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Bivariate Limit Theorems for Record Values Based on Random Sample Sizes

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Abstract

In this paper, the class of limit distribution functions (df’s) of the joint upper record values with random sample size is fully characterized. Necessary and sufficient conditions, as well as the domains of attraction of the limit df’s are obtained. As an application of this result, the sufficient conditions for the weak convergence of the random of record quasi-ranges, record quasi-midranges, record extremal quasi-quotients and record extremal quasi-products are obtained. Moreover, the classes of the non-degenerate limit df’s of these statistics are derived.

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Acknowledgments

The authors are grateful to the Editor-in-Chief, Professor Dipak K. Dey, and the referees for suggestions and comments that improved the presentation substantially.

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Correspondence to M. A. Abd Elgawad.

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Elgawad, M.A.A., Barakat, H.M. & Yan, T. Bivariate Limit Theorems for Record Values Based on Random Sample Sizes. Sankhya A 82, 50–67 (2020). https://doi.org/10.1007/s13171-019-00167-2

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Keywords and phrases.

  • Weak convergence
  • Random sample size
  • Joint record values
  • Record functions

AMS (2000) subject classification.

  • Primary 60F05
  • 62E20
  • Secondary 62E15