Journal of Statistical Physics

, Volume 157, Issue 4–5, pp 915–930 | Cite as

An Application of the Renormalization Group Method to Stable Limit Laws

  • Dong LiEmail author
  • Yakov G. Sinai


Let \((X_n)_{n=1}^{\infty }\) be a sequence of independent identically distributed random variables. We study the normalized partial sums and the corresponding renormalization group flow in the space of probability densities. We prove the convergence to stable limit laws under suitable assumptions on the initial density.


Stable limit law Renormalization group Random variable  



Y. G. Sinai was supported by NSF Grant DMS 1265547. D. Li was supported in part by an Nserc discovery grant. We thank the anonymous referees for very helpful comments and suggestions.


  1. 1.
    Li, D., Sinai, Y.G.: Blow ups of complex solutions of the 3D Navier-Stokes system and renormalization group method. J. Eur. Math. Soc. (JEMS) 10(2), 267–313 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Li, D., Sinai, Y.G.: Asymptotic behavior of generalized convolutions. Regul. Chaotic Dyn. 14(2), 248–262 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. 3.
    Christoph, G., Wolf, W.: Convergence Theorems with a Stable Limit Law. Akademie Verlag, Berlin (1992)zbMATHGoogle Scholar
  4. 4.
    Gnedenko, B.V., Kolmogorov, A.N.: Sums of Independent Random Variables. Addison-Wesley, Reading, MA (1954)zbMATHGoogle Scholar
  5. 5.
    De Haan, L., Peng, L.: Exact rates of convergence to a stable law. J. London Math. Soc. 59(2), 1134–1152 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Juozulynas, A., Paulauskas, V.: Some remarks on the rate of convergence to stable limit laws. Liet. Mat. Rink. 38(4), 439–455 (1998). (Translation in Lithuanian Math. J. 38 (1998), no. 4, 335–347 (1999)Google Scholar
  7. 7.
    Kuske, R., Keller, J.B.: Rate of convergence to a stable limit law. SIAM J. Appl. Math. 61(4), 1308–1323 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd (ed.) vol. 2. John Wiley, New York (1971)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  3. 3.Landau Institute of Theoretical PhysicsMoscowRussia

Personalised recommendations