Sharp estimate on the supremum of a class of sums of small i.i.d. random variables

We take a class of functions FF with polynomial covering numbers on a measurable space (X,X)(X,X) together with a sequence of independent, identically distributed XX-space valued random variables ξ1,…,ξnξ1,…,ξn, and give a good estimate on the tail distribution of supf∈F∑j=1nf(ξj) if the expected values E|f(ξ1)|E|f(ξ1)| are very small for all f∈Ff∈F. In a subsequent paper (Major, in press) we give a sharp bound for the supremum of normalized sums of i.i.d. random variables in a more general case. But the proof of that estimate is based on the results in this work.