Theoretical Statistics and Mathematics Unit, ISI Delhi

Convergence of Lower Records and Infinite divisiblity

by Arup Bose, Sreela Gangopadhyay, Anish Sarkar and Arindam Sengupta

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

isid/ms/2003/09 [fulltext]

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