Probability Theory and Related Fields

, Volume 156, Issue 3–4, pp 665–705

On fixed points of a generalized multidimensional affine recursion

Open Access
Article

Abstract

Let G be a multiplicative subsemigroup of the general linear group Gl \({(\mathbb{R}^d)}\) which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a G-valued random matrix A, we consider the following generalized multidimensional affine equation
$$R\stackrel{\mathcal{D}}{=} \sum_{i=1}^N A_iR_i+B,$$
where N ≥ 2 is a fixed natural number, A1, . . . , AN are independent copies of \({A, B \in \mathbb{R}^d}\) is a random vector with positive entries, and R1, . . . , RN are independent copies of \({R \in \mathbb{R}^d}\) , which have also positive entries. Moreover, all of them are mutually independent and \({\stackrel{\mathcal{D}}{=}}\) stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc’h and Le Page (Simplicité de spectres de Lyapounov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004; On matricial renewal theorems and tails of stationary measures for affine stochastic recursions, Preprint, 2011) and Kesten’s renewal theorem (Kesten in Ann Probab 2:355–386, 1974), that under appropriate conditions, there exists χ >  0 such that \({{\mathbb{P}(\{\langle R, u \rangle > t\})\asymp t^{-\chi}}}\) , as t → ∞, for every unit vector \({u \in \mathbb{S}^{d-1}}\) with positive entries.

Keywords

Heavy tailed random variables Renewal theory Stationary measures Markov chains Spectral theory 

Mathematics Subject Classification

60H25 60K05 60J80 

References

  1. 1.
    Bogachev V.I.: Measure Theory. Springer, Berlin (2006)Google Scholar
  2. 2.
    Breiman L.: The strong law of large numbers for a class of Markov chains. Ann. Math. Stat. 31(3), 801–803 (1960)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Buraczewski, D., Damek, E., Guivarc’h, Y.: On multidimensional Mandelbrot’s cascades. Preprint (2010). http://www.math.uni.wroc.pl/~dbura/publications/man110707.pdf
  4. 4.
    Buraczewski, D., Damek, E., Mentemeier, S., Mirek, M.: Heavy tailed solutions of multivariate smoothing transforms. Preprint (2012)Google Scholar
  5. 5.
    Goldie Ch.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Guivarc’h, Y., Le Page, É.: Simplicité de spectres de Lyapounov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. Random Walks and Geometry, pp. 181–259. Walter de Gruyter GmbH & Co. KG, Berlin (2004)Google Scholar
  7. 7.
    Guivarc’h, Y., Le Page, É.: On matricial renewal theorems and tails of stationary measures for affine stochastic recursions. Preprint (2011)Google Scholar
  8. 8.
    Guivarc’h, Y., Raugi, A.: Products of Random Matrices: Convergence Theorems. Random Matrices and Their Applications (Brunswick, Maine, 1984). Contemporary Mathematics, vol. 50, pp. 31–54. American Mathematical Society, Providence (1986)Google Scholar
  9. 9.
    Guivarc’h Y., Urban R.: Semigroups actions on tori and stationary measures on projective spaces. Stud. Math. 171(1), 33–66 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hennion H.: Limit theorems for products of positive random matrices. Ann. Probab. 25(4), 1545–1587 (1997)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hennion H., Hervé L.: Stable laws and products of positive random matrices. J. Theor. Probab. 21(4), 966–981 (2008)MATHCrossRefGoogle Scholar
  12. 12.
    Horn R.A., Johnson Ch.R.: Topics in Matrix Analysis. Cambridge University Press, London (1994)MATHGoogle Scholar
  13. 13.
    Jelenković, P.R., Olvera-Cravioto, M.: Information ranking and power laws on trees. Adv. Appl. Probab. (2010, in press). http://arxiv.org/abs/0905.1738
  14. 14.
    Jelenković, P.R., Olvera-Cravioto, M.: Implicit renewal theorem and power tails on trees (2010). http://arxiv.org/abs/1006.3295
  15. 15.
    Jelenković, P.R., Olvera-Cravioto, M.: Implicit renewal theorem for trees with general weights (2010). http://arxiv.org/abs/1012.2165
  16. 16.
    Kesten H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kesten H.: Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2, 355–386 (1974)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Liu Q.: Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107 (2001)MATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WroclawWroclawPoland

Personalised recommendations