# zbMATH — the first resource for mathematics

Extremes of $$q$$-Ornstein-Uhlenbeck processes. (English) Zbl 1405.60072
Summary: Two limit theorems are established on the extremes of a family of stationary Markov processes, known as $$q$$-Ornstein-Uhlenbeck processes with $$q \in(- 1, 1)$$. Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown-Resnick-type limit theorem on the minimum process of i.i.d. copies of the $$q$$-Ornstein-Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-$$\min$$-stable process arises in the limit.

##### MSC:
 60G70 Extreme value theory; extremal stochastic processes 60J25 Continuous-time Markov processes on general state spaces
GAUSSIAN
Full Text:
##### References:
 [1] Albin, J. M.P., On extremal theory for stationary processes, Ann. Probab., 18, 1, 92-128, (1990) · Zbl 0704.60029 [2] Aldous, D., (Probability Approximations Via the Poisson Clumping Heuristic, Applied Mathematical Sciences, vol. 77, (1989), Springer-Verlag New York) · Zbl 0679.60013 [3] Biane, P., Processes with free increments, Math. Z., 227, 1, 143-174, (1998) · Zbl 0902.60060 [4] Billingsley, P., (Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, (1999), John Wiley & Sons Inc., A Wiley-Interscience Publication New York) [5] Blumenthal, R. M.; Getoor, R. K., (Markov Processes and Potential Theory, Pure and Applied Mathematics, vol. 29, (1968), Academic Press New York-London) · Zbl 0169.49204 [6] Bozejko, M.; Kümmerer, B.; Speicher, R., $$q$$-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys., 185, 1, 129-154, (1997) · Zbl 0873.60087 [7] Brown, M., A property of Poisson processes and its application to macroscopic equilibrium of particle systems, Ann. Math. Statist., 41, 1935-1941, (1970) · Zbl 0233.60057 [8] Brown, B. M.; Resnick, S. I., Extreme values of independent stochastic processes, J. Appl. Probab., 14, 4, 732-739, (1977) · Zbl 0384.60055 [9] Bryc, W.; Matysiak, W.; Szabłowski, P. J., Probabilistic aspects of al-Salam-chihara polynomials, Proc. Amer. Math. Soc., 133, 4, 1127-1134, (2005), (electronic) · Zbl 1074.33015 [10] Bryc, W.; Matysiak, W.; Wesołowski, J., Quadratic harnesses, $$q$$-commutations, and orthogonal martingale polynomials, Trans. Amer. Math. Soc., 359, 11, 5449-5483, (2007) · Zbl 1129.60068 [11] Bryc, W.; Wang, Y., The local structure of $$q$$-Gaussian processes, Probab. Math. Statist., 36, 2, 235-252, (2016) · Zbl 1357.60081 [12] Bryc, W.; Wesołowski, J., Conditional moments of $$q$$-meixner processes, Probab. Theory Related Fields, 131, 3, 415-441, (2005) · Zbl 1118.60065 [13] Cheng, D.; Xiao, Y., Excursion probability of Gaussian random fields on sphere, Bernoulli, 22, 2, 1113-1130, (2016) · Zbl 1337.60102 [14] Cheng, D.; Xiao, Y., The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments, Ann. Appl. Probab., 26, 2, 722-759, (2016) · Zbl 1339.60055 [15] Dȩbicki, K.; Hashorva, E.; Ji, L., Extremes of a class of nonhomogeneous Gaussian random fields, Ann. Probab., 44, 2, 984-1012, (2016) · Zbl 1341.60044 [16] de Haan, L., A spectral representation for MAX-stable processes, Ann. Probab., 12, 4, 1194-1204, (1984) · Zbl 0597.60050 [17] de Haan, L.; Ferreira, A., (Extreme Value Theory, Springer Series in Operations Research and Financial Engineering, (2006), Springer New York), An introduction · Zbl 1101.62002 [18] de Haan, L.; Pickands III, J., Stationary MIN-stable stochastic processes, Probab. Theory Related Fields, 72, 4, 477-492, (1986) · Zbl 0577.60034 [19] Dieker, A. B.; Mikosch, T., Exact simulation of Brown-resnick random fields at a finite number of locations, Extremes, 18, 2, 301-314, (2015) · Zbl 1319.60108 [20] Dombry, C.; Eyi-Minko, F., Regular conditional distributions of continuous MAX-infinitely divisible random fields, Electron. J. Probab., 18, 7, 1-21, (2013) · Zbl 1287.60066 [21] Engelke, S.; Kabluchko, Z., MAX-stable processes and stationary systems of Lévy particles, Stochastic Process. Appl., 125, 11, 4272-4299, (2015) · Zbl 1330.60065 [22] Engelke, S.; Kabluchko, Z.; Schlather, M., An equivalent representation of the Brown-resnick process, Statist. Probab. Lett., 81, 8, 1150-1154, (2011) · Zbl 1234.60056 [23] Engelke, S.; Kabluchko, Z.; Schlather, M., Maxima of independent, non-identically distributed Gaussian vectors, Bernoulli, 21, 1, 38-61, (2015) · Zbl 1322.60073 [24] Ethier, S. N.; Kurtz, T. G., (Markov Processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986), John Wiley & Sons, Inc. New York), Characterization and convergence · Zbl 0592.60049 [25] Falconer, K. J., The local structure of random processes, J. Lond. Math. Soc. (2), 67, 3, 657-672, (2003) · Zbl 1054.28003 [26] Giné, E.; Hahn, M. G.; Vatan, P., MAX-infinitely divisible and MAX-stable sample continuous processes, Probab. Theory Related Fields, 87, 2, 139-165, (1990) · Zbl 0688.60031 [27] Hashorva, E.; Ji, L., Extremes of $$\alpha(\mathbf{t})$$-locally stationary Gaussian random fields, Trans. Amer. Math. Soc., 368, 1, 1-26, (2016) · Zbl 1408.60041 [28] Ismail, M. E.H., (Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, vol. 98, (2009), Cambridge University Press Cambridge) [29] Kabluchko, Z., Extremes of space-time Gaussian processes, Stochastic Process. Appl., 119, 11, 3962-3980, (2009) · Zbl 1181.60075 [30] Kabluchko, Z., Spectral representations of sum- and MAX-stable processes, Extremes, 12, 4, 401-424, (2009) · Zbl 1224.60120 [31] Kabluchko, Z.; Schlather, M., Ergodic properties of MAX-infinitely divisible processes, Stochastic Process. Appl., 120, 3, 281-295, (2010) · Zbl 1205.60101 [32] Kabluchko, Z.; Schlather, M.; de Haan, L., Stationary MAX-stable fields associated to negative definite functions, Ann. Probab., 37, 5, 2042-2065, (2009) · Zbl 1208.60051 [33] Kabluchko, Z.; Stoev, S., Stochastic integral representations and classification of sum- and MAX-infinitely divisible processes, Bernoulli, 22, 1, 107-142, (2016) · Zbl 1339.60065 [34] Khoshnevisan, D., Escape rates for Lévy processes, Studia Sci. Math. Hungar., 33, 1-3, 177-183, (1997) · Zbl 0910.60061 [35] Leadbetter, M. R.; Lindgren, G.; Rootzén, H., (Extremes and Related Properties of RandOm Sequences and Processes, Springer Series in Statistics, (1983), Springer-Verlag New York) · Zbl 0518.60021 [36] Ledford, A. W.; Tawn, J. A., Statistics for near independence in multivariate extreme values, Biometrika, 83, 1, 169-187, (1996) · Zbl 0865.62040 [37] LePage, R.; Woodroofe, M.; Zinn, J., Convergence to a stable distribution via order statistics, Ann. Probab., 9, 4, 624-632, (1981) · Zbl 0465.60031 [38] Oesting, M.; Kabluchko, Z.; Schlather, M., Simulation of Brown-resnick processes, Extremes, 15, 1, 89-107, (2012) · Zbl 1329.60157 [39] Penrose, M. D., Minima of independent Bessel processes and of distances between Brownian particles, J. Lond. Math. Soc. (2), 43, 2, 355-366, (1991) · Zbl 0686.60086 [40] Penrose, M. D., Semi-MIN-stable processes, Ann. Probab., 20, 3, 1450-1463, (1992) · Zbl 0762.60040 [41] Pickands III, J., Asymptotic properties of the maximum in a stationary Gaussian process, Trans. Amer. Math. Soc., 145, 75-86, (1969) · Zbl 0206.18901 [42] Piterbarg, V. I., (Asymptotic Methods in the Theory of Gaussian Processes and Fields, Translations of Mathematical Monographs, vol. 148, (1996), American Mathematical Society Providence RI), Translated from the Russian By V. V. Piterbarg, Revised By the Author [43] Resnick, S. I., (Extreme Values, Regular Variation, and Point Processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4, (1987), Springer-Verlag New York) [44] Revuz, D.; Yor, M., (Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, (1999), Springer-Verlag Berlin) · Zbl 0917.60006 [45] Stoev, S. A., On the ergodicity and mixing of MAX-stable processes, Stochastic Process. Appl., 118, 9, 1679-1705, (2008) · Zbl 1184.60013 [46] Stoev, S. A.; Taqqu, M. S., Extremal stochastic integrals: a parallel between MAX-stable processes and $$\alpha$$-stable processes, Extremes, 8, 4, 237-266, (2005), 2006 · Zbl 1142.60355 [47] Szabłowski, P. J., $$q$$-Wiener and ($$\alpha, q$$)-Ornstein-Uhlenbeck processes. A generalization of known processes, Theory Probab. Appl., 56, 4, 634-659, (2012) · Zbl 1276.60045 [48] Wang, Y., Large jumps of $$q$$-Ornstein-Uhlenbeck processes, Statist. Probab. Lett., 118, 110-116, (2016) · Zbl 1347.60117 [49] Weintraub, K. S., Sample and ergodic properties of some MIN-stable processes, Ann. Probab., 19, 2, 706-723, (1991) · Zbl 0788.60050 [50] Xiao, Y., Asymptotic results for self-similar Markov processes, (Asymptotic Methods in Probability and Statistics, (Ottawa, on, 1997), (1998), North-Holland Amsterdam), 323-340 · Zbl 0936.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.