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Extremal behaviour of models with multivariate random recurrence representation. (English) Zbl 1118.60060
The authors consider a \(q\)-dimensional stochastic recurrence equation \(Y_{n}=A_{n}Y_{n-1}+{\zeta}_{n}\), \(n\in {\mathbb N}\), for some iid sequence \(\{(A_{n},{\zeta}_{n})\}_{n\in{\mathbb N}}\) of random \(q\times q\)-matrices \(A_{n}\) and \(q\)-dimensional vectors \({\zeta}_{n}\). The results of the paper concern the extremal behavior of the process \(y_{n}=z_{*}^{\prime} Y_{n}\), \(n\in{\mathbb N}\), for \(z_{*}\in{\mathbb R}^{q}\) with \(| z_{*}| =1\). They can be considered as an extension of results by L. de Haan, S. I. Resnick, H. Rootzén and C. G. de Vries [Stochastic Processes Appl. 32, No. 2, 213–224 (1989; Zbl 0679.60029)] and B. Basrak, R. A. Davis and T. Mikosch [ibid. 99, No. 1, 95–115 (2002; Zbl 1060.60033)] to the case where \(A_{n}\) can be general \(q\times q\)-matrices. The paper starts with results on the existence of a stationary solution of the process \(\{Y_{n}\}_{n\in{\mathbb N}}\). The main results present a description of the limit distribution of properly normalized running maxima \(M_{n}=\max\{y_{1},\dots,y_{n}\}\), and give an explicit representation of the compound Poisson limit of the point processes of exceedances of \(\{y_{n}\}_{n\in{\mathbb N}}\) over high thresholds in terms of its Poisson intensity and its jump distribution which represents the cluster behavior of such models on high levels. The obtained results are complemented by examples showing their consequences for a random coefficient autoregressive process.

MSC:
60J05 Discrete-time Markov processes on general state spaces
60H25 Random operators and equations (aspects of stochastic analysis)
60G70 Extreme value theory; extremal stochastic processes
90B05 Inventory, storage, reservoirs
91B84 Economic time series analysis
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[1] Basrak, B.; Davis, R.D.; Mikosch, T., A characterization of multivariate regular variation, Ann. appl. probab., 12, 908-920, (2002) · Zbl 1070.60011
[2] Basrak, B.; Davis, R.D.; Mikosch, T., Regular variation of GARCH processes, Stochastic. process. appl., 99, 95-115, (2002) · Zbl 1060.60033
[3] Borkovec, M., Extremal behaviour of the autoregressive process with ARCH (1) errors, Stochastic. process. appl., 85, 189-207, (2000) · Zbl 0991.62069
[4] Diaconis, P.; Freedman, D., Iterated random functions, SIAM rev., 41, 45-76, (1999) · Zbl 0926.60056
[5] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling extremal events for insurance and finance, (1997), Springer Berlin · Zbl 0873.62116
[6] Feigin, P.D.; Tweedie, R.D., Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments, J. time ser. anal., 6, 1-14, (1985) · Zbl 0572.62069
[7] Goldie, C.M., Implicit renewal theory and tails of solutions of random equations, Ann. appl. probab., 1, 126-166, (1991) · Zbl 0724.60076
[8] Goldie, C.M.; Maller, R., Stability of perpetuities, Ann. probab., 28, 1195-1218, (2000) · Zbl 1023.60037
[9] Haan, L.; de Resnick, S.I.; Rootzén, H.; de Vries, C.G., Extremal behaviour of solutions to a stochastic difference equation, with applications to ARCH processes, Stochastic. process. appl., 32, 213-224, (1989) · Zbl 0679.60029
[10] Hult, H.; Lindskog, F., On kesten’s counterexample to the cramér – wold device for regular variation, Bernoulli, 12, 1, 133-142, (2006) · Zbl 1108.60015
[11] Kallenberg, O., Random measures, (1983), Akademie-Verlag Berlin · Zbl 0288.60053
[12] Kesten, H., Random difference equations and renewal theory for products of random matrices, Acta math., 131, 207-248, (1973) · Zbl 0291.60029
[13] Klüppelberg, C.; Pergamenchtchikov, S., Renewal theory for functionals of a Markov chain with compact state space, Ann. probab., 31, 2270-2300, (2003) · Zbl 1048.60065
[14] Klüppelberg, C.; Pergamenchtchikov, S., The tail of the stationary distribution of a random coefficient AR(\(q\)) model, Ann. appl. probab., 14, 971-1005, (2004) · Zbl 1094.62114
[15] Leadbetter, M.R.; Lindgren, G.; Rootzén, H., Extremes and related properties of random sequences and processes, (1983), Springer New York · Zbl 0518.60021
[16] Meyn, S.; Tweedie, R., Markov chains and stochastic stability, (1993), Springer New York · Zbl 0925.60001
[17] Mikosch, T.; Starica, C., Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process, Ann. statist., 28, 1427-1451, (2000), An extended version is available at · Zbl 1105.62374
[18] E. Le Page, Théorèmes de renouvellement pour les produits de matrixs aléatoires. equations aux différences aléatoires, in: Publ. Sém. Math., Université de Rennes, 1983
[19] Resnick, S.I., Extreme values, regular variation, and point processes, (1987), Springer New York · Zbl 0633.60001
[20] Resnick, S., On the foundations of multivariate heavy-tail analysis, J. appl. probab., 41A, 191-212, (2004) · Zbl 1049.62056
[21] Rootzén, H., Maxima and exceedances of stationary Markov chains, Ann. appl. prob., 20, 371-390, (1988) · Zbl 0654.60023
[22] De Sapporta, B.; Guivarc’h, Y.; Le Page, E., On the multidimensional stochastic equation \(Y(n + 1) = a(n) Y(n) + b(n)\), C. R. acad. sci., 339, 7, 499-502, (2004)
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