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Records for time-dependent stationary Gaussian sequences

Part of: Stochastic processes

Published online by Cambridge University Press:  04 May 2020

Michael Falk ,
Amir Khorrami Chokami  and
Simone A. Padoan
Michael Falk*
Affiliation:
University of Würzburg
Amir Khorrami Chokami*
Affiliation:
Polytechnic of Turin
Simone A. Padoan*
Affiliation:
Bocconi University
*
*Postal address: Institute of Mathematics, University of Würzburg, Am Hubland, D-97074, Würzburg, Germany. Email address: [email protected]
**Postal address: Department of Mathematical Sciences, Polytechnic of Turin, Corso Duca degli Abruzzi 24, 10129, Turin, Italy. Email address: [email protected]
***Postal address: Department of Decision Sciences, Bocconi University of Milan, via Roentgen 1, 20136 Milano, Italy. Email address: [email protected]
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Abstract

For a zero-mean, unit-variance stationary univariate Gaussian process we derive the probability that a record at the time n, say $X_n$ , takes place, and derive its distribution function. We study the joint distribution of the arrival time process of records and the distribution of the increments between records. We compute the expected number of records. We also consider two consecutive and non-consecutive records, one at time j and one at time n, and we derive the probability that the joint records $(X_j,X_n)$ occur, as well as their distribution function. The probability that the records $X_n$ and $(X_j,X_n)$ take place and the arrival time of the nth record are independent of the marginal distribution function, provided that it is continuous. These results actually hold for a strictly stationary process with Gaussian copulas.

Keywords

Arrival timeclosed skew-normal distributionGaussian processincrementsjoint recordsrecordstrictly stationary process

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G10: Stationary processes 60G40: Stopping times; optimal stopping problems; gambling theory 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 78 - 96
Copyright
© Applied Probability Trust 2020

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