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Convergence of lower records and infinite divisibility

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

Arup Bose ,
Sreela Gangopadhyay ,
Anish Sarkar  and
Arindam Sengupta
Arup Bose*
Affiliation:
Indian Statistical Institute, Kolkata
Sreela Gangopadhyay*
Affiliation:
Indian Statistical Institute, Kolkata
Anish Sarkar*
Affiliation:
Indian Statistical Institute, Delhi
Arindam Sengupta*
Affiliation:
Indian Institute of Technology, Guwahati
*
∗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, 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.
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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.

Keywords

Lower recordgeometric distributioninfinite divisibilityasymptotic distributionLévy measureself-decomposablegeneralized gamma convolution

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms 62E15: Exact distribution theory 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 865 - 880
Copyright
Copyright © Applied Probability Trust 2003 

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References

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