Normal approximation for hierarchical structures

Details Normal approximation for hierarchical structures Normal approximation for hierarchical structures

2005-03-24

arxiv.org/abs/math/0503549v1

Annals of Applied Probability 2004, Vol. 14, No. 4, 1950-1969

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051604000000440 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051604000000440

Given F:[a,b]^k\to [a,b] and a nonconstant X_0 with P(X_0\in [a,b])=1, define the hierarchical sequence of random variables {X_n}_{n\ge 0} by X_{n+1}=F(X_{n,1},...,X_{n,k}), where X_{n,i} are i.i.d. as X_n. 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 C\gamma^n for 0<\gamma<1 is obtained on the Wasserstein distance between the standardized distribution of X_n 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.

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