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

By Alexander Iksanov. 2003

63 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8bbc6f0bce918b487f11a0fcc4fc3ef95c96b1e6

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Citations in the corpus

Generalised Ornstein-Uhlenbeck Processes and the Convergence of Levy Integrals (2005, background): “For an application of the discrete time results to branching random walks, see Iksanov [25] and his references.”
Asymptotics for Weighted Random Sums (2011, background): “Such weighted random sums appear in the context of weighted branching processes and fixed-point equations of smoothing transforms (see [15, 12, 1]), and more recently, in the analysis of information ranking algorithms, e.”
Fixed points of the smoothing transform: two-sided solutions (2010, background): “3), there is a substantial literature, [2,6,7, 9,10,14,19,22], and relatively complete results.”
A survey of max-type recursive distributional equations (2004, background): “See [38] for the case where N may be infinite.”
Fixed Points of the Smoothing Transform: the Boundary Case (2005, background): “…on these and related questions; see, for example, Kahane and Peyrière (1976), Biggins (1977), Durrett and Liggett (1983), Pakes (1992), Rösler (1992), Biggins and Kyprianou (1997), Liu (1998), Liu (2000), Iksanov and Jurek (2002), Iksanov (2004), Caliebe and Rösler (2003) and Caliebe (2003).”
On Distributional Properties of Perpetuities (2008, background): “This connection was first observed in [14] and further exploited in [17].”