤1supx∈X|f(x)|≤1, Ef(ξ1)=0Ef(ξ1)=0 and Ef(ξ1)2<σ2Ef(ξ1)2<σ2 hold with some 0≤σ≤10≤σ≤1 for all f∈Ff∈F. Roughly speaking this estimate states that under some natural conditions the above supremum is not much larger than the largest element taking part in it. The proof heavily depends on the main result of paper Major (2015). We also present an example that shows that our results are sharp, and compare them with results of earlier papers.