ArticlesNo Access

RÉNYI FUNCTION FOR MULTIFRACTAL RANDOM FIELDS

NIKOLAI N. LEONENKO

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK

Search for more papers by this author

and

NARN-RUEIH SHIEH

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, P. R. China

Corresponding author. On leave from National Taiwan University.

Search for more papers by this author

https://doi.org/10.1142/S0218348X13500096Cited by:14 (Source: Crossref)

Abstract

This paper presents the basic scheme and the log-normal, log-gamma and log-negative-inverted-gamma scenarios to establish the Rényi function for infinite products of homogeneous isotropic random fields on Rⁿ; in particular for random fields on the sphere in R³. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

Keywords:

References

U. Frisch , Turbulence ( Cambridge University Press , 1995 ) . Crossref, Google Scholar
D. Harte , Multifractals: Theory and Applications ( Chapman and Hall , 2001 ) . Crossref, Google Scholar
R. Riedi, Theory and Applications of Long-Range Dependence, eds. P. Doukhan, G. Oppenheim and M. Taqqu (Birkhäuser, Boston, 2003) pp. 625–716. Google Scholar
B. Mandelbrot , J. Fluid Mech. 62 , 331 ( 1974 ) . Crossref, Web of Science, Google Scholar
J.-P. Kahane , Ann. Sc. Math. Québec 9 , 105 ( 1985 ) . Google Scholar
J.-P. Kahane , Chin. Ann. Math. 8B , 1 ( 1987 ) . Google Scholar
S. J. Taylor , Fractals 3 , 737 ( 1995 ) . Link, Web of Science, Google Scholar
M. Molchan , Comm. Math. Physics 179 , 681 ( 1996 ) . Crossref, Web of Science, Google Scholar
S. Jaffard , Probab. Theory Relat. Fields 114 , 207 ( 1999 ) . Crossref, Web of Science, Google Scholar
J. Barral and B. Mandelbrot , Prob. Th. Rel. Fields 124 , 409 ( 2002 ) . Crossref, Web of Science, Google Scholar
E. Bacry and J. F. Muzy , Comm. Math. Physics 236 , 449 ( 2003 ) . Crossref, Web of Science, Google Scholar
N.-R. Shieh and S. J. Taylor , J. Appl. Prob. 39 , 100 ( 2002 ) . Crossref, Web of Science, Google Scholar
P. Mörters and N.-R. Shieh , J. Appl. Prob. 41 , 1223 ( 2004 ) . Crossref, Web of Science, Google Scholar
P. Mörters and N.-R. Shieh 42, eds. K. S. Lau, Z. P. Xin and S. T. Yau (AMS, 2008) pp. 655–662. Google Scholar
P. Mannersalo , I. Norros and R. Riedi , Adv. Appl. Prob. 34 , 888 ( 2002 ) . Crossref, Web of Science, Google Scholar
V. V. Anh , N. N. Leonenko and N.-R. Shieh , Adv. Appl. Probab. 40 , 1129 ( 2008 ) . Crossref, Web of Science, Google Scholar
V. V. Anh, N. N. Leonenko and N.-R. Shieh, Math. Commun. 13(2), 133 (2008). Web of Science, Google Scholar
V. V. Anh, N. N. Leonenko and N.-R. Shieh, Bernoulli 15(2), 508 (2009). Crossref, Web of Science, Google Scholar
V. V. Anh, N. N. Leonenko and N.-R. Shieh, Stoch. Anal. Appl. 27(3), 475 (2009). Crossref, Web of Science, Google Scholar
V. V. Anh, N. N. Leonenko and N.-R. Shieh, Dependence in Probability and Statistics, Springer Lecture Notes in Statistics 200, eds. P. Doukhanet al. (Springer-Verlag, Berlin, Heidelberg, 2010) pp. 103–122. Crossref, Google Scholar
M. Matsui and N.-R. Shieh, Electron. J. Probab. 14(23), 594 (2009). Crossref, Web of Science, Google Scholar
M. Matsui and N.-R. Shieh , Stochastics 1 ( 2012 ) . Google Scholar
V. V. Anh et al. , Nonlinearity 23 , 823 ( 2010 ) . Crossref, Web of Science, Google Scholar
D. Marinucci, Statist. Sci. 19(2), 294 (2004). Crossref, Web of Science, Google Scholar
D. Marinucci and G. Domenico , Representation, Limit Theorems and Cosmological Applications , London Mathematical Society Lecture Note Series 389 ( Cambridge University Press , Cambridge , 2011 ) . Google Scholar
N. N. Leonenko , Limit Theorems for Random Fields with Singular Spectrum , Mathematics and Its Applications 465 ( Kluwer Academic Publishers , Dordrecht , 1999 ) . Crossref, Google Scholar
M. Ya. Kelbert, N. N. Leonenko and M. D. Ruiz-Medina, Adv. Appl. Probab. 37(1), 108 (2005). Crossref, Web of Science, Google Scholar
Y. Saito , J. Phys. Soc. Japan 61 , 403 ( 1992 ) . Crossref, Google Scholar
H. Joe , Multivariate Models and Dependent Concepts ( Chapman and Hall , London , 1997 ) . Crossref, Google Scholar
O. E. Barndorf-Nilsen and Yu. Shmigel, Spatio-temporal modeling based on Lévy processes, and its applications to turbulence, (Russian) Uspekhi Mat. Nauk 59 (2004) 63–90; translation in Russian Math. Surveys 59 (2004) 65–90 . Google Scholar
N.-R. Shieh , Fractals 17 , 505 ( 2009 ) . Link, Web of Science, Google Scholar