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One- versus multi-component regular variation and extremes of Markov trees

Published online by Cambridge University Press:  24 September 2020

Johan Segers Open the ORCID record for Johan Segers [Opens in a new window]
Johan Segers*
Affiliation:
Université catholique de Louvain
*
*Postal address: Université catholique de Louvain, LIDAM/ISBA, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. Email: [email protected]
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Abstract

A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

Keywords

Conditional independencegraphical modelHüsler–Reiss distributionmax-linear modelMarkov treemultivariate Pareto distributionPickands dependence functionregular variationroot change formulatail measuretail treetime change formula
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 855 - 878
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Copyright
© Applied Probability Trust 2020

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References

Asadi, P., Davison, A. C. and Engelke, S. (2015). Extremes on river networks. Ann. Appl. Statist. 9, 2023–2050.10.1214/15-AOAS863CrossRefGoogle Scholar
Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stoch. Proc. Appl. 119, 1055–1080.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Bortot, P. and Coles, S. (2000). A sufficiency property arising from the characterization of extremes of a Markov chain. Bernoulli 6, 183–190.10.2307/3318638CrossRefGoogle Scholar
Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. R. Statist. Soc. B [Statist. Methodology] 53, 377–392.Google Scholar
Das, B. and Resnick, S. I. (2011). Conditioning on an extreme component: Model consistency with regular variation on cones. Bernoulli 17, 226–252.CrossRefGoogle Scholar
Dombry, C., Hashorva, E. and Soulier, P. (2018). Tail measure and spectral tail process of regularly varying time series. Ann. Appl. Prob. 28, 3884–3921.10.1214/18-AAP1410CrossRefGoogle Scholar
Dombry, C. and Ribatet, M. (2015). Functional regular variations, Pareto processes and peaks over threshold. Statist. Interface 8, 9–17.CrossRefGoogle Scholar
Einmahl, J. H. J., Kiriliouk, A. and Segers, J. (2018). A continuous updating weighted least squares estimator of tail dependence in high dimensions. Extremes 21, 205–233.10.1007/s10687-017-0303-7CrossRefGoogle Scholar
Engelke, S. and Hitz, A. S. (2019). Graphical models for extremes. To appear in J. R. Statist. Soc. B [Statist. Methodology]. Preprint available at https://arxiv.org/abs/1812.01734 .Google Scholar
Ferreira, A. and de Haan, L. (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli 20, 1717–1737.CrossRefGoogle Scholar
Garralda Guillem, A. I. (2000). Structure de dépendance des lois de valeurs extrêmes bivariées. C. R. Acad. Sci. Paris 330, 593–596.CrossRefGoogle Scholar
Gissibl, N. and Klüppelberg, C. (2018). Max-linear models on directed acyclic graphs. Bernoulli 24, 2693–2720.CrossRefGoogle Scholar
Gissibl, N., Klüppelberg, C. and Otto, M. (2018). Tail dependence of recursive max-linear models with regularly varying noise variables. Econometrics Statist. 6, 149–167.CrossRefGoogle Scholar
Gudendorf, G. and Segers, J. (2010). Extreme-value copulas. In Copula Theory and Its Applications, eds. P. Jaworski, F. Durante, W. K. Härdle, and T. Rychlik, Springer, Berlin, Heidelberg, pp. 127–145.10.1007/978-3-642-12465-5_6CrossRefGoogle Scholar
Heffernan, J. E. and Resnick, S. I. (2007). Limit laws for random vectors with an extreme component. Ann. Appl. Prob. 17, 537–571.CrossRefGoogle Scholar
Hitz, S. A. and Evans, J. R. (2016). One-component regular variation and graphical modeling of extremes. 53, 733–746.10.1017/jpr.2016.37CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N. S.) 80, 121–140.10.2298/PIM0694121HCrossRefGoogle Scholar
Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Prob. Lett. 7, 283–286.CrossRefGoogle Scholar
Janssen, A. (2018). Spectral tail processes and max-stable approximations of multivariate regularly varying time series. Stoch. Proc. Appl. 129, 1993–2009.10.1016/j.spa.2018.06.010CrossRefGoogle Scholar
Janssen, A. and Segers, J. (2014). Markov tail chains. J. Appl. Prob. 51, 1133–1153.10.1239/jap/1421763332CrossRefGoogle Scholar
Kiriliouk, A., Rootzén, H., Segers, J. and Wadsworth, J. L. (2019). Peaks over thresholds modeling with multivariate generalized Pareto distributions. Technometrics 61, 123–135.CrossRefGoogle Scholar
Lauritzen, S. L. (1996). Graphical Models. Oxford Science Publications, Oxford.Google Scholar
Lee, D. and Joe, H. (2018). Multivariate extreme value copulas with factor and tree dependence structures. Extremes 21, 147–176.CrossRefGoogle Scholar
Lindskog, F., Resnick, S. I. and Roy, J. (2014). Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Prob. Surveys 11, 270–314.CrossRefGoogle Scholar
Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Prob. 4, 529–548.CrossRefGoogle Scholar
Planinić, H. and Soulier, P. (2018). The tail process revisited. Extremes 21, 551–579.10.1007/s10687-018-0312-1CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.10.1007/978-0-387-75953-1CrossRefGoogle Scholar
Resnick, S. I. (2006). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer-Verlag, New York.Google Scholar
Resnick, S. I. and Zeber, D. (2014). Transition kernels and the conditional extreme value model. Extremes 17, 263–287.CrossRefGoogle Scholar
Rootzén, H., Segers, J. and Wadsworth, J. L. (2018). Multivariate generalized Pareto distributions: Parametrizations, representations, and properties. J. Multivariate Anal. 165, 117–131.CrossRefGoogle Scholar
Samorodnitsky, G. and Owada, T. (2012) Tail measures of stochastic processes or random fields with regularly varying tails. Preprint.Google Scholar
Segers, J. (2007). Multivariate regular variation of heavy-tailed Markov chains. Tech. rep. DP0703, Institut de statistique, Université catholique de Louvain. Available at https://arxiv.org/abs/0701411 .Google Scholar
Segers, J., Zhao, Y. and Meinguet, T. (2017). Polar decomposition of regularly varying time series in star-shaped metric spaces. Extremes 20, 539–566.10.1007/s10687-017-0287-3CrossRefGoogle Scholar
Smith, R. L. (1992). The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45.CrossRefGoogle Scholar
Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84, 249–268.CrossRefGoogle Scholar
Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar
Wainwright, M. J. and Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1, 1–305.CrossRefGoogle Scholar