On the limit behaviour of the counting process of -records and ties for the maximum in discrete random sequences

On the limit behaviour of the counting process of -records and ties for the maximum in discrete random sequences

Autores: Raúl Gouet Bañares , Francisco Javier López Lorente , Gerardo Sanz Sáiz
Localización: XXX Congreso Nacional de Estadística e Investigación Operativa y de las IV Jornadas de Estadística Pública: actas, 2007, ISBN 978-84-690-7249-3
Idioma: inglés
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Resumen
- Given a sequence of independent, identically distributed random variables, taking nonnegative integer values, we consider the asymptotic behavior of the counting processes of -records (observations greater than the previous maximum plus a fixed real number ) and ties for the maximum (observations equal to the previous maximum).
  
  We pay special attention to weak records, because of their direct relation with ties for the maximum. Although discrete records and weak records have attracted attention in recent years, most papers are focused on record values rather than their counting process.
  
  Using a martingale approach, we obtain laws of large numbers and central limit theorems. The conditions for the existence of such limits as well as the normalizing sequences depend on the tail behavior of the distribution of the underlying random variable. Applications to well-known distributions are given.