TY - JOUR
T1 - On the minimum of several random variables
AU - Gordon, Y.
AU - Litvak, A. E.
AU - Schütt, C.
AU - Werner, E.
PY - 2006/12
Y1 - 2006/12
N2 - 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.
AB - 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.
KW - Expectations
KW - Exponential distribution
KW - Moments
KW - Normal distribution
KW - Order statistics
UR - http://www.scopus.com/inward/record.url?scp=33845618694&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=33845618694&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-06-08453-X
DO - 10.1090/S0002-9939-06-08453-X
M3 - Article
AN - SCOPUS:33845618694
SN - 0002-9939
VL - 134
SP - 3665
EP - 3675
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 12
ER -