Normal approximation for hierarchical structures

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.Comment: 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

Bounds on the constant in the mean central limit theorem

Let $X_1,\...,X_n$ be independent with zero means, finite variances $\sigma_1^2,\...,\sigma_n^2$ and finite absolute third moments. Let $F_n$ be the distribution function of $(X_1+\...+X_n)/\sigma$, where $\sigma^2=\sum_{i=1}^n\sigma_i^2$, and $\Phi$ that of the standard normal. The $L^1$-distance between $F_n$ and $\Phi$ then satisfies $$\Vert F_n-\Phi\Vert_1\le\frac{1}{\sigma^3}\sum_{i=1}^nE|X_i|^3.$$ In particular, when $X_1,\...,X_n$ are identically distributed with variance $\sigma^2$, we have \Vert F_n-\Phi\Vert_1\le\frac{E|X_1|^3}{\sigma^3\sqrt{n}}\qquad for all $n\in\mathbb{N}$, corresponding to an $L^1$-Berry--Esseen constant of 1.Comment: Published in at http://dx.doi.org/10.1214/10-AOP527 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Berry-Esseen bound with applications to vertex degree counts in the Erd\H{o}s-R\'{e}nyi random graph

Applying Stein's method, an inductive technique and size bias coupling yields a Berry-Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdos-Renyi random graph of a given degree.Comment: Published in at http://dx.doi.org/10.1214/12-AAP848 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Zero biasing and growth processes

The tools of zero biasing are adapted to yield a general result suitable for analyzing the behavior of certain growth processes. The main theorem is applied to prove central limit theorems, with explicit error terms in the L^1 metric, for certain statistics of the Jack measure on partitions and for the number of balls drawn in a Polya-Eggenberger urn process.Comment: 21 pages. Error in one term of the bound of the main theorem has been corrected, resulting in some changes to the bound for urn proces

A Berry-Esseen bound for the uniform multinomial occupancy model

The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that \sup_{z \in \mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left( \frac{1+(\frac{n}{m})^3}{\sigma_{n,m}} \right) \quad \mbox{for all $n \ge d$ and $m \ge 2$,} where $W_{n,m}$ and $\sigma_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variable. Asymptotically, the bound is optimal up to constants if $n$ and $m$ tend to infinity together in a way such that $n/m$ stays bounded.Comment: Typo corrected in abstrac