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On multivariate records from random vectors with independent components

Part of: Stochastic processes Distribution theory - Probability Multivariate analysis

Published online by Cambridge University Press:  28 March 2018

M. Falk ,
A. Khorrami Chokami  and
S. A. Padoan
M. Falk*
Affiliation:
University of Würzburg
A. Khorrami Chokami*
Affiliation:
Bocconi University of Milan
S. A. Padoan*
Affiliation:
Bocconi University of Milan
*
* Postal address: Institute of Mathematics, University of Würzburg, Am Hubland, D-97074, Würzburg, Germany. Email address: michael.falk@uni-wuerzburg.de
** Postal address: Department of Decision Sciences, Bocconi University of Milan, via Roentgen 1, 20136 Milano, Italy.
** Postal address: Department of Decision Sciences, Bocconi University of Milan, via Roentgen 1, 20136 Milano, Italy.
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Abstract

Let X1, X2, . . . be independent copies of a random vector X with values in ℝd and a continuous distribution function. The random vector Xn is a complete record, if each of its components is a record. As we require X to have independent components, crucial results for univariate records clearly carry over. But there are substantial differences as well. While there are infinitely many records in the d = 1 case, they occur only finitely many times in the series if d ≥ 2. Consequently, there is a terminal complete record with probability 1. We compute the distribution of the random total number of complete records and investigate the distribution of the terminal record. For complete records, the sequence of waiting times forms a Markov chain, but unlike the univariate case now the state at ∞ is an absorbing element of the state space.

Keywords

Multivariate recordcomplete recordterminal recordwaiting timeMarkov chain

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60E05: Distributions 62H05: Characterization and structure theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 43 - 53
Copyright
Copyright © Applied Probability Trust 2018 

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References

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