On the minimum of several random variables

For a given sequence of real numbers a₁, ..., a_n, we denote the kth smallest one by k-min_1≤i≤na_i. 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 real numbers 0 < x₁ ≤ ... ≤ x_n and every k ≤ n, one has c max_1≤j≤k k + 1 - j/∑_i=jⁿ 1/x _i ≤ double-struck E sign k- min_1≤i≤n |x _iξ_i| ≤ C ln(k + 1) max_1≤j≤k k + 1 - j/∑_i=jⁿ 1/x_i, where ξ₁, ..., ξ_n are independent random variables from the class A. Moreover, if k = 1, then the left-hand side estimate does not require independence of the ξ_i's. We provide similar estimates for the moments of k-min _1≤i≤n|x_iξ_i| as well.