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Convergence of lower records and infinite divisibility

  • Arup Bose (a1), Sreela Gangopadhyay (a1), Anish Sarkar (a2) and Arindam Sengupta (a3)

Abstract

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

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Corresponding author

Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, 203 B.T. Road, Kolkata 700108, India.
∗∗ Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India. Email address: anish@isid.ac.in
∗∗∗ Postal address: Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India.

References

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[1] Arnold, B. C. and Villaseñor, J. A. (1999). The asymptotic distributions of sums of records. Extremes 1, 351363.
[2] Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities (Lecture Notes Statist. 76). Springer, New York.
[3] Bose, A., Gangopadhyay, S., Sarkar, A., and Sengupta, A. (2001). Asymptotic properties of sums of upper records. Submitted.
[4] Chung, K. L. (1974). A Course in Probability Theory, 2nd edn. Academic Press, New York.
[5] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. (2) 44, 423453.
[6] Hudson, W. N., and Tucker, H. G. (1975). Equivalence of infinitely divisible distributions. Ann. Prob. 3, 7079.
[7] Lukacs, E. (1960). Characteristic Functions. Griffin, London.
[8] Resnick, S. I. (1964). Limit laws for record values. Stoch. Process. Appl. 1, 6782.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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