On the minimum of several random variables

For a given sequence of real numbers a1, . . . , an we denote the k-th smallest one by k-min1≤i≤n ai. Let A be a class of random variables satisfying certain distribution conditions (the class contains N(0, 1) Gaussian random variables). We show that there exist two absolute positive constants c and C such that for every sequence of positive real numbers x1, . . . , xn and every k ≤ n one has c max 1≤j≤k k + 1− j ∑n i=j 1/xi ≤ E kmin 1≤i≤n |xiξi| ≤ C ln(k + 1) max 1≤j≤k k + 1− j ∑n i=j 1/xi , where ξ1, . . . , ξn are independent random variables from the class A. Moreover, if k = 1 then the left hand side estimate does not require independence of the ξi’s. We provide similar estimates for the moments of k-min1≤i≤n |xiξi| as well.