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

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

$$L(z)=\lim_{T\to \infty}\frac{1}{T}\log \mathsf{E}\exp\bigl\{z\mathop{\mathrm{meas}}\bigl(\Xi\cap [0,T]\bigr)\bigr\}\quad \hbox{for}\ |z|<\varepsilon(a,\lambda_{0}).$$