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Oscillations for order statistics of some discrete processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 September 2020

Andrea Ottolini
Andrea Ottolini*
Affiliation:
Stanford University
*
*Postal address: 450 Jane Stanford Way, Building 380, Stanford, CA94305-2125, USA. Email: ottolini@stanford.edu
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Abstract

Suppose k balls are dropped into n boxes independently with uniform probability, where n, k are large with ratio approximately equal to some positive real $\lambda$ . The maximum box count has a counterintuitive behavior: first of all, with high probability it takes at most two values $m_n$ or $m_n+1$ , where $m_n$ is roughly $\frac{\ln n}{\ln \ln n}$ . Moreover, it oscillates between these two values with an unusual periodicity. In order to prove this statement and various generalizations, it is first shown that for $X_1,\ldots,X_n$ independent and identically distributed discrete random variables with common distribution F, under mild conditions, the limiting distribution of their maximum oscillates in three possible families, depending on the tail of the distribution. The result stated at the beginning follows from the ensemble equivalence for the order statistics in various allocations problems, obtained via conditioning limit theory. Results about the number of ties for the maximum, as well as applications, are also provided.

Keywords

Extreme value theoryrandom allocationsconditioning limit theory

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 703 - 719
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Copyright
© Applied Probability Trust 2020

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