Leonenko, Nikolai; Olenko, Andriy Tauberian and Abelian theorems for long-range dependent random fields. (English) Zbl 1307.60068 Methodol. Comput. Appl. Probab. 15, No. 4, 715-742 (2013). Summary: This paper surveys abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and spectral densities at zero. The use of the theorems and their limitations are demonstrated through applications to some new and less-known examples of covariance functions of long-range dependent random fields. Cited in 28 Documents MSC: 60G60 Random fields 62E20 Asymptotic distribution theory in statistics 40E05 Tauberian theorems Keywords:random field; covariance function; Abelian theorems; Tauberian theorems; long-range dependence PDFBibTeX XMLCite \textit{N. Leonenko} and \textit{A. Olenko}, Methodol. Comput. Appl. Probab. 15, No. 4, 715--742 (2013; Zbl 1307.60068) Full Text: DOI arXiv References: [1] Anderson DN (1992) A multivariate Linnik distribution. Stat Probab Lett 14:333-336 · Zbl 0754.60022 [2] Andrews GE, Askey R, Roy R (1999) Special functions. Cambridge University Press, Cambridge · Zbl 0920.33001 [3] Beran J, Ghosh S, Schell D (2009) On least squares estimation for long-memory lattice processes. J Multivar Anal 100:2178-2194 · Zbl 1175.62101 [4] Bingham NH (1972) A Tauberian theorem for integral transforms of Hankel type. J Lond Math Soc 5:493-503 · Zbl 0244.44004 [5] Bingham, NH; Inoue, A.; Kono, N. (ed.); Shieh, NR (ed.), An Abel-Tauber theorem for Hankel transforms, 83-90 (1997), River Edge · Zbl 0998.44003 [6] Bingham NH, Inoue A (1997) The Drasin-Shea-Jordan theorem for Fourier and Hankel transforms. Q J Math Oxf Ser 48(2):279-307 · Zbl 0889.42005 [7] Bingham NH, Inoue A (1999) Ratio Mercerian theorems with applications to Hankel and Fourier transforms. Proc Lond Math Soc 79:626-648 · Zbl 1030.44005 [8] Bingham NH, Inoue A (2000) Tauberian and Mercerian theorems for systems of kernels. J Math Anal Appl 252:177-197 · Zbl 0984.44005 [9] Bingham NH, Inoue A (2000) Extension of the Drasin-Shea-Jordan theorem. J Math Soc Jpn 52:545-559 · Zbl 0964.44002 [10] Bingham NH (2008) Tauberian theorems and large deviations. Stochastica 80:143-149 · Zbl 1138.60311 [11] Bingham NH, Goldie CM, Teugels JL (1989) Regular variation. Cambridge University Press, Cambridge [12] Bochner S (1933) Monotone funktionen, Stieltjessche integrale und harmonische analyse. Math Ann 108:378-410 · JFM 59.0272.01 [13] Donoghue WJ (1969) Distributions and Fourier transforms. Academic Press, New York · Zbl 0188.18102 [14] Doukhan P, León JR, Soulier P (1996) Central and non-central limit theorems for quadratic forms of a strongly dependent Gaussian field. Braz J Probab Stat 10:205-223 · Zbl 0881.60023 [15] Doukhan P, Oppenheim G, Taqqu MS (eds) (2003) Long-range dependence: theory and applications. Birkhauser, Boston · Zbl 1005.00017 [16] Erdogan MB, Ostrovskii IV (1998) Analytic and asymptotic properties of generalized Linnik probability densities. J Math Anal Appl 217:555-578 · Zbl 0893.60004 [17] Fang KT, Kotz S, Ng K (1990) Symmetric multivariate and related distributions. Chapman & Hall, London · Zbl 0699.62048 [18] Gneiting T, Schlather M (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev 46:269-282 · Zbl 1062.60053 [19] Halidov IA (1978) Some problems of the theory of correlation functions. Vestnik Leningr Univ Mat Meh Astron 3:63-68 · Zbl 0404.60044 [20] Inoue A, Kikuchi H (1999) Abel-Tauber theorems for Hankel and Fourier transforms and a problem of Boas. Hokkaido Math J 