uu.seUppsala University Publications
Change search
ReferencesLink to record
Permanent link

Direct link
The functional equation of the smoothing transform
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
2012 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 40, no 5, 2069-2105 p.Article in journal (Refereed) Published
Abstract [en]

Given a sequence T = (T-i)(i >= 1) of nonnegative random variables, a function f on the positive halfline can be transformed to E Pi(i >= 1) f (tT(i)). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on [0, infinity), the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, X =(d) Sigma(i >= 1) TiXi, where =(d) denotes equality of the corresponding laws, and X-1, X-2, ... is a sequence of i.i.d. copies of X independent of T. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation X =(d) inf{X-i/T-i:i >= 1, T-i > 0}. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.

Place, publisher, year, edition, pages
2012. Vol. 40, no 5, 2069-2105 p.
Keyword [en]
Branching process, branching random walk, Choquet-Deny-type functional equation, endogeny, fixed point, general branching process, multiplicative martingales, smoothing transformation, stochastic fixed-point equation, Weibull distribution, weighted branching
National Category
Natural Sciences
URN: urn:nbn:se:uu:diva-188569DOI: 10.1214/11-AOP670ISI: 000311005600007OAI: oai:DiVA.org:uu-188569DiVA: diva2:578524
Available from: 2012-12-18 Created: 2012-12-17 Last updated: 2013-03-04Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text
By organisation
Mathematical Statistics
In the same journal
Annals of Probability
Natural Sciences

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 124 hits
ReferencesLink to record
Permanent link

Direct link