• Bernoulli
  • Volume 23, Number 4B (2017), 3469-3507.

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

Nikolai Leonenko, M. Dolores Ruiz-Medina, and Murad S. Taqqu

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A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in $d$-dimensional space. As a particular case, integrals of non-linear functions of chi-squared random fields, with Laguerre rank being equal to one and two, are studied. When the Laguerre rank is equal to one, the characteristic function of the limit random variable, given by a Rosenblatt-type distribution, is obtained. When the Laguerre rank is equal to two, a multiple Wiener–Itô stochastic integral representation of the limit distribution is derived and an infinite series representation, in terms of independent random variables, is obtained for the limit.

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Bernoulli, Volume 23, Number 4B (2017), 3469-3507.

Received: January 2015
Revised: January 2016
First available in Project Euclid: 23 May 2017

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Hermite expansion Laguerre expansion multiple Wiener–Itô stochastic integrals non-central limit results reduction theorems series expansions


Leonenko, Nikolai; Ruiz-Medina, M. Dolores; Taqqu, Murad S. Non-central limit theorems for random fields subordinated to gamma-correlated random fields. Bernoulli 23 (2017), no. 4B, 3469--3507. doi:10.3150/16-BEJ853.

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  • [1] Anh, V.V. and Leonenko, N.N. (1999). Non-Gaussian scenarios for the heat equation with singular initial conditions. Stochastic Process. Appl. 84 91–114.
  • [2] Anh, V.V., Leonenko, N.N. and Ruiz-Medina, M.D. (2013). Macroscaling limit theorems for filtered spatiotemporal random fields. Stoch. Anal. Appl. 31 460–508.
  • [3] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 93. Cambridge: Cambridge Univ. Press.
  • [4] Bateman, H. and Erdelyi, A. (1953). Higher Transcenental Functions II. New York: McGraw-Hill.
  • [5] Berman, S.M. (1982). Local times of stochastic processes with positive definite bivariate densities. Stochastic Process. Appl. 12 1–26.
  • [6] Berman, S.M. (1984). Sojourns of vector Gaussian processes inside and outside spheres. Z. Wahrsch. Verw. Gebiete 66 529–542.
  • [7] Brelot, M. (1960). Lectures on Potential Theory. Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy. Lectures on Mathematics 19. Bombay: Tata Institute of Fundamental Research.
  • [8] 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.
  • [9] Caetano, A.M. (2000). Approximation by functions of compact support in Besov–Triebel–Lizorkin spaces on irregular domains. Studia Math. 142 47–63.
  • [10] Chen, Z.-Q., Meerschaert, M.M. and Nane, E. (2012). Space–time fractional diffusion on bounded domains. J. Math. Anal. Appl. 393 479–488.
  • [11] Dautray, R. and Lions, J.-L. (1990). Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Berlin: Springer.
  • [12] Dobrushin, R.L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
  • [13] Doukhan, P., León, J.R. and Soulier, P. (1996). Central and non-central limit theorems for quadratic forms of a strongly dependent Gaussian field. Rebrape 10 205–223.
  • [14] Engliš, M. and Ali, S.T. (2015). Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics. J. Math. Phys. 56 072109, 22.
  • [15] Finlay, R. and Seneta, E. (2007). A gamma activity time process with noninteger parameter and self-similar limit. J. Appl. Probab. 44 950–959.
  • [16] Fox, R. and Taqqu, M.S. (1985). Noncentral limit theorems for quadratic forms in random variables having long-range dependence. Ann. Probab. 13 428–446.
  • [17] Fuglede, B. (2005). Dirichlet problems for harmonic maps from regular domains. Proc. Lond. Math. Soc. (3) 91 249–272.
  • [18] Gajek, L. and Mielniczuk, J. (1999). Long- and short-range dependent sequences under exponential subordination. Statist. Probab. Lett. 43 113–121.
  • [19] Heyde, C.C. and Leonenko, N.N. (2005). Student processes. Adv. in Appl. Probab. 37 342–365.
  • [20] Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. London: Chapman & Hall.
  • [21] Lancaster, H.O. (1958). The structure of bivariate distributions. Ann. Math. Stat. 29 719–736.
  • [22] Lancaster, H.O. (1963). Correlations and canonical forms of bivariate distributions. Ann. Math. Stat. 34 532–538.
  • [23] Leonenko, N. (1999). Limit Theorems for Random Fields with Singular Spectrum. Mathematics and Its Applications 465. Dordrecht: Kluwer Academic.
  • [24] Leonenko, N. and Olenko, A. (2014). Sojourn measures of Student and Fisher–Snedecor random fields. Bernoulli 20 1454–1483.
  • [25] Leonenko, N.N., Meerschaert, M.M. and Sikorskii, A. (2013). Fractional Pearson diffusions. J. Math. Anal. Appl. 403 532–546.
  • [26] Leonenko, N.N., Ruiz-Medina, M.D. and Taqqu, M. (2015). Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence. Available at arXiv:1501.02247 [math.ST].
  • [27] Lim, S.C. and Teo, L.P. (2010). Analytic and asymptotic properties of multivariate generalized Linnik’s probability densities. J. Fourier Anal. Appl. 16 715–747.
  • [28] Linde, A. and Mukhanov, V.F. (1997). Non-Gaussian isocurvature perturbations from inflation. Phys. Rev. D 56 535.
  • [29] Lukacs, E. (1970). Characteristic Functions, 2nd ed. New York: Hafner Publishing.
  • [30] Major, P. (1981). Multiple Wiener–Itô Integrals: With Applications to Limit Theorems. Lecture Notes in Math. 849. Berlin: Springer.
  • [31] Marinucci, D. and Peccati, G. (2011). Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. London Mathematical Society Lecture Note Series 389. Cambridge: Cambridge Univ. Press.
  • [32] Mielniczuk, J. (2000). Some properties of random stationary sequences with bivariate densities having diagonal expansions and nonparametric estimators based on them. J. Nonparametr. Stat. 12 223–243.
  • [33] Novikov, D., Schmalzing, J. and Mukhanov, V.F. (2000). On non-Gaussianity in the cosmic microwave background. Astronom. Astrophys. Lib. 364 17–25.
  • [34] Osswald, H. (2012). Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion: An Introduction. Cambridge Tracts in Mathematics 191. Cambridge: Cambridge Univ. Press.
  • [35] Rajput, B.S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
  • [36] Sarmanov, O.V. (1963). Investigation of stationary Markov processes by the method of eigenfunction expansion. Translated in Selected Translations in Mathematical Statistics and Probability Theory Amer. Math. Amer. Math. Soc., Providence 4 245–269.
  • [37] Simon, B. (2005). Trace Ideals and Their Applications, 2nd ed. Mathematical Surveys and Monographs 120. Providence, RI: Amer. Math. Soc.
  • [38] Stein, E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton, NJ: Princeton Univ. Press.
  • [39] Taqqu, M.S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • [40] Taqqu, M.S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
  • [41] Taylor, J.E. and Worsley, K.J. (2007). Detecting sparse signals in random fields, with an application to brain mapping. J. Amer. Statist. Assoc. 102 913–928.
  • [42] Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18. New York: North-Holland.
  • [43] Veillette, M.S. and Taqqu, M.S. (2013). Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 982–1005.
  • [44] Wong, E. and Thomas, J.B. (1962). On polynomial expansions of second-order distributions. J. Soc. Indust. Appl. Math. 10 507–516.
  • [45] Worsley, K.J. (2001). Testing for signals with unknown location and scale in a $\chi^{2}$ random field, with an application to fMRI. Adv. in Appl. Probab. 33 773–793.