Skip to main content

Convergence to Stable Laws for Multidimensional Stochastic Recursions: The Case of Regular Matrices

Abstract

Given a sequence (M n , Q n ) n ≥ 1 of i.i.d. random variables with generic copy (M, Q) ∈ GL(d, ℝ) ×ℝd , we consider the random difference equation (RDE)

$$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$

n ≥ 1, and assume the existence of κ > 0 such that

$$ \lim_{n \to \infty} \left(\mathbb{E}\ensuremath{\left\| {M_1 \cdots M_n}^\kappa \right\|}\right)^{\frac{1}{n}} = 1 . $$

We prove, under suitable assumptions, that the sequence S n  = R 1 + ... + R n , appropriately normalized, converges in law to a multidimensional stable distribution with index κ. As a by-product, we show that the unique stationary solution R of the RDE is regularly varying with index κ, and give a precise description of its tail measure.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alsmeyer, G., Mentemeier, S.: Tail behaviour of stationary solutions of random difference equations: the case of regular matrices. J. Differ. Equ. Appl. (2012, to appear). doi:10.1080/10236198.2011.571383

    MathSciNet  Google Scholar 

  2. 2.

    Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12(3), 908–920 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Benda, M.: A central limit theorem for contractive stochastic dynamical systems. J. Appl. Probab. 35(1), 200–205 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Boman, J., Lindskog, F.: Support theorems for the Radon transform and Cramér-Wold theorems. J. Theor. Probab. 22(3), 683–710 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bougerol, P., Picard, N.: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20(4), 1714–1730 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Buraczewski, D., Damek, E., Guivarc’h, Y.: Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Relat. Fields 148, 333–402 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Buraczewski, D., Damek, E., Guivarc’h, Y.: On multidimensional Mandelbrot’s cascades. arXiv:1109.1845v1 (2011, submitted)

  8. 8.

    Buraczewski, D., Damek, E., Guivarc’h, Y., Hulanicki, A., Urban, R.: Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Relat. Fields 145(3–4), 385–420 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Buraczewski, D., Damek, E., Mirek, M.: Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems. Stoch. Process. Their Appl. 122(1), 42–67 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    de Haan, L., Resnick, S.I., Rootzén, H., de Vries, C.G.: Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stoch. Process. Their Appl. 32(2), 213–224 (1989)

    MATH  Article  Google Scholar 

  11. 11.

    de Saporta, B., Guivarc’h, Y., Le Page, Y.: On the multidimensional stochastic equation Y n + 1 = A n Y n  + B n . C. R. Math. Acad. Sci. Paris 339(7), 499–502 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Diaconis, P., Freedman, D.: Iterated random functions. SIAM Rev. 41, 45–76 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Gao, Z., Guivarc’h, Y., Le Page, E.: Spectral gap properties and convergence to stable laws for affine random walks on ℝd. arXiv:1108.3146v2 (2011, submitted)

  14. 14.

    Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Guivarc’h, Y., Le Page, E.: On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Ergod. Theory Dyn. Syst. 28(2), 423–446 (2008)

    MATH  Google Scholar 

  16. 16.

    Guivarc’h, Y., Le Page, E.: Spectral gap properties and asymptotics of stationary measures for affine random walks. arXiv:1204.6004v1 (2012, submitted)

  17. 17.

    Hennion, H., Hervé, L.: Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics, vol. 1766. Springer, Berlin (2001)

    MATH  Book  Google Scholar 

  18. 18.

    Hennion, H., Hervé, L.: Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32(3A), 1934–1984 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Hervé, L., Pène, F.: The Nagaev-Guivarc’h method via the Keller-Liverani theorem. Bull. Soc. Math. Fr. 138(3), 415–489 (2010)

    MATH  Google Scholar 

  20. 20.

    Keller, G., Liverani, C.: Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 28(1), 141–152 (1999)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Klüppelberg, C., Pergamenchtchikov, S.: The tail of the stationary distribution of a random coefficient AR(q) model. Ann. Appl. Probab. 14(2), 971–1005 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Klüppelberg, C., Pergamenchtchikov, S.: Extremal behaviour of models with multivariate random recurrence representation. Stoch. Process. Their Appl. 117(4), 432–456 (2007)

    MATH  Article  Google Scholar 

  24. 24.

    Le Page, É.: Théorèmes de renouvellement pour les produits de matrices aléatoires. Séminaires de probabilités Rennes. Publication des Séminaires de Mathématiques, Univ. Rennes I, pp. 1–116 (1983)

  25. 25.

    Mirek, M.: Heavy tail phenomenon and convergence to stable laws for iterated lipschitz maps. Probab. Theory Relat. Fields 151, 705–734 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Mirek, M.: On fixed points of a generalized multidimensional affine recursion. Probab. Theory Relat. Fields. arXiv:1111.1756v1 (2012, accepted)

  27. 27.

    Nagaev, S.V.: Some limit theorems for stationary markov chains. Theory Probab. Appl. 2(4), 378–406 (1957)

    Article  Google Scholar 

  28. 28.

    Wu, W.B., Woodroofe, M.: A central limit theorem for iterated random functions. J. Appl. Probab. 37(3), 748–755 (2000)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sebastian Mentemeier.

Additional information

E. Damek was partially supported by MNiSW grant N N201 393937, M. Mirek was partially supported by MNiSW grant N N201 392337, J. Zienkiewicz was partially supported by MNiSW grant N N201 397137, S. Mentemeier was supported by the Deutsche Forschungsgemeinschaft (SFB 878).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Damek, E., Mentemeier, S., Mirek, M. et al. Convergence to Stable Laws for Multidimensional Stochastic Recursions: The Case of Regular Matrices. Potential Anal 38, 683–697 (2013). https://doi.org/10.1007/s11118-012-9292-y

Download citation

Keywords

  • Weak limit theorems
  • Random difference equations
  • Stable laws
  • Stochastic recursions
  • Multivariate regular variation

Mathematics Subject Classifications (2010)

  • Primary 60F05; Secondary 60J05
  • 60E07
  • 60H25