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Bivariate Limit Theorems for Record Values Based on Random Sample Sizes

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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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  1. Ahsanullah, M. (1995). Record Statistics. Nova science publishers, Inc. 6.

  2. Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. Wiley, New York.

  3. Barakat, H. M. (2007). Measuring the asymptotic dependence between generalized order statistics. J. Statist. Theory App. (JSTA).6, 2, 106–117.

  4. Barakat, H. M. (2012). Asymptotic behavior of the record value sequence. J. Korean Statist. Soc41, 369–374.

  5. Barakat, H. M. and ABD Elgawad, M. A. (2017a). Asymptotic behavior of the joint record values, with applications. Statist. Probab. Lett.124, 13–21.

  6. Barakat, H. M., ABD Elgawad, M. A. and Yan, T. (2017b). Asymptotic behavior of record values with random indices. ProbStat. Forum10, 16–22.

  7. Barlevy, G. and Nagaraja, H. N. (2006). Characterizations in a random record model with a nonidentically distributed initial record. J. Appl. Probab.43, 1119–1136.

  8. Feller, W. (1979). An Introduction to Probability Theory and its Applications. Vol. 2, John Wiley & Sons. Inc. (Wiley Eastern University Edition).

  9. Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd edn. FL, Krieger.

  10. Resnick, S. I. (1973). Limit laws for record values. Stochastic Process. Appl.1, 67–82.

  11. Silvestrov, D.S. (2004). Limit Theorems for Randomly Stopped Stochastic Processes. Springer, Berlin.

  12. Tata, M. N. (1969). On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb12, 9–20.

  13. Yang, M. C. K. (1975). On the distribution of the inter-record times in an increasing population. J. Appl. Probab.12, 148–154.

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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