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Elementary fixed points of the BRW smoothing transforms with infinite number of summands

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  • Iksanov, Aleksander M.

Abstract

The branching random walk (BRW) smoothing transform T is defined as , where given realizations {Xi}i=1L of a point process, U1,U2,... , are conditionally independent identically distributed random variables, and 0[less-than-or-equals, slant]Prob{L=[infinity]}[less-than-or-equals, slant]1. Given [alpha][set membership, variant](0,1], [alpha]-elementary fixed points are fixed points of T whose Laplace-Stieltjes transforms [phi] satisfy lims-->+0 (1-[phi](s))/s[alpha]=const>0. If [alpha]=1, these are the fixed points with finite mean. We show exactly when elementary fixed points exist. In this case these are the only fixed points of T and are unique up to a multiplicative constant. These results do not need any extra moment conditions. In particular, a distributional version of Biggins' martingale convergence theorem is proved in full generality. Essentially we apply recent results due to Lyons (Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, Vol. 84, Springer, Berlin, 1997, p. 217) and Goldie and Maller (Ann. Probab. 28 (2000) 1195), as the key point of our approach is a close connection between fixed points with finite mean and perpetuities. As a by-product, we lift from our general results the solution to a Pitman-Yor problem. Finally, we study the tail behaviour of some fixed points with finite mean.

Suggested Citation

  • Iksanov, Aleksander M., 2004. "Elementary fixed points of the BRW smoothing transforms with infinite number of summands," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 27-50, November.
    • Handle: RePEc:eee:spapps:v:114:y:2004:i:1:p:27-50
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    References listed on IDEAS

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    1. Ludwig Baringhaus & Rudolf Grübel, 1997. "On a Class of Characterization Problems for Random Convex Combinations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(3), pages 555-567, September.
    2. Iksanov, A.M.Aleksander M. & Kim, Che-Soong, 2004. "On a Pitman-Yor problem," Statistics & Probability Letters, Elsevier, vol. 68(1), pages 61-72, June.
    3. Aleksander M. Iksanov & Che Soong Kim, 2004. "New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 46(2), pages 313-321, June.
    4. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    5. Caliebe, Amke & Rösler, Uwe, 2003. "Fixed points with finite variance of a smoothing transformation," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 105-129, September.
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    Cited by:

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    2. Decrouez, Geoffrey & Hambly, Ben & Jones, Owen Dafydd, 2015. "The Hausdorff spectrum of a class of multifractal processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1541-1568.
    3. Meiners, Matthias, 2009. "Weighted branching and a pathwise renewal equation," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2579-2597, August.
    4. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
    5. Bassetti, Federico & Matthes, Daniel, 2014. "Multi-dimensional smoothing transformations: Existence, regularity and stability of fixed points," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 154-198.
    6. Gerold Alsmeyer & Alex Iksanov & Uwe Rösler, 2009. "On Distributional Properties of Perpetuities," Journal of Theoretical Probability, Springer, vol. 22(3), pages 666-682, September.

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