Abstract
Limit distributions of solutions of the multidimensional Bürgers equation are found in the case where an initial condition is a random field of type χ2 of degreek with a long-range dependence.
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J. Whitham,Linear and Nonlinear Waves [Russian translation], Mir, Moscow (1977).
M. Rosenblatt, “Remark on the Bürgers equation,”J. Math. Phys.,9, 1129–1136 (1968).
A. V. Bulinskii and S. A. Molchanov, “Asymptotic Gaussian property of a solution of the Bürgers equation with random initial data,”Teor. Veroyatn. Primen.,36, No. 3, 217–235 (1991).
L. Giraitis, S. A. Molchanow, and Surgailis D., “Long memory shot noises and limit theorems with applications to Bürger's equations. New direction in time series analysis. Pt. II,”IMA Volumes Math. Appl.,46, 153–171 (1993).
N. N. Leonenko and E. Orsingher,Limit Theorems for Solutions of Bürgers Equation with Gaussian and Non-Gaussian Initial Conditions, Preprint No. 91.1, University of Rome, Rome (1991).
N. N. Leonenko, E. Orsingher, and K. V. Rybasov, “Limit distributions of solutions of the many-dimensional Bürgers equation with random initial data. I, II,”Ukr. Mat. Zh.,46, No. 7, 870–877; No. 8, 1003–1010 (1994).
R. L. Dobrushin and P. Major, “Non-central limit theorems for nonlinear functionals of Gaussian fields,”Z. Wahrscheinlichkeitstheor. Verw. Geb.,50, 1–28 (1979).
A. V. Ivanov and N. N. Leonenko,Statistical Analysis of Random Fields, Kluwer AP, Dordrecht (1989).
P. Major, “Multiple Wiener-Itô integrals,”Lect. Notes Math.,849, (1981).
Ya. G. Sinai, “Two results concerning the asymptotic behaviour of solutions of the Bürgers equations with force,”J. Statist. Phys.,64, 1–12 (1991).
M. S. Taqqu, “Convergence of integrated processes of arbitrary Hermite rank,”Z. Wahrscheinlichkeitstheor. Verw. Geb.,50, No. 1, 55–83 (1979).
S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev,Nonlinear Random Waves in Dispersion-Free Media [in Russian], Nauka, Moscow (1990).
N. N. Leonenko and A. Ya. Olenko, “Tauberian and Abelian theorems for the correlation function of a homogeneous isotropic random field,”Ukr. Mat. Zh.,43, No. 12, 1652–1664 (1991).
A. Ya. Olenko,Several Aspects of the Correlation and Spectral Random Field Theory [in Russian], Author's Abstract of the Candidate Degree Thesis (Physics and Mathematics), Kiev (1991).
H. Bateman and A. Erdelyi,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1953).
S. M. Berman, “High level sojourns for strongly dependent Gaussian processes,”Z. Wahrscheinlichkeitstheor. Verw. Geb.,50, 223–236 (1979).
H. O. Lancaster, “The structure of bivariate distributions,”Ann. Math. Statist.,29, 719–736 (1958); Correction,Ann. Math. Statist.,35, 1988 (1964).
S. D. Wicksell, “On correlation functions of type III,”Biometrika,25, 121–133 (1933).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 330–336, March, 1995.
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Leonenko, N.N., Zhanbing, L. & Rybasov, K.V. Non-Gaussian limit distributions of solutions of the many-dimensional Bürgers equation with random initial data. Ukr Math J 47, 385–392 (1995). https://doi.org/10.1007/BF01056300
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DOI: https://doi.org/10.1007/BF01056300