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On multivariate records from random vectors with independent components

  • M. Falk (a1), A. Khorrami Chokami (a2) and S. A. Padoan (a2)
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.

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Corresponding author
* 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.
*** Email address: amir.khorrami@phd.unibocconi.it
**** Email address: simone.padoan@unibocconi.it
References
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Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley, New York.
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.
Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Krieger, Melbourne, FL.
Goldie, C. M. and Resnick, S. (1989). Records in a partially ordered set. Ann. Prob. 17, 678699.
Goldie, C. M. and Resnick, S. I. (1995). Many multivariate records. Stoch. Process. Appl. 59, 185216.
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
Zott, M. (2016). Extreme value theory in higher dimensions: max-stable processes and multivariate records. Doctoral thesis. University of Würzburg.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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