Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Joint exceedances of random products

Anja Janßen and Holger Drees

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Abstract

We analyze the joint extremal behavior of $n$ random products of the form $\prod_{j=1}^{m}X_{j}^{a_{ij}}$, $1\leq i\leq n$, for non-negative, independent regularly varying random variables $X_{1},\ldots,X_{m}$ and general coefficients $a_{ij}\in\mathbb{R}$. Products of this form appear for example if one observes a linear time series with gamma type innovations at $n$ points in time. We combine arguments of linear optimization and a generalized concept of regular variation on cones to show that the asymptotic behavior of joint exceedance probabilities of these products is determined by the solution of a linear program related to the matrix $\mathbf{A}=(a_{ij})$.

Résumé

Nous étudions le comportement extrémal multivarié de $n$ produits aléatoires de la forme $\prod_{j=1}^{m}X_{j}^{a_{ij}}$, $1\leq i\leq n$, pour des variables aléatoires positives, indépendantes et identiquement distribuées $X_{1},\ldots,X_{m}$ et des coefficients $a_{ij}\in\mathbb{R}$ quelconques. De tels produits apparaissent notamment lorsqu’on observe un échantillon de taille $n$ issu d’une série temporelle linéaire dont les innovations sont de type gamma. En combinant des arguments d’optimisation linéaire et le concept de variations régulières étendu à des cônes, nous montrons que le comportement asymptotique des probabilités de dépassement de seuil multiples pour de tels produits est déterminé par la solution d’un problème de programmation linéaire associé à la matrice $\mathbf{A}=(a_{ij})$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 437-465.

Dates
Received: 4 August 2015
Revised: 3 November 2016
Accepted: 27 November 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030835

Digital Object Identifier
doi:10.1214/16-AIHP811

Mathematical Reviews number (MathSciNet)
MR3765896

Zentralblatt MATH identifier
06880061

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60B10: Convergence of probability measures 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]

Keywords
Extreme value theory Linear programming M-convergence Random products Regular variation

Citation

Janßen, Anja; Drees, Holger. Joint exceedances of random products. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 437--465. doi:10.1214/16-AIHP811. https://projecteuclid.org/euclid.aihp/1519030835


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