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Records from stationary observations subject to a random trend

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  21 March 2016

Raúl Gouet ,
F. Javier López  and
Gerardo Sanz
Raúl Gouet*
Affiliation:
Universidad de Chile
F. Javier López*
Affiliation:
Universidad de Zaragoza
Gerardo Sanz*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Dpto. Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807, CNRS), Universidad de Chile, Beauchef 851, 8370456 Santiago, Chile. Email address: [email protected]
∗∗ Postal address: Dpto. Métodos Estadísticos and BIFI, Facultad de Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna, 12, 50009 Zaragoza, Spain.
∗∗ Postal address: Dpto. Métodos Estadísticos and BIFI, Facultad de Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna, 12, 50009 Zaragoza, Spain.
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Abstract

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We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Yn = Xn + Tn, n ≥ 1, where (Xn)n ∈ Z is a stationary ergodic sequence of random variables and (Tn)n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

Keywords

Recordstationary processergodic theoremstrong convergencerandom trendasymptotic normality

MSC classification

Primary: 60G10: Stationary processes 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 1175 - 1189
Copyright
Copyright © Applied Probability Trust 2015 

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