On geometric records: rate of appearance and magnitude
| Autor: | Raúl Gouet, Gerardo Sanz, F. J. Lopez |
|---|---|
| Rok vydání: | 2012 |
| Předmět: |
Statistics and Probability
Discrete mathematics Sequence media_common.quotation_subject Asymptotic distribution Magnitude (mathematics) Statistical and Nonlinear Physics Continuous mapping theorem Infinity Distribution function Limit (mathematics) Statistics Probability and Uncertainty Extreme value theory Mathematics media_common |
| Zdroj: | JOURNAL OF STATISTICAL MECHANICS Artículos CONICYT CONICYT Chile instacron:CONICYT |
| ISSN: | 1742-5468 |
| Popis: | We study the long-term behavior of geometric records from a sequence {Xn}n ? 1 of independent, nonnegative, random observations, with common continuous distribution function F. Given a parameter k > 1, the nth observation Xn is a geometric record if Xn > kmax{X1,..., Xn ? 1}, that is, if Xn is k times greater than all preceding observations. This concept was introduced by Eliazar in 2005 (Physica A 348 181), where the question of waiting times was addressed. We consider the number Nn of geometric records among X1,..., Xn, and show that Nn increases to a finite random limit , for very light-tailed F. For medium and heavy-tailed F, we prove that Nn diverges to infinity, establish its growth rate and give conditions for asymptotic normality. We also analyze the magnitude of geometric records, pointing out an unexpected relationship with models of paralyzable counters in particle physics. Our results are presented in a discrete-time setting but we show how they can be translated into continuous time. Examples of applications to common families of distributions, such as Fr?chet systems, are also provided. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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