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Choquet random sup-measures with aggregations

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Abstract

A variation of Choquet random sup-measures is introduced. These random sup-measures are shown to arise as the scaling limits of empirical random sup-measures of a general aggregated model. Because of the aggregations, the finite-dimensional distributions of introduced random sup-measures do not necessarily have classical extreme-value distributions. Examples include the recently introduced stable-regenerative random sup-measures as a special case.

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References

  • Bai, S.: Representations of Hermite processes using local time of intersecting stationary stable regenerative sets. J. Appl. Probab. 57(4), 1234–1251 (2020)

    Article  MathSciNet  Google Scholar 

  • Bai, S., Owada, T., Wang, Y.: A functional non-central limit theorem for multiple-stable processes with long-range dependence. Stochastic Process. Appl. 130(9), 5768–5801 (2020)

    Article  MathSciNet  Google Scholar 

  • Basrak, B., Planinić, H., Soulier, P.: An invariance principle for sums and record times of regularly varying stationary sequences. Probab. Theory Related Fields 172(3-4), 869–914 (2018)

    Article  MathSciNet  Google Scholar 

  • Basrak, B., Segers, J.: Regularly varying multivariate time series. Stochastic Process. Appl. 119(4), 1055–1080 (2009)

    Article  MathSciNet  Google Scholar 

  • Bertoin, J.: Subordinators: examples and applications. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), volume 1717 of Lecture Notes in Math., pp 1–91. Springer, Berlin (1999)

  • Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999). A Wiley-Interscience Publication

    Book  Google Scholar 

  • Boucheron, S., Lugosi, G., Massart, P.: Concentration inequalities. Oxford University Press, Oxford (2013). A nonasymptotic theory of independence, With a foreword by Michel Ledoux

    Book  Google Scholar 

  • Caravenna, F., Sun, R., Zygouras, N.: The continuum disordered pinning model. Probab. Theory Related Fields 164(1-2), 17–59 (2016)

    Article  MathSciNet  Google Scholar 

  • Cohen, S., Dombry, C.: Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions. J. Math. Kyoto Univ. 49 (2), 267–286 (2009)

    MathSciNet  MATH  Google Scholar 

  • Cohen, S., Samorodnitsky, G.: Random rewards, fractional Brownian local times and stable self-similar processes. Ann. Appl. Probab. 16(3), 1432–1461 (2006)

    Article  MathSciNet  Google Scholar 

  • Davis, R. A., Hsing, T.: Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23(2), 879–917 (1995)

    Article  MathSciNet  Google Scholar 

  • Davis, R. A., Mikosch, T.: The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26(5), 2049–2080 (1998)

    Article  MathSciNet  Google Scholar 

  • Dombry, C., Guillotin-Plantard, N.: Discrete approximation of a stable self-similar stationary increments process. Bernoulli 15(1), 195–222 (2009)

    Article  MathSciNet  Google Scholar 

  • Dombry, C., Hashorva, E., Soulier, P.: Tail measure and spectral tail process of regularly varying time series. Ann. Appl. Probab. 28(6), 3884–3921 (2018)

    Article  MathSciNet  Google Scholar 

  • Doney, R. A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107(4), 451–465 (1997)

    Article  MathSciNet  Google Scholar 

  • Durieu, O., Wang, Y.: A family of random sup-measures with long-range dependence. Electron. J. Probab. 23(107), 1–24 (2018)

    MathSciNet  MATH  Google Scholar 

  • Enriquez, N.: A simple construction of the fractional Brownian motion. Stochastic Process. Appl. 109(2), 203–223 (2004)

    Article  MathSciNet  Google Scholar 

  • Giacomin, G.: Random polymer models. Imperial College Press, London (2007)

    Book  Google Scholar 

  • Kaj, I., Taqqu, M. S.: Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In: In and out of Equilibrium. 2, volume 60 of Progr. Probab., pp 383–427. Basel, Birkhäuser (2008)

  • Kallenberg, O.: Foundations of modern probability. Probability and its Applications (New York). Springer, New York (1997)

    MATH  Google Scholar 

  • Karlin, S.: Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17, 373–401 (1967)

