On the minimum of several random variables

Y. Gordon, A. E. Litvak, C. Schütt, E. Werner

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24 Scopus citations


For a given sequence of real numbers a1, ..., an, we denote the kth smallest one by k-min1≤i≤nai. 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 ≤ ... ≤ xn and every k ≤ n, one has c max1≤j≤k k + 1 - j/∑i=jn 1/x i ≤ double-struck E sign k- min1≤i≤n |x iξi| ≤ C ln(k + 1) max1≤j≤k k + 1 - j/∑i=jn 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-min 1≤i≤n|xiξi| as well.

Original languageEnglish (US)
Pages (from-to)3665-3675
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number12
StatePublished - Dec 2006
Externally publishedYes


  • Expectations
  • Exponential distribution
  • Moments
  • Normal distribution
  • Order statistics


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