Escort mean values and the characterization of power-law-decaying probability densities

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
American Institute of Physics
Google Scholar
Research Projects
Organizational Units
Journal Issue
Escort mean values or q-moments constitute useful theoretical tools for describing basic features of some probability densities such as those which asymptotically decay like power laws. They naturally appear in the study of many complex dynamical systems, particularly those obeying nonextensive statistical mechanics, a current generalization of the Boltzmann–Gibbs theory. They recover standard mean values or moments for q=1. Here we discuss the characterization of a nonnegative probability density by a suitable set of all its escort mean values together with the set of all associated normalizing quantities, provided that all of them converge. This opens the door to a natural extension of the well-known characterization, for the q=1 instance, of a distribution in terms of the standard moments, provided that all of them have finite values. This question would be specially relevant in connection with probability densities having divergent values for all nonvanishing standard moments higher than a given one e.g., probability densities asymptotically decaying as power laws, for which the standard approach is not applicable. The Cauchy–Lorentz distribution, whose second and higher even order moments diverge, constitutes a simple illustration of the interest of this investigation. In this context, we also address some mathematical subtleties with the aim of clarifying some aspects of an interesting nonlinear generalization of the Fourier transform, namely, the so called q-Fourier transform.
© 2009 American Institute of Physics. One of us (C.T.) has benefited from interesting related conversations with H. J. Hilhorst, S. Umarov, and E. M. F. Curado, and acknowledges partial financial support by CNPq and Faperj (Brazilian agencies). Another one (R.F.A.-E.) acknowledges the financial supports of Ministerio de Educación y Ciencia (Project No. FPA2004-02602) and of Ministerio de Ciencia e Innovación (Project No. FIS2008-01323), Spain. This work was partially supported (A.R.P.) by the Projects FQM-2445 and FQM-207 of the Junta de Andalucía (Spain, EU).
Unesco subjects
1. A. Pluchino, A. Rapisarda, C. Tsallis, Europhys. Lett., 80, 26002 (2007). 2. A. Pluchino, A. Rapisarda, C. Tsallis, Physica A, 387, 3121 (2008). 3. C. Tsallis, J. Stat. Phys., 52, 479 (1988). 4. C. Tsallis, Phys. World, 10, 42 (1997). 5. C. Tsallis, Braz. J. Phys., 29, 1 (1999). 6. P. Douglas, S. Bergamini, F. Renzoni, Phys. Rev. Lett., 96, 110601 (2006). 7. B. Liu, J. Goree, Phys. Rev. Lett., 100, 055003 (2008). 8. A. Upadhyaya, J.-P. Rieu, J. A. Glazier, Y. Sawada, Physica A, 293, 549 (2001). 9. K. E. Daniels, C. Beck, E. Bodenschatz, Physica D, 193, 208 (2004). 10. Nonextensive Entropy—Interdisciplinary Applications, edited by M. Gell-Mann and C. Tsallis (Oxford University Press, Oxford, 2004). 11. Nonextensive Statistical Mechanics: New Trends, New perspectives, Europhysics News, edited by J. P. Boon and C. Tsallis (European Physical Society, 2005, Vol. 36). 12. C. Tsallis, Entropy, in Encyclopedia of Complexity and Systems Science (Springer, Berlin, 2009). 13. C. Tsallis, Introduction to Nonextensive Statistical Mechanics—Approaching a Complex World (Springer, New York, 2009). 14. C. Beck, Phys. Rev. Lett., 87, 180601 (2001). 15. C. Beck, Europhys. Lett., 57, 329 (2002). 16. H. Touchette, C. Beck, Phys. Rev. E, 71, 016131 (2005). 17. C. Beck, Phys. Rev. Lett., 98, 064502 (2007). 18. A. R. Plastino, A. Plastino, Physica A, 222, 347 (1995). 19. C. Tsallis, D. J. Bukman, Phys. Rev. E, 54, R2197 (1996). 20. T. D. Frank, Nonlinear Fokker-Planck Equations (Springer-Verlag, Berlin, 2005). 21. E. K. Lenzi, L. C. Malacarne, R. S. Mendes, I. T. Pedron, Physica A, 319, 245 (2003). 22. F. D. Nobre, E. M. F. Curado, G. Rowlands, Physica A, 334, 109 (2004). 23. V. Schwammle, E. M. F. Curado, F. D. Nobre, Eur. Phys. J. B, 58, 159 (2007). 24. V. Schwammle, F. D. Nobre, E. M. F. Curado, Phys. Rev. E, 76, 041123 (2007). 25. L. Borland, Phys. Rev. Lett., 89, 098701 (2002). 26. E. T. Jaynes, in Papers on Probability, Statistics and Statistical Physics, edited by R. D. Rosenkrantz (Reidel, Dordrecht, 1987). 27. E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, 2005). 28. C. Tsallis, R. S. Mendes, A. R. Plastino, Physica A, 261, 534 (1998). 29. C. Beck, F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993). 30. S. Abe, Phys. Rev. E, 68, 031101 (2003). 31. S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math., 76, 307 (2008). 32. S. Umarov, C. Tsallis, Phys. Lett. A, 372, 4874 (2008). 33. J. A. Shohat, J. D. Tamarkin, The Problem of Moments, Am. Math. Soc. Mathematical Surveys (American Mathematical Society, New York, 1943), Vol. II. 34. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis (Hafner, New York, 1965 (translated by N. Kemmer). 35. T. S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978). 36. A. G. Bakan, Proc. Am. Math. Soc., 130, 3545 (2002). 37. R. F. Álvarez-Estrada, Ann. Phys., 11, 357 (2002) --- Ann. Phys., 15, 379 (2006). 38. R. F. Álvarez-Estrada, Eur. Phys. J. A, 31, 761 (2007). 39. C. Tsallis, S. M. D. Queiros, AIP Conf. Proc., 965, 8 (2007) -- S. M. D. Queiros, C. Tsallis, ibid., 965, 21 (2007). 40. The so-called moment problem in theory of probabilities is a mathematically quite complex one. The discussion is normally done separately for various classes of support of the probability distribution, namely, for the [0,1] support (Hausdorff moment problem), the [0,∞) support (Stieltjes moment problem), and the (∞,−∞) support (Hamburger moment problem). General necessary and sufficient conditions are still elusive (Refs. 33–36).