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 = i≥1 ( i + X i ) defined by a Poisson process = i≥1 δ X i with bounded intensity function λ(t) ≤ λ 0 and a sequence of independent copies 1 , 2 , . . . of a random compact subset 0 of the real line R 1 whose diameter 0 possesses a finite exponential moment E exp{a 0 }. We first study the higher-order covariance functions E ξ(t 1 )ξ(t 2 ) · · · ξ(t