Advances in Applied Probability

Records from stationary observations subject to a random trend

Raúl Gouet, F. Javier López, and Gerardo Sanz

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Abstract

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)nZ 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.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 4 (2015), 1175-1189.

Dates
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1449859805

Digital Object Identifier
doi:10.1239/aap/1449859805

Mathematical Reviews number (MathSciNet)
MR3433301

Zentralblatt MATH identifier
1333.60051

Subjects
Primary: 60G10: Stationary processes 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems

Keywords
Record stationary process ergodic theorem strong convergence random trend asymptotic normality

Citation

Gouet, Raúl; López, F. Javier; Sanz, Gerardo. Records from stationary observations subject to a random trend. Adv. in Appl. Probab. 47 (2015), no. 4, 1175--1189. doi:10.1239/aap/1449859805. https://projecteuclid.org/euclid.aap/1449859805


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