Minima of sequences of Gaussian random variables

Abstract

For a given sequence of real numbers a 1 , … , a n we denote the k-th smallest one by k - min 1 ⩽ i ⩽ n a i . We show that there exist two absolute positive constants c and C such that for every sequence of positive real numbers x 1 , … , x n and every k ⩽ 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 g i | ⩽ C ln ( k + 1 ) max 1 ⩽ j ⩽ k k + 1 − j ∑ i = j n 1 / x i , where g i ∈ N ( 0 , 1 ) , i = 1 , … , n , are independent Gaussian random variables. Moreover, if k = 1 then the left hand side estimate does not require independence of the g i s. Similar estimates hold for E k - min 1 ⩽ i ⩽ n | x i g i | p as well. To cite this article: Y. Gordon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).