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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Record times


Authors: J. Galambos and E. Seneta
Journal: Proc. Amer. Math. Soc. 50 (1975), 383-387
MSC: Primary 60F05
DOI: https://doi.org/10.1090/S0002-9939-1975-0368111-2
MathSciNet review: 0368111
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Abstract: Let $ {X_1},{X_2}, \cdots $ be independent and identically distributed random variables with a continuous distribution function, $ L(n) \geq 2$ is called a record time if $ {X_{L(n)}}$ is strictly larger than any previous $ {X_j}$, and we put $ L(1) = 1$. The sequence $ 1 = L(1) < L(2) < L(3) < \cdots $ of record times is a strictly increasing sequence of random variables. In the present note we investigate the sequence $ \{ L(n)\} $ through the ratios $ U(n) = L(n)/L(n - 1),n \geq 2$. We use an integer valued approximation $ T(n)$ to $ U(n)$, defined as the smallest integer such that $ U(n) \leq T(n)$. These approximations turn out to be independent and identically distributed. This fact makes it possible to deduce several limit laws for $ U(n)$ and for $ \Delta (n) = L(n) - L(n - 1),n \geq 2$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0368111-2
Keywords: Continuous distribution, record times, ratios of record times, limit laws, maximum
Article copyright: © Copyright 1975 American Mathematical Society