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An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line

An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line We consider a not necessarily stationary one-dimensional Boolean model Ξ=∪ i≥1(Ξ i +X i ) defined by a Poisson process $\Psi=\sum_{i\ge 1}\delta_{X_{i}}$ with bounded intensity function λ(t)≤λ 0 and a sequence of independent copies Ξ 1,Ξ 2,… of a random compact subset Ξ 0 of the real line ℝ1 whose diameter ‖Ξ 0‖ possesses a finite exponential moment $\mathsf{E}\exp\{a\|\Xi_{0}\|\}$ . We first study the higher-order covariance functions $\mathop{\mathsf{E}}\limits^{\frown}\xi(t_{1})\xi(t_{2})\cdots \xi(t_{k})$ of the {0,1}-valued stochastic process $\xi(t)=\mathbf{1}_{\Xi^{c}(t)},\ t\in \mathbb{R}^{1}$ , and derive exponential estimates of them as well as of the mixed cumulants Cum k (ξ(t 1),ξ(t 2),…,ξ(t k )). From this, we derive Cramér-type large deviations relations and a Berry–Esseen bound for the distribution of empirical total length meas(Ξ∩[0,T]) of Ξ within [0,T] as T grows large. Second, we prove that the family of events {ξ(t)=1}={t ∉ Ξ}, t∈ℝ1, satisfies an almost-Markov-type mixing condition with an exponentially decaying mixing rate. In case of a stationary Boolean model, i.e. λ(t)≡λ 0, these properties enable us to show the existence and analyticity of the thermodynamic limit $$L(z)=\lim_{T\to \infty}\frac{1}{T}\log \mathsf{E}\exp\bigl\{z\mathop{\mathrm{meas}}\bigl(\Xi\cap [0,T]\bigr)\bigr\}\quad \hbox{for}\ |z|<\varepsilon(a,\lambda_{0}).$$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line

Acta Applicandae Mathematicae , Volume 96 (3) – Mar 23, 2007

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References (23)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Springer Science + Business Media B.V.
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-007-9105-2
Publisher site
See Article on Publisher Site

Abstract

We consider a not necessarily stationary one-dimensional Boolean model Ξ=∪ i≥1(Ξ i +X i ) defined by a Poisson process $\Psi=\sum_{i\ge 1}\delta_{X_{i}}$ with bounded intensity function λ(t)≤λ 0 and a sequence of independent copies Ξ 1,Ξ 2,… of a random compact subset Ξ 0 of the real line ℝ1 whose diameter ‖Ξ 0‖ possesses a finite exponential moment $\mathsf{E}\exp\{a\|\Xi_{0}\|\}$ . We first study the higher-order covariance functions $\mathop{\mathsf{E}}\limits^{\frown}\xi(t_{1})\xi(t_{2})\cdots \xi(t_{k})$ of the {0,1}-valued stochastic process $\xi(t)=\mathbf{1}_{\Xi^{c}(t)},\ t\in \mathbb{R}^{1}$ , and derive exponential estimates of them as well as of the mixed cumulants Cum k (ξ(t 1),ξ(t 2),…,ξ(t k )). From this, we derive Cramér-type large deviations relations and a Berry–Esseen bound for the distribution of empirical total length meas(Ξ∩[0,T]) of Ξ within [0,T] as T grows large. Second, we prove that the family of events {ξ(t)=1}={t ∉ Ξ}, t∈ℝ1, satisfies an almost-Markov-type mixing condition with an exponentially decaying mixing rate. In case of a stationary Boolean model, i.e. λ(t)≡λ 0, these properties enable us to show the existence and analyticity of the thermodynamic limit $$L(z)=\lim_{T\to \infty}\frac{1}{T}\log \mathsf{E}\exp\bigl\{z\mathop{\mathrm{meas}}\bigl(\Xi\cap [0,T]\bigr)\bigr\}\quad \hbox{for}\ |z|<\varepsilon(a,\lambda_{0}).$$

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Mar 23, 2007

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