Abstract
We prove theorems of Tauberian and Abelian types for nonintegrable correlation functions of homogeneous isotropic random fields and use them to study asymptotic distributions of local functionals of Gaussian fields.
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Literature cited
M. I. Yadrenko, Spectral Theory of Random Fields [in Russian], Vishcha Shkola, Kiev University, Kiev (1980).
N. N. Leonenko and A. V. Ivanov, Statistical Analysis of Random Fields [in Russian], Vishcha Shkola, Kiev University, Kiev (1986).
N. H. Binghem, “A Tauberian theorem for integral transforms of Hankel type,” J. London Math. Soc.,5, No. 3, 493–503 (1972).
G. Laue, “Tauberian and Abelian theorems for characteristic functions,” Teor. Veroyatn. Mat. Stat.,37, 78–92 (1987).
R. N. Miroshin, Intersection of Curves by Gaussian Processes [in Russian], Leningrad State Univ. (1981).
V. S. Vladimirov, Yu. N. Drozhzhinov, and B.I. Zav'yalov, Multidimensional Tauberian Theorems for Generalized Functions [in Russian], Nauka, Moscow (1986).
W. Feller, Introduction to Probability Theory and Its Applications [Russian translation], Vol. 2, Mir, Moscow (1984).
D. V. Widder, The Laplace Transform, Princeton University Press, Princeton (1969).
H. Watson, Theory of Bessel Functions [Russian translation], Vol. 1, Izdatel'stvo Inostrannoi Literatury, Moscow (1949).
G. Laue, “Zur theorie der charakteristischen Funktionen nichnegativer zufallsgroben,” Trans. 6th Prague Conf. Inform. Theory, Prague (1973), pp. 529–546.
H. Bateman and A. Erdelyi, Tables of Integral Transforms. Bessel Transforms. Integrals of Special Functions [Russian translation], Vol. 2, Nauka, Moscow (1970).
E. Titchmarsh, Introduction to the Theory of Fourier Integrals [in Russian], OGIZ, Moscow (1948).
H. Bateman and A. Erdelyi, Tables of Integral Transforms. Fourier, Laplace, and Mellin Transforms [Russian translation], Vol. 1, Nauka, Moscow (1969).
G. Doetsch, A Guide to Practical Application of Laplace Transforms [Russian translation], Nauka, Moscow (1965).
R. L. Dobrushin and P. Major, “Noncentral limit theorem for nonlinear functional of Gaussian field,” Z. Wahrscheinlichkeits Theor. Verw. Geb.,50, No. 1, 27–52 (1979).
M. S. Taqqu, “Convergence of integrated processes of arbitrary Hermite rang,” Z. Wahrscheinlichkeits Theor. Verw. Geb.,50, No. 1, 53–83 (1979).
P. Major, “Multiple Wiener-Ito integrals,” Lect. Notes Math.,849 (1981).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1652–1664, December, 1991.
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Leonenko, N.N., Olenko, A.Y. Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field. Ukr Math J 43, 1539–1548 (1991). https://doi.org/10.1007/BF01066693
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DOI: https://doi.org/10.1007/BF01066693