28:577-596 · Zbl 0939.40003 [21] Klykavka B (2011) Tauberian theorems for random fields with singularity in spectrum. Dissertation, Kyiv University [22] Laue G (1987) Tauberian and Abelian theorems for characteristic functions. Theory Probab Math Stat 37:78-92 · Zbl 0634.60018 [23] Lavancier F (2005) Les champs aléatoires á longue mémoire. Dissertation, Université de Lille [24] Lavancier, F.; Bertail, P. (ed.); Doukhan, P. (ed.); Soulier, P. (ed.), Long memory random fields, 195-220 (2006), New York · Zbl 1113.60053 [25] Lavancier F (2007) Invariance principles for non-isotropic long memory random fields. Stat Inference Stoch Process 10:255-282 · Zbl 1143.62058 [26] Lavancier F (2008) The V/S test of long-range dependence in random fields. Electron J Statist 2:1373-1390 · Zbl 1320.62209 [27] Leonenko NN (1999) Limit theorems for random fields with singular spectrum. Kluwer Academic, Dordrecht [28] Leonenko NN, Ivanov AV (1989) Statistical analysis of random fields. Kluwer Academic, Dordrecht [29] Leonenko NN, Olenko A (1991) Tauberian and Abelian theorems for correlation functions of homogeneous isotropic random field. Ukr Math J 43:1652-1664 [30] Leonenko NN, Olenko A (1993) Tauberian theorems for correlation functions and limit theorems for spherical averages of random fields. Random Oper Stoch Equ 1:57-67 · Zbl 0842.60049 [31] Lim SC, Teo LP (2010) Analytic and asymptotic properties of multivariate generalized Linnik’s probability densities. J Fourier Anal Appl 16:715-747 · Zbl 1202.60028 [32] Linnik JuV (1953) Linear forms and statistical criteria, I, II. Ukr Math J 5:207-290 · Zbl 0052.36701 [33] Olenko A (1991) Some problems in correlation and spectral theory of random fields. Dissertation, Kyiv University · Zbl 1143.62058 [34] Olenko A (1996) Tauberian and Abelian theorems for strongly dependent random fields. Ukr Math J 48:368-383 [35] Olenko A (2005) Tauberian theorems for random fields with the OR asymptotics I. Theory Probab Math Stat 73:120-133 · Zbl 1115.60056 [36] Olenko A (2006) Tauberian theorems for random fields with OR asymptotics II. Theory Probab Math Stat 74:81-97 · Zbl 1150.60027 [37] Ostrovskii IV (1995) Analytic and asymptotic properties of multivariate Linnik’s distribution. Math Phys Anal Geom 2:436-455 · Zbl 0849.60012 [38] Pitman EJG (1968) On the behaviour of the characteristic function of a probability distribution in the neighborhood of the origin. J Aust Math Soc 8:423-443 · Zbl 0164.48502 [39] Schoenberg J (1938) Metric spaces and completely monotone functions. Ann Math 39:811-841 · JFM 64.0617.03 [40] Seneta E (1976) Regularly varying functions. Springer, Berlin · Zbl 0324.26002 [41] Soni K, Soni RP (1974) A Tauberian theorem related to the modified Hankel transform. Bull Aust Math Soc 11:167-180 · Zbl 0285.44005 [42] Soni K, Soni RP (1975) Slowly varying functions and asymptotic behaviour of a class of integral transforms, III. J Math Anal Appl 33:23-34 · Zbl 0314.44005 [43] Yadrenko MI (1983) Spectral theory of random fields. Optimization Software Inc, New York [44] Yaglom AM (1957) Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab Appl 2:273-320 [45] Yakimiv AL (2005) Probabilistic applications of Tauberian theorems. VSP, Leiden · Zbl 1114.60001 [46] Wainger S (1965) Special trigonometric series in k-dimensions. AMS, Providence · Zbl 0136.36601 [47] Wolfe SJ (1973) On the local behavior of characteristic functions. Ann Probab 1:862-866 · Zbl 0274.60015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.