Fixed points of smoothing transformation in random environment

Abstract

At each time n\in \mathbb{N},\mathrm{l}\mathrm{e}\mathrm{t}{\overline{Y}}^{(n)}(\xi )=(y_1^{(n)}(\xi ),y_2^{(n)}(\xi ),\cdots ) be a random sequence of non-negative numbers that are ultimately zero in a random environment\xi \text{=}（{\xi }_n{）}_{n\in \mathbb{N}}. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.\xi ,Z(\xi )\stackrel{\text{d}}{=}{\sum }_{i\in \mathbb{N}+}{y}_i^{(\mathrm{0})}(\xi )Z_i^{(\mathrm{1})}(\xi )，where ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝ are random variables in random environment which satisfy that for any environment\xi; under P_{\xi }; ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝are independent of each other and {\bar{Y}}^{(\mathrm{0})}(\xi ), and have the same conditional distribution P_{\xi }(Z_i^{(1)}(\xi )\in \cdot )=P_{T\xi }(Z(T\xi )\in \cdot ) where T is the shift operator. This extends the classical results of J. D. Biggins [J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.

Keywords

Smoothing transformation / functional equation / branching random walk / random environment / martingales

References

[1] Alsmeyer G, Biggins J D, Meiners M. The functional equation of the smoothing transform. Ann Probab, 2012, 40: 2069–2105

[2] Biggins J D. Martingale convergence in the branching random walk. J Appl Probab, 1977, 14: 25–37

[3] Biggins J D, Kyprianou A E. Seneta-Heyde norming in the branching random walk. Ann Probab, 1997, 25: 337–360

[4] Biggins J D, Kyprianou A E. Measure change in multitype branching. Adv in Appl Probab, 2004, 36: 544–581

[5] Biggins J D, Kyprianou A E. Fixed points of the smoothing transform: the boundary case. Electron J Probab, 2005, 10: 609–631

[6] Caliebe A. Symmetric fixed points of a smoothing transformation. Adv in Appl Probab, 2003, 35: 377–394

[7] Caliebe A, Rösler U. Fixed points with finite variance of a smoothing transform. Stochastic Process Appl, 2003, 107: 105–129

[8] Doney R A. A limit theorem for a class of supercritical branching processes. J Appl Probab, 1972, 9: 707–724

[9] Durrett R, Liggett M. Fixed points of the smoothing transform. Z Wahrsch Gebiete, 1983, 64: 275–301

[10] Iksanov A M. Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stochastic Process Appl, 2004, 114: 27–50

[11] Iksanov A M, Jurek Z J. On fixed points of Poisson shot noise transform. Adv in Appl Probab, 2002, 34: 798–825

[12] Kuhlbusch D. On weighted branching processes in random environment. Stochastic Process Appl, 2004, 109: 113–144

[13] Liu Q S. Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv in Appl Probab, 1998, 30: 85–112

[14] Liu Q S. On generalized multiplicative cascades. Stochastic Process Appl, 2000, 86: 263–286

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