Sharp probability estimates for generalized Smirnov statistics
نویسندگان
چکیده
منابع مشابه
Sharp Probability Estimates for Generalized Smirnov Statistics
We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0, 1] random variables stays to one side of a given line.
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(1.2) log2 pj ≤ αj + β (1 ≤ j ≤ ω(n)), where α ≥ 0 and log2 y denotes log log y. The distribution of integers satisfying (1.1) is important in the study of the distribution of divisors of integers (see [3]; Ch. 2 of [4]). We present here estimates for Nk(x;α, β) = #{n ≤ x : ω(n) = k, (1.1)}, Mk(x;α, β) = #{n ≤ x : ω(n) = k, (1.2)}. It is a relatively simple matter, at least heuristically, to re...
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Abstract: Let N be a (large) positive integer, let b be an integer satisfying 1 < b < N that is relatively prime to N , and let r be the order of b modulo N . Finally, let QC be a quantum computer whose input register has the size specified in Shor’s original description of his order-finding algorithm. We prove that when Shor’s algorithm is implemented on QC, then the probability P of obtaining...
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Sharp lower and upper bounds for quasiconvex moments of generalized order statistics are proven by the use of the rearranged Moriguti’s inequality. Even in the second moment case, the method yields improvements of known quantile and moment bounds for the expectation of order and record statistics based on independent identically distributed random variables. The bounds are attainable providing ...
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ژورنال
عنوان ژورنال: Monatshefte für Mathematik
سال: 2008
ISSN: 0026-9255,1436-5081
DOI: 10.1007/s00605-007-0516-y