Electronic Journal of Probability

Fixed points of the multivariate smoothing transform: the critical case

Konrad Kolesko and Sebastian Mentemeier

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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

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