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On records and related processes for sequences with trends

Part of: Special processes Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

K. Borovkov
K. Borovkov*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville 3052, Australia. Email address: kostya@ams.unimelb.edu.au.
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Abstract

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.

Keywords

Sequence with linear trendrecordsMarkov chains

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60K99: None of the above, but in this section

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 668 - 681
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported by ARC Grant S69711423.

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