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Functional limit theorems for linear statistics from sequential ranks
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  • Published: November 1986

Functional limit theorems for linear statistics from sequential ranks

  • E. V. Khmaladze1 &
  • A. M. Parjanadze2 

Probability Theory and Related Fields volume 73, pages 585–595 (1986)Cite this article

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  • 11 Citations

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Summary

Let X 1 ,..., X n be a sequence of continuously distributed independent random variables. The normalized ranks R kn and sequential ranks S k , k=1,...,n, are defined by

$${\text{R}}_{{\text{kn}}} = \frac{1}{{\text{n}}}\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} ,} {\text{ S}}_{\text{k}} = \frac{1}{{\text{k}}}\sum\limits_{{\text{j = }}1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} .} $$

The subject of the present paper is the asymptotic behavior, as n→∞, of the process

$$\frac{1}{{\sqrt {\text{n}} }}\sum\limits_{{\text{k}} \leqq {\text{nt}}} {{\text{a}}({\text{S}}_{\text{k}} ),} {\text{ }}0 \leqq {\text{t}} \leqq 1,$$

for a∈L 2 (0, 1), \(\int\limits_0^1 {{\text{adn}} = 0} \). For suitable a, the limiting law of that process is expressed as solution of a stochastic equation under the hypothesis of identically distributed X 1,..., X n as well as under a class of contiguous alternatives, which contains the occurrence of a change point in the series of measurements.

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References

  1. Parjanadze, A.M.: Application of sequential ranks for a change point detection (in Russian). Bulletin of the Acad. of Sci. of the Georgian SSR. 116, 241–243 (1984)

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  2. Barndorff-Nielsen, O.: On limit behaviour of extreme order statistics. Ann. Math. Stat. 34, 992–1002 (1963)

    MATH  MathSciNet  Google Scholar 

  3. Billingsley, P.: Convergence of probability measures. New York-London-Sydney-Toronto. Wiley: 1968

    Google Scholar 

  4. Hájek, J., Šhidak, Z.: Theory of rank tests. Prague: Academia 1967

    Google Scholar 

  5. Bickel, P., Wichura, M.: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42, 1656–1670 (1971)

    MathSciNet  Google Scholar 

  6. Liptser, R.Sh., Shiryayev, A.N.: Statistics of random processes. I. General theory. Berlin Heidelberg New York: Springer 1977

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Author information

Authors and Affiliations

  1. Matematičeskii Institut im. V.A. Steklova AN SSSR, ul. Vavilova, 42, 117966 GSP-1, Moskva, USSR

    E. V. Khmaladze

  2. Matematičeskii Institut im. A.M. Razmadze AN GSSR, Plekhanov ave., 150A, 380012, Tbilisi, USSR

    A. M. Parjanadze

Authors
  1. E. V. Khmaladze
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  2. A. M. Parjanadze
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Cite this article

Khmaladze, E.V., Parjanadze, A.M. Functional limit theorems for linear statistics from sequential ranks. Probab. Th. Rel. Fields 73, 585–595 (1986). https://doi.org/10.1007/BF00324854

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  • Received: 02 September 1985

  • Issue Date: November 1986

  • DOI: https://doi.org/10.1007/BF00324854

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Keywords

  • Stochastic Process
  • Asymptotic Behavior
  • Probability Theory
  • Limit Theorem
  • Mathematical Biology
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