    MathSciNet  MATH  Google Scholar 

  • Kulik, R., Soulier, P.: Heavy-tailed time series. Springer, Berlin (2020)

    Book  Google Scholar 

  • Lacaux, C., Samorodnitsky, G.: Time-changed extremal process as a random sup measure. Bernoulli 22(4), 1979–2000 (2016)

    Article  MathSciNet  Google Scholar 

  • Mikosch, T., Samorodnitsky, G.: Scaling limits for cumulative input processes. Math. Oper. Res. 32(4), 890–918 (2007)

    Article  MathSciNet  Google Scholar 

  • Molchanov, I.: Theory of Random Sets, volume 87 of Probability Theory and Stochastic Modelling. Springer, London (2017). Second edition of [MR2132405]

    Google Scholar 

  • Molchanov, I., Strokorb, K.: Max-stable random sup-measures with comonotonic tail dependence. Stochastic Process. Appl. 126(9), 2835–2859 (2016)

    Article  MathSciNet  Google Scholar 

  • Norberg, T.: Random capacities and their distributions. Probab. Theory Related Fields 73(2), 281–297 (1986)

    Article  MathSciNet  Google Scholar 

  • O’Brien, G. L., Torfs, P. J. J. F., Vervaat, W.: Stationary self-similar extremal processes. Probab. Theory Related Fields 87(1), 97–119 (1990)

    Article  MathSciNet  Google Scholar 

  • Owada, T., Samorodnitsky, G.: Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43(1), 240–285 (2015)

    Article  MathSciNet  Google Scholar 

  • Pipiras, V., Taqqu, M. S.: Long-range Dependence and Self-similarity, volume 45 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2017)

    MATH  Google Scholar 

  • Pipiras, V., Taqqu, M. S., Levy, J. B.: Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli 10(1), 121–163 (2004)

    Article  MathSciNet  Google Scholar 

  • Pitman, J., Yor, M.: Bessel processes and infinitely divisible laws. In: Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980), volume 851 of Lecture Notes in Math., pp 285–370. Springer, Berlin (1981)

  • Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59(4), 425–457 (1982)

    Article  MathSciNet  Google Scholar 

  • Resnick, S. I.: Extreme Values, Regular Variation, and Point Processes, volume 4 of Applied Probability. A Series of the Applied Probability Trust. Springer, New York (1987)

    Google Scholar 

  • Rosiński, J., Samorodnitsky, G.: Classes of mixing stable processes. Bernoulli 2(4), 365–377 (1996)

    Article  MathSciNet  Google Scholar 

  • Samorodnitsky, G.: Stochastic Processes and Long Range Dependence. Springer, Cham (2016)

    Book  Google Scholar 

  • Samorodnitsky, G., Wang, Y.: Extremal theory for long range dependent infinitely divisible processes. Ann. Probab. 47(4), 2529–2562 (2019)

    Article  MathSciNet  Google Scholar 

  • Sibuya, M.: Generalized hypergeometric, digamma and trigamma distributions. Ann. Inst. Statist. Math. 31(3), 373–390 (1979)

    Article  MathSciNet  Google Scholar 

  • Treszczotko, Ł.: Random walks in doubly random scenery. Electron. Commun. Probab. 23(66), 11 (2018)

    MathSciNet  MATH  Google Scholar 

  • Vervaat, W.: Random upper semicontinuous functions and extremal processes. In: Probability and Lattices, volume 110 of CWI Tract, pp 1–56. Math. Centrum, Centrum Wisk. Inform., Amsterdam (1997)

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Acknowledgements

YW thanks Shuyang Bai, Olivier Durieu, Ilya Molchanov, Gennady Samorodnitsky and Na Zhang for very helpful discussions, and the Associate Editor and two anonymous referees for helpful comments and suggestions. YW’s research was partially supported by Army Research Office grants W911NF-17-1-0006 and W911NF-20-1-0139 at University of Cincinnati.

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  1. Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH, 45221-0025, USA

    Yizao Wang

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  1. Yizao Wang
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Wang, Y. Choquet random sup-measures with aggregations. Extremes 25, 25–54 (2022). https://doi.org/10.1007/s10687-021-00425-3

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