2004/11/01 by Larry Goldstein
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60F05 #msc:60G18 #msc:82D30
paper · pdf · doi:10.1214/105051604000000440
published as Annals of Applied Probability 2004, Vol. 14, No. 4, 1950-1969 · Published at http://dx.doi.org/10.1214/105051604000000440 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2004/11/01 · arxiv created 2005/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given F:[a,b]k→[a,b] and a nonconstant X0 with P(X0∈[a,b])=1, define the hierarchical sequence of random variables Xnn≥0 by Xn+1=F(Xn,1,…,Xn,k), where Xn,i are i.i.d. as Xn. Such sequences arise from hierarchical structures which have been extensively studied in the physics literature to model, for example, the conductivity of a random medium. Under an averaging and smoothness condition on nontrivial F, an upper bound of the form Cγn for 0<γ<1 is obtained on the Wasserstein distance between the standardized distribution of Xn and the normal. The results apply, for instance, to random resistor networks and, introducing the notion of strict averaging, to hierarchical sequences generated by certain compositions. As an illustration, upper bounds on the rate of convergence to the normal are derived for the hierarchical sequence generated by the weighted diamond lattice which is shown to exhibit a full range of convergence rate behavior.