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Metrika

, Volume 74, Issue 1, pp 33–54 | Cite as

A Cramér-type large deviation theorem for sums of functions of higher order non-overlapping spacings

  • Sherzod M. MirakhmedovEmail author
  • Syed Ikram A. Tirmizi
  • Muhammad Naeem
Article

Abstract

Let U 1, U 2, . . . , U n–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as \({G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}\) with notation U 0 = 0, U n = 1, where \({N^\prime=\left\lfloor n/s\right\rfloor}\) is the integer part of n/s. Let \({ N=\left\lceil n/s\right\rceil}\) be the smallest integer greater than or equal to n/s, f m (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic \({f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}\) is proved.

Keywords

Cramér condition Exponential distribution Uniform spacings Uniform distribution Large deviation 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Sherzod M. Mirakhmedov
    • 1
    Email author
  • Syed Ikram A. Tirmizi
    • 1
  • Muhammad Naeem
    • 1
  1. 1.Ghulam Ishaq Khan Institute of EngineeringScience and TechnologyTopiPakistan

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