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Shift-invariant homogeneous classes of random fields

Published 1 Nov 2021 in math.PR and stat.AP | (2111.00792v4)

Abstract: Given an $Rd$-valued random field (rf) $Z(t),t\in T$ and an $\alpha$-homogeneous mapping $\kappa$ we define the corresponding equivalent class of rf's (denoted by $K_\alpha$) which include representers of the same tail measure $\nu_Z$. When $T$ is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of $Z$. This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant $K_\alpha$'s. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps $U$ acting on elements of $K_\alpha$ turns out to be crucial for properties of functionals of $Z$. Applications of our findings concern max-stable and symmetric $\alpha$-stable rf's, their maximal indices as well as their random shift-representations.

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References (61)
  1. I. Molchanov, M. Schmutz, and K. Stucki, “Invariance properties of random vectors and stochastic processes based on the zonoid concept,” Bernoulli, vol. 20, no. 3, pp. 1210–1233, 2014.
  2. E. Hashorva, “Representations of max-stable processes via exponential tilting,” Stochastic Process. Appl., vol. 128, no. 9, pp. 2952–2978, 2018.
  3. H.-P. Scheffler and S. Stoev, “Implicit extremes and implicit max–stable laws,” Extremes, vol. 20, pp. 265–299, 2017.
  4. D. Kremer, “Implicit max-stable extremal integrals,” Extremes, vol. 24, no. 1, pp. 1–35, 2021.
  5. H. Planinić and P. Soulier, “The tail process revisited,” Extremes, vol. 21, no. 4, pp. 551–579, 2018.
  6. R. Kulik and P. Soulier, Heavy tailed time series. Springer, Cham, 2020.
  7. C. Dombry, E. Hashorva, and P. Soulier, “Tail measure and spectral tail process of regularly varying time series,” Ann. Appl. Probab., vol. 28, no. 6, pp. 3884–3921, 2018.
  8. B. Basrak and H. Planinić, “Compound Poisson approximation for regularly varying fields with application to sequence alignment,” Bernoulli, vol. 27, no. 2, pp. 1371–1408, 2021.
  9. P. Soulier, “The tail process and tail measure of continuous time regularly varying stochastic processes,” Extremes, vol. 25, no. 1, pp. 107–173, 2022.
  10. E. Hashorva and A. Kume, “Multivariate max-stable processes and homogeneous functionals,” Statist. Probab. Lett., vol. 173, p. 109066, 2021.
  11. L. de Haan, “A spectral representation for max-stable processes,” Ann. Probab., vol. 12, no. 4, pp. 1194–1204, 1984.
  12. S. A. Stoev and M. S. Taqqu, “Extremal stochastic integrals: a parallel between max-stable processes and α𝛼\alphaitalic_α-stable processes,” Extremes, vol. 8, no. 4, pp. 237–266, 2005.
  13. I. Molchanov and K. Stucki, “Stationarity of multivariate particle systems,” Stochastic Process. Appl., vol. 123, no. 6, pp. 2272–2285, 2013.
  14. E. Hashorva, “On extremal index of max-stable random fields,” Lith. Math. J., vol. 61, no. 2, pp. 217–238, 2021.
  15. L. de Haan and J. Pickands, III, “Stationary min-stable stochastic processes,” Probab. Theory Relat. Fields, vol. 72, no. 4, pp. 477–492, 1986.
  16. M. Bladt, E. Hashorva, and G. Shevchenko, “Tail measures and regular variation,” Electron. J. Probab., vol. 27, pp. Paper No. 64, 43, 2022.
  17. G. Samorodnitsky, Stochastic processes and long range dependence. Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2016.
  18. T. Owada and G. Samorodnitsky, “Tail measures of stochastic processes of random fields with regularly varying tails. Technical report,” 2012.
