Electronic Journal of Probability

Fixed points of the multivariate smoothing transform: the critical case

Konrad Kolesko and Sebastian Mentemeier

Full-text: Open access

Abstract

Given a sequence T_1, T_2, ... of random d-by-d matrices with nonnegative entries, suppose there is a random vector X with nonnegative entries, such that the sum T_1 X_1 + T_2 X_2 +...  has the same law as X, with X_1, X_2, ... being i.i.d.copies of X, independent of T_1, T_2, ... Then (the law of) X is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case d=1, a function m is introduced, such that the existence of $\alpha \in (0,1]$ with $m(\alpha)=1$ and $m'(\alpha) \le 0$ guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case $m'(\alpha)=0$ and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 52, 24 pp.

Dates
Accepted: 11 May 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067158

Digital Object Identifier
doi:10.1214/EJP.v20-4022

Mathematical Reviews number (MathSciNet)
MR3347921

Zentralblatt MATH identifier
1326.60021

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G44: Martingales with continuous parameter

Keywords
Multivariate Smoothing Transform Branching Random Walk Derivative Martingale Harris Recurrence Products of Random Matrices Markov Random Walk

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kolesko, Konrad; Mentemeier, Sebastian. Fixed points of the multivariate smoothing transform: the critical case. Electron. J. Probab. 20 (2015), paper no. 52, 24 pp. doi:10.1214/EJP.v20-4022. https://projecteuclid.org/euclid.ejp/1465067158


Export citation

References

  • Aidekon, Elie. Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 (2013), no. 3A, 1362–1426.
  • Aidekon, Elie; Shi, Zhan. The Seneta-Heyde scaling for the branching random walk. Ann. Probab. 42 (2014), no. 3, 959–993.
  • Alsmeyer, Gerold. Recurrence theorems for Markov random walks. Probab. Math. Statist. 21 (2001), no. 1, Acta Univ. Wratislav. No. 2298, 123–134.
  • Alsmeyer, Gerold; Meiners, Matthias. Fixed points of inhomogeneous smoothing transforms. J. Difference Equ. Appl. 18 (2012), no. 8, 1287–1304.
  • Alsmeyer, Gerold; Mentemeier, Sebastian. Tail behaviour of stationary solutions of random difference equations: the case of regular matrices. J. Difference Equ. Appl. 18 (2012), no. 8, 1305–1332.
  • Athreya, K. B.; Ney, P. A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 (1978), 493–501.
  • Basrak, Bojan; Davis, Richard A.; Mikosch, Thomas. A characterization of multivariate regular variation. Ann. Appl. Probab. 12 (2002), no. 3, 908–920.
  • Biggins, J. D.; Kyprianou, A. E. Seneta-Heyde norming in the branching random walk. Ann. Probab. 25 (1997), no. 1, 337–360.
  • Biggins, J. D.; Kyprianou, A. E. Measure change in multitype branching. Adv. in Appl. Probab. 36 (2004), no. 2, 544–581.
  • Biggins, J. D.; Kyprianou, A. E. Fixed points of the smoothing transform: the boundary case. Electron. J. Probab. 10 (2005), no. 17, 609–631.
  • Biggins, J. D.; Rahimzadeh Sani, A. Convergence results on multitype, multivariate branching random walks. Adv. in Appl. Probab. 37 (2005), no. 3, 681–705.
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
  • Bochner, Salomon. Harmonic analysis and the theory of probability. University of California Press, Berkeley and Los Angeles, 1955. viii+176 pp.
  • Ding, Jian; Zeitouni, Ofer. Extreme values for two-dimensional discrete Gaussian free field. Ann. Probab. 42 (2014), no. 4, 1480–1515.
  • Breiman, Leo. The strong law of large numbers for a class of Markov chains. Ann. Math. Statist. 31 1960 801–803.
  • Buraczewski, Dariusz. On tails of fixed points of the smoothing transform in the boundary case. Stochastic Process. Appl. 119 (2009), no. 11, 3955–3961.
  • Buraczewski, Dariusz; Damek, Ewa; Guivarc'h, Yves; Mentemeier, Sebastian. On multidimensional Mandelbrot cascades. J. Difference Equ. Appl. 20 (2014), no. 11, 1523–1567.
  • Buraczewski, Dariusz; Kolesko, Konrad. Linear stochastic equations in the critical case. J. Difference Equ. Appl. 20 (2014), no. 2, 188–209.
  • Buraczewski, D. and Mentemeier, S., (2015+). Precise Large Deviation Results for Products of Random Matrices, to appear in Ann. Inst. H. Poincaré Probab. Statist.
  • Durrett, Richard; Liggett, Thomas M. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275–301.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Goldie, Charles M. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991), no. 1, 126–166.
  • Guivarc'h, Yves. Sur une extension de la notion de loi semi-stable. (French) [On an extension of the notion of semistable law] Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), no. 2, 261–285.
  • Iksanov, A. and Meiners, M. (2015). Fixed points of multivariate smoothing transforms with scalar weights, ALEA 12, 69–114.
  • Jagers, Peter. Branching processes with biological applications. Wiley Series in Probability and Mathematical Statistics-Applied Probability and Statistics. Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. xiii+268 pp. ISBN: 0-471-43652-6
  • Kesten, Harry. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207–248.
  • Kyprianou, A. E. Martingale convergence and the stopped branching random walk. Probab. Theory Related Fields 116 (2000), no. 3, 405–419.
  • Kyprianou, A. E. Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab. 35 (1998), no. 4, 795–801.
  • Kyprianou, Andreas E.; Rahimzadeh Sani, A. Martingale convergence and the functional equation in the multi-type branching random walk. Bernoulli 7 (2001), no. 4, 593–604.
  • Liu, Quansheng. Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 (1998), no. 1, 85–112.
  • Mentemeier, S. (2013). On multivariate stochastic fixed point equations: The smoothing transform and random difference equations. Ph.D. thesis, Westfälische Wilhelms-Universität Münster.
  • Mentemeier, S. (2015+). The Fixed Points of the Multivariate Smoothing Transform to appear in Probab. Theory Related Fields, DOI: 10.1007/s00440-015-0615-y.
  • Nerman, Olle. On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57 (1981), no. 3, 365–395.
  • Nummelin, E. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 (1978), no. 4, 309–318.