Convergence of the population dynamics algorithm in the Wasserstein metric. (English) Zbl 1441.60053

The author is interested in weighted branching processes as solutions of stochastic fixed-point equations in distribution (SFPE) of the form \(R \stackrel{\mathcal D}= \Phi(Q,N,\{C_i\},\{R_i\})\), where \((Q,N,\{C_i\}\) is a random vector of real-valued elements, \(N\) a natural number, and \(\{R_i\}\) a sequence of i.i.d. copies of \(R\), independent of \((Q,N,\{C_i\})\). In case of non-uniqueness these solutions can be characterized by the only attracting one among them, which can thus be constructed by iterating the SFPE. The focus here is on the “population dynamics algorithm” described in [M. Mézard and A. Montanari, Information, physics and computation. Oxford: Oxford University Press (2009; Zbl 1163.94001)]. Using an iterative bootstrap technique, it is superior to naïve Monte Carlo methods. However, as the samples thus obtained are neither independent nor exactly distributed according to the target distribution, convergence of the algorithm must be investigated. On the basis of Lipschitz conditions and suitable boundedness assumptions, the author proves convergence in the Wasserstein metric [C. Villani, Optimal transport. Old and new. Berlin: Springer (2009; Zbl 1156.53003)] of order \(p\geq 1\). She also shows consistency of the estimators based on the sample pool produced by the algorithm. Illustrating the broad spectrum of applications, four explicit examples are given: The linear SFPE or “smoothing transform”, the maximum SFPE or “high order Lindley equation”, the discounted tree-sum SFPE, and the “free-entropy” SFPE. For further examples the reader is referred to [G. Alsmeyer et al., Ann. Probab. 40, No. 5, 2069–2105 (2012; Zbl 1266.39022)].


60H25 Random operators and equations (aspects of stochastic analysis)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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