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A martingale approach to strong convergence of the number of records

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Raúl Gouet ,
F. Javier López  and
Miguel San Miguel
Raúl Gouet*
Affiliation:
Universidad de Chile
F. Javier López*
Affiliation:
Universidad de Zaragoza
Miguel San Miguel*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile. Email address: rgouet@dim.uchile.cl
∗∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
∗ Postal address: Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile. Email address: rgouet@dim.uchile.cl
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Abstract

Let (Xn) be a sequence of independent, identically distributed random variables, with common distribution function F, possibly discontinuous. We use martingale arguments to connect the number of upper records from (Xn) with sums of minima of related random variables. From this relationship we derive a general strong law for the number of records for a wide class of distributions F, including geometric and Poisson.

Keywords

Extremesmartingalesrecords

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 864 - 873
Copyright
Copyright © Applied Probability Trust 2001 

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