Uniform estimates for order statistics and Orlicz functions
نویسندگان
چکیده
We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ1, . . . , ξn with log-concave distribution and scalars x1, . . . , xn, for every k ≤ n we provide estimates for E kmin1≤i≤n |xiξi| and E k-max1≤i≤n |xiξi| in terms of the value k and the appropriate Orlicz norm ‖(1/x1, . . . , 1/xn)‖M , associated with the distribution function of the random variable |ξ1|. For example, if ξ1 is the standard N(0, 1) Gaussian random variable, then the corresponding Orlicz function is M(s) = √ 2 π ∫ s 0 e − 1 2t2 dt. We would like to emphasize that our estimates do not depend on the length n of the sequence.
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