• Bernoulli
  • Volume 20, Number 3 (2014), 1454-1483.

Sojourn measures of Student and Fisher–Snedecor random fields

Nikolai Leonenko and Andriy Olenko

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Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher–Snedecor random fields. Some simulation results are also presented.

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Bernoulli, Volume 20, Number 3 (2014), 1454-1483.

First available in Project Euclid: 11 June 2014

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excursion set first Minkowski functional Fisher–Snedecor random fields heavy-tailed limit theorems random field sojourn measure Student random fields


Leonenko, Nikolai; Olenko, Andriy. Sojourn measures of Student and Fisher–Snedecor random fields. Bernoulli 20 (2014), no. 3, 1454--1483. doi:10.3150/13-BEJ529.

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  • [1] Adler, R.J., Samorodnitsky, G. and Taylor, J.E. (2010). Excursion sets of three classes of stable random fields. Adv. in Appl. Probab. 42 293–318.
  • [2] Adler, R.J. and Taylor, J.E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. New York: Springer.
  • [3] Ahmad, O. and Pinoli, J.C. (2013). On the linear combination of the Gaussian and student’s $t$ random field and the integral geometry of its excursion sets. Statist. Probab. Lett. 83 559–567.
  • [4] Arcones, M.A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242–2274.
  • [5] Arcones, M.A. (2000). Distributional limit theorems over a stationary Gaussian sequence of random vectors. Stochastic Process. Appl. 88 135–159.
  • [6] Azaïs, J.M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Hoboken, NJ: Wiley.
  • [7] Bardet, J.M. and Surgailis, D. (2013). Moment bounds and central limit theorems for Gaussian subordinated arrays. J. Multivariate Anal. 114 457–473.
  • [8] Berman, S.M. (1984). Sojourns of vector Gaussian processes inside and outside spheres. Z. Wahrsch. Verw. Gebiete 66 529–542.
  • [9] Berman, S.M. (1992). Sojourns and Extremes of Stochastic Processes. The Wadsworth & Brooks/Cole Statistics/Probability Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software.
  • [10] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [11] Borovkov, K. and McKinlay, S. (2012). The uniform law for sojourn measures of random fields. Statist. Probab. Lett. 82 1745–1749.
  • [12] Braverman, M. (1997). Suprema and sojourn times of Lévy processes with exponential tails. Stochastic Process. Appl. 68 265–283.
  • [13] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
  • [14] Bulinski, A., Spodarev, E. and Timmermann, F. (2012). Central limit theorems for the excursion set volumes of weakly dependent random fields. Bernoulli 18 100–118.
  • [15] Cao, J. (1999). The size of the connected components of excursion sets of $\chi^{2}$, $t$ and $F$ fields. Adv. in Appl. Probab. 31 579–595.
  • [16] Cao, J. and Worsley, K. (1999). The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Probab. 9 1021–1057.
  • [17] Cook, J.D. (2009). Upper and lower bounds for the normal distribution function. Available at
  • [18] Dehling, H. and Taqqu, M.S. (1989). The empirical process of some long-range dependent sequences with an application to $U$-statistics. Ann. Statist. 17 1767–1783.
  • [19] Dobrushin, R.L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
  • [20] Doukhan, P., Lang, G. and Surgailis, D. (2002). Asymptotics of weighted empirical processes of linear fields with long-range dependence. Ann. Inst. Henri Poincaré Probab. Stat. 38 879–896.
  • [21] Gradshteyn, I.S. and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, 7th ed. (A. Jeffrey and D. Zwillinger, eds.). Amsterdam: Elsevier/Academic Press.
  • [22] Hariz, S.B. (2002). Limit theorems for the non-linear functional of stationary Gaussian processes. J. Multivariate Anal. 80 191–216.
  • [23] Ivanov, A.V. and Leonenko, N.N. (1989). Statistical Analysis of Random Fields. Mathematics and Its Applications (Soviet Series) 28. Dordrecht: Kluwer Academic.
  • [24] Ivanov, A.V., Leonenko, N.N., Ruiz-Medina, M.D. and Savich, I.N. (2013). Limit theorems for weighted non-linear transformations of Gaussian processes with singular spectra. Ann. Probab. 41 1088–1114.
  • [25] Kratz, M.F. (2006). Level crossings and other level functionals of stationary Gaussian processes. Probab. Surv. 3 230–288.
  • [26] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. New York: Springer.
  • [27] Leonenko, N. (1999). Limit Theorems for Random Fields with Singular Spectrum. Mathematics and Its Applications 465. Dordrecht: Kluwer Academic.
  • [28] Leonenko, N. and Olenko, A. (2013). Tauberian and Abelian theorems for long-range dependent random fields. Methodol. Comput. Appl. Probab. To appear. Available at
  • [29] Leonenko, N.N. (1987). Limit distributions of the characteristics of exceeding of a level by a Gaussian random field. Math. Notes 41 339–345.
  • [30] Leonenko, N.N. (1988). On the accuracy of the normal approximation of functionals of strongly correlated Gaussian random fields. Math. Notes 43 161–171.
  • [31] Leonenko, N.N. and Olenko, A.Y. (1991). Tauberian and Abelian theorems for the correlation function of a homogeneous isotropic random field. Ukrain. Mat. Zh. 43 1652–1664 (in Russian) (transl. Ukr. Math. J. 43 (1991) 1539–1548).
  • [32] Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Probab. 40 1069–1104.
  • [33] Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40 262–293.
  • [34] Maejima, M. (1985). Sojourns of multidimensional Gaussian processes with dependent components. Yokohama Math. J. 33 121–130.
  • [35] Maejima, M. (1986). Some sojourn time problems for $2$-dimensional Gaussian processes. J. Multivariate Anal. 18 52–69.
  • [36] Marinucci, D. (2004). Testing for non-Gaussianity on cosmic microwave background radiation: A review. Statist. Sci. 19 294–307.
  • [37] Meschenmoser, D. and Shashkin, A. (2011). Functional central limit theorem for the volume of excursion sets generated by associated random fields. Statist. Probab. Lett. 81 642–646.
  • [38] Novikov, D., Schmalzing, J. and Mukhanov, V.F. (2000). On non-Gaussianity in the cosmic microwave background. Astronom. Astrophys. 364 17–25.
  • [39] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
  • [40] Olenko, A.Y. (2005). A Tauberian theorem for fields with the OR spectrum. I. Teor. Ĭmovīr. Mat. Stat. 73 120–133 (in Ukrainian) (transl. Theory Probab. Math. Statist. 73 (2006) 135–149).
  • [41] Peccati, G. and Taqqu, M.S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams: A Survey With Computer Implementation. Bocconi & Springer Series 1. Milan: Springer.
  • [42] Schlather, M. (2013). RandomFields: Simulation and analysis of random fields in R. Available at
  • [43] Seneta, E. (1976). Regularly Varying Functions. Lecture Notes in Math. 508. Berlin: Springer.
  • [44] Taqqu, M.S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • [45] Taqqu, M.S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
  • [46] Taqqu, M.S. (1986). Sojourn in an elliptical domain. Stochastic Process. Appl. 21 319–326.
  • [47] Worsley, K.J. (1994). Local maxima and the expected Euler characteristic of excursion sets of $\chi ^{2}$, $F$ and $t$ fields. Adv. in Appl. Probab. 26 13–42.