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Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 1, Pages 147–152 (Mi tvp3069)  

This article is cited in 3 scientific papers (total in 3 papers)

Short Communications

Limit distribution of a number of coinciding intervals

N. V. Klykova

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $X_1,…,X_T$ be independent random variables uniformly distributed on the set $\{1,…,N\}$, let $X_{(1)},…,X_{(2)}\le…\le X_{(T)}$ be their order statistics and $\zeta(T,N)$ be a number of pairs $(i,j)$, $1\le i<j\le T-1$, such that $X_{(i+1)}-X_{(i)}=X_{(j+1)}-X_{(j)}$. We give a full proof of the convergence theorem of the distribution $\zeta(T,N)$ to the Poisson distribution with parameter $\lambda$ for $T,N\to\infty$, $T^3/4N\to\lambda$. Heuristic proof of this statement was given in [D. Aldous, Probability Approximation via the Poisson Clumping Heuristic, Springer-Verlag, Berlin, Heidelberg, 1989].

Keywords: birthday problem, set of order statistics, spacings.

DOI: https://doi.org/10.4213/tvp3069

Full text: PDF file (511 kB)

English version:
Theory of Probability and its Applications, 2003, 47:1, 151–156

Bibliographic databases:

Received: 12.03.2001

Citation: N. V. Klykova, “Limit distribution of a number of coinciding intervals”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 147–152; Theory Probab. Appl., 47:1 (2003), 151–156

Citation in format AMSBIB
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\paper Limit distribution of a number of coinciding intervals
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\pages 147--152
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\jour Theory Probab. Appl.
\yr 2003
\vol 47
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\pages 151--156
\crossref{https://doi.org/S0040585X97979512}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Kruglov, “Limit distributions of the number of vectors satisfying a linear relation”, Discrete Math. Appl., 18:5 (2008), 465–481  mathnet  crossref  crossref  mathscinet  zmath  elib
    2. V. I. Kruglov, “Puassonovskaya approksimatsiya dlya raspredeleniya chisla “parallelogrammov” v sluchainoi vyborke iz $\mathbb Z_N^q$”, Matem. vopr. kriptogr., 3:2 (2012), 63–78  mathnet  crossref
    3. V. A. Kopyttsev, V. G. Mikhailov, “An estimate of the approximation accuracy in B. A. Sevastyanov's limit theorem and its application in the problem of random inclusions”, Discrete Math. Appl., 25:3 (2015), 149–156  mathnet  crossref  crossref  mathscinet  isi  elib  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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