On the minimum of several random variables

Abstract

For a given sequence of real numbers a 1 , … , a n a_{1}, \dots , a_{n} , we denote the k k th smallest one by k - min 1 ≤ i ≤ n a i {k\mbox {-}\min } _{1\leq i\leq n}a_{i} . Let A \mathcal {A} be a class of random variables satisfying certain distribution conditions (the class contains N ( 0 , 1 ) N(0, 1) Gaussian random variables). We show that there exist two absolute positive constants c c and C C such that for every sequence of real numbers 0 > x 1 ≤ … ≤ x n 0> x_{1}\leq \ldots \leq x_{n} and every k ≤ n k\leq n , one has \[ c max 1 ≤ j ≤ k   k + 1 − j ∑ i = j n 1 / x i ≤ E k - min 1 ≤ i ≤ n | x i ξ i | ≤ C ln ⁡ ( k + 1 ) max 1 ≤ j ≤ k   k + 1 − j ∑ i = j n 1 / x i , c \max _{1 \leq j \leq k}\ \frac {k+1-j}{\sum _{i=j}^n 1/x_i } \leq \mathbb E \, \, k\mbox {-}\min _{1\leq i\leq n} |x_{i} \xi _{i}| \leq C\, \ln (k+1)\, \max _{1 \leq j \leq k}\ \frac {k+1-j}{\sum _{i=j}^n 1/x_i}, \] where ξ 1 , … , ξ n \xi _1, \dots , \xi _n are independent random variables from the class A \mathcal {A} . Moreover, if k = 1 k=1 , then the left-hand side estimate does not require independence of the ξ i \xi _i ’s. We provide similar estimates for the moments of k - min 1 ≤ i ≤ n | x i ξ i | {k\mbox {-}\min }_{1\leq i\leq n} |x_{i} \xi _{i}| as well.