The functional equation of the smoothing transform
By Gerold Alsmeyer, J. Biggins, M. Meiners. 2009
83 citations recorded by the source corpus.
Given a sequence T = (Ti)i� 1 of non-negative random variables, a function f on the positive halfline can be transformed to E Q i� 1 f(tTi). 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 [0,1 ), the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, X d
Subfields: Artificial Intelligence, Finance, Management Science and Operations Research
View Paper: https://www.semanticscholar.org/paper/7aea2e29f541771700da30ff9eecd10bdd42bc7e
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In Artificial Intelligence
In Finance
In Management Science and Operations Research
Citations in the corpus
- Branching random walks (2015, background): “A key ingredient is to study fixed points of the so-called smoothing transforms (Durrett and Liggett [103], Alsmeyer [20], Alsmeyer, Biggins and Meiners [21]).”
- Fixed points of the smoothing transform: two-sided solutions (2010, methodology): “2) in [2], we infer that μα := E ∑ v∈G> 1 e −αS(v)δS(v) is the distribution of the first ladder height of the random walk (Sn)n≥0, that is, μα (·) = P(Sσ ∈ ·) where σ := inf{n ≥ 0 : Sn > 0}.”