  19. C. Dombry and F. Eyi-Minko, “Stationary max-stable processes with the markov property,” Stochastic Processes and their Applications, vol. 124, no. 6, pp. 2266–2279, 2014.
  20. A. Ehlert and M. Schlather, “Capturing the multivariate extremal index: bounds and interconnections,” Extremes, vol. 11, no. 4, pp. 353–377, 2008.
  21. J. Rosiński, “Minimal integral representations of stable processes,” Probability and Mathematical Statistics, vol. 26, no. 1, pp. 121–142, 2006.
  22. Y. Wang, P. Roy, and S. A. Stoev, “Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions,” Ann. Probab., vol. 41, no. 1, pp. 206–228, 2013.
  23. Y. Wang and S. A. Stoev, “On the association of sum- and max-stable processes,” Statist. Probab. Lett., vol. 80, no. 5-6, pp. 480–488, 2010.
  24. M. Oesting, Z. Kabluchko, and M. Schlather, “Simulation of Brown-Resnick processes,” Extremes, vol. 15, no. 1, pp. 89–107, 2012.
  25. B. M. Brown and S. I. Resnick, “Extreme values of independent stochastic processes,” J. Appl. Probab., vol. 14, pp. 732–739, 1977.
  26. W. F. Eddy, “The distribution of the convex hull of a Gaussian sample,” J. Appl. Probab., vol. 17, no. 3, pp. 686–695, 1980.
  27. Z. Kabluchko, M. Schlather, and L. de Haan, “Stationary max-stable fields associated to negative definite functions,” Ann. Probab., vol. 37, pp. 2042–2065, 2009.
  28. M. Lifshits, Lectures on Gaussian processes. SpringerBriefs in Mathematics, Springer, Heidelberg, 2012.
  29. S. A. Stoev, “On the ergodicity and mixing of max-stable processes,” Stochastic Process. Appl., vol. 118, no. 9, pp. 1679–1705, 2008.
  30. S. Engelke and Z. Kabluchko, “Max-stable processes associated with stationary systems of independent Lévy particles,” Stochastic Process. Appl., vol. 125, no. 11, pp. 4272–4299, 2015.
  31. H. Planinić, “Palm theory for extremes of stationary regularly varying time series and random fields,” Extremes, pp. 1–38, 2022.
  32. B. Basrak and J. Segers, “Regularly varying multivariate time series,” Stochastic Process. Appl., vol. 119, no. 4, pp. 1055–1080, 2009.
  33. Y. Cissokho and R. Kulik, “Estimation of cluster functionals for regularly varying time series: sliding blocks estimators,” Electron. J. Stat., vol. 15, no. 1, pp. 2777–2831, 2021.
  34. Y. Cissokho and R. Kulik, “Estimation of cluster functionals for regularly varying time series: Runs estimators,” Electronic Journal of Statistics, vol. 16, no. 1, pp. 3561–3607, 2022.
  35. Z. Chen and R. Kulik, “Limit theorems for unbounded cluster functionals of regularly varying time series,” arXiv preprint arXiv:2308.16270, 2023.
  36. Z. Chen and R. Kulik, “Asymptotic expansions for blocks estimators: POT framework,” arXiv preprint arXiv:2309.03163, 2023.
  37. Y. Wang and S. A. Stoev, “On the structure and representations of max-stable processes,” Adv. in Appl. Probab., vol. 42, no. 3, pp. 855–877, 2010.
  38. C. Dombry and Z. Kabluchko, “Ergodic decompositions of stationary max-stable processes in terms of their spectral functions,” Stochastic Process. Appl., vol. 127, no. 6, pp. 1763–1784, 2017.
  39. C. Dörr and M. Schlather, “Characterization theorems for pseudo-variograms,” J. Appl. Probab., vol. 60, no. 4, pp. 1219–1231, 2023.
  40. Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2004.
  41. C. Dombry, “Representation of homogeneous measures and tail measures of regularly varying time series,” https://www.birs.ca/cmo-workshops/2018/18w5086/files/Dombry_Cle%CC%81ment.pdf, 2018.
  42. J. Segers, Y. Zhao, and T. Meinguet, “Polar decomposition of regularly varying time series in star-shaped metric spaces,” Extremes, vol. 20, no. 3, pp. 539–566, 2017.
  43. L. Wu and G. Samorodnitsky, “Regularly varying random fields,” Stochastic Process. Appl., vol. 130, no. 7, pp. 4470–4492, 2020.
  44. E. Hashorva, “Cluster random fields and random-shift representations,” arXiv preprint arXiv:2206.15064, 2022.
  45. Z. Kabluchko, “Spectral representations of sum- and max-stable processes,” Extremes, vol. 12, pp. 401–424, 2009.
  46. P. Roy, “Nonsingular group actions and stationary S⁢α⁢S𝑆𝛼𝑆S\alpha Sitalic_S italic_α italic_S random fields,” Proc. Amer. Math. Soc., vol. 138, no. 6, pp. 2195–2202, 2010.
  47. K. Dȩbicki and E. Hashorva, “Approximation of supremum of max-stable stationary processes & Pickands constants,” J. Theoret. Probab., vol. 33, no. 1, pp. 444–464, 2020.
  48. J. Rosiński, “On the structure of stationary stable processes,” Ann. Probab., vol. 23, no. 3, pp. 1163–1187, 1995.
  49. G. Samorodnitsky, “Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes,” Ann. Probab., vol. 32, no. 2, pp. 1438–1468, 2004.
  50. Z. Kabluchko and Y. Wang, “Limiting distribution for the maximal standardized increment of a random walk,” Stochastic Process. Appl., vol. 124, no. 9, pp. 2824–2867, 2014.
  51. S. A. Stoev, “Max–stable processes: Representations, ergodic properties and statistical applications,” Dependence in Probability and Statistics, Lecture Notes in Statistics 200, Doukhan, P., Lang, G., Surgailis, D., Teyssiere, G. (Eds.), vol. 200, pp. 21–42, 2010.
  52. K. Dȩbicki and E. Hashorva, “On extremal index of max-stable stationary processes,” Probab. Math. Statist., vol. 37, no. 2, pp. 299–317, 2017.
  53. K. Dȩbicki, E. Hashorva, and Z. Michna, “On the continuity of Pickands constants,” J. Appl. Probab., vol. 59, no. 1, pp. 187–201, 2022.
  54. G. Samorodnitsky, “Maxima of continuous-time stationary stable processes,” Adv. in Appl. Probab., vol. 36, no. 3, pp. 805–823, 2004.
  55. P. Roy and G. Samorodnitsky, “Stationary symmetric α𝛼\alphaitalic_α-stable discrete parameter random fields,” J. Theoret. Probab., vol. 21, no. 1, pp. 212–233, 2008.
  56. J. L. Doob, Stochastic processes. Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1953 original, A Wiley-Interscience Publication.
  57. K. Bisewski, E. Hashorva, and G. Shevchenko, “The harmonic mean formula for random processes,” Stochastic Analysis and Applications, pp. 1–13, 2022.
  58. Springer-Verlag, Berlin-New York, english ed., 1980. Translated from the Russian by Samuel Kotz.
  59. D. Ferger, “Arginf-sets of multivariate cadlag processes and their convergence in hyperspace topologies,” Theory of Stochastic Processes, vol. 20, no. 2, pp. 13–41, 2015.
  60. A. Janßen, S. Neblung, and S. Stoev, “Tail-dependence, exceedance sets, and metric emobeddings,” Extremes, vol. 26, no. 4, pp. 747–785, 2023.
  61. R. Yuen, S. Stoev, and D. Cooley, “Distributionally robust inference for extreme value-at-risk,” Insurance: Mathematics and Economics, vol. 92, pp. 70–89, 2020.
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