Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T20:49:57.191Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley, New York. CrossRefGoogle Scholar
Falk, M., Khorrami Chokami, A. and Padoan, S. A. (2017). Some results on joint record events. Preprint. Available at https://arxiv.org/abs/1707.06254. Google Scholar
Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Krieger, Melbourne, FL. Google Scholar
Goldie, C. M. and Resnick, S. (1989). Records in a partially ordered set. Ann. Prob. 17, 678–699. Google Scholar
Goldie, C. M. and Resnick, S. I. (1995). Many multivariate records. Stoch. Process. Appl. 59, 185–216. Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. CrossRefGoogle Scholar
Zott, M. (2016). Extreme value theory in higher dimensions: max-stable processes and multivariate records. Doctoral thesis. University of Würzburg. Google Scholar