Fixed points of inhomogeneous smoothing transforms
2012 (English)In: Journal of difference equations and applications (Print), ISSN 1023-6198, Vol. 18, no 8, 1287-1304 p.Article in journal (Refereed) Published
We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation X=C-d + Sigma(i >= 1) TiXi, where =(d) means equality in distribution, (C, T-1, T-2, . . .) is a given sequence of non-negative random variables and X-1, X-2, . . . is a sequence of i.i.d. copies of the non-negative random variable X independent of (C, T-1, T-2, . . .). In this situation, X (or, more precisely, the distribution of X) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Furthermore, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C = 0. Using this correspondence and the known theory on the homogeneous equation, we present a full characterization of the set of fixed points under mild assumptions.
Place, publisher, year, edition, pages
2012. Vol. 18, no 8, 1287-1304 p.
branching random walk, fixed point, multiplicative martingales, smoothing transform, stochastic fixed-point equation, weighted branching process
IdentifiersURN: urn:nbn:se:uu:diva-180298DOI: 10.1080/10236198.2011.589514ISI: 000306746300003OAI: oai:DiVA.org:uu-180298DiVA: diva2:549076