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Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions

Journal of the Indian Society for Probability and Statistics volume 22, pages 357–373 (2021)Cite this article

Abstract

Let \(\left\{ X_n\right\} \) be a sequence of independent and identically distributed non-negative valued random variables defined over a common probability space and let the common distribution function F be continuous. Suppose that F belongs to the domain of partial attraction of a max-semi Frechet law. Define for any \(r\ge 1\) and \( n\ge r\), \( M_{r,[nt]}\) = the rth highest among \(\left\{ X_1,X_2,\ldots ,X_{[nt]}\right\} \) if \( \frac{r}{n} \le t \le 1,\) and = min \(\left\{ X_1,X_2,\ldots ,X_r\right\} \) if \(0\le t \le \frac{r}{n}.\) We establish below, a functional law of the iterated logarithm for the sequence \((M_{r,[nt]}, 0\le t \le 1)\), properly normalized, under the \(M_1\)-topology.

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Acknowledgements

This work is carried out as a part of the project under the emeritus fellowship of University Grants Commission (UGC), New Delhi, India. The author is grateful to UGC, India, for the support.

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  1. Department of Studies in Statistics, University of Mysore, Mysore, 570006, India

    R. Vasudeva

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  1. R. Vasudeva
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Vasudeva, R. Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions. J Indian Soc Probab Stat 22, 357–373 (2021). https://doi.org/10.1007/s41096-021-00108-z

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  • DOI: https://doi.org/10.1007/s41096-021-00108-z

Keywords

  • Max-semi Fréchet law
  • Domain of partial attraction
  • Functional law of the iterated logarithm
  • Almost sure limit function

Mathematics Subject Classification

  • 60F15
  • 60F17
  • 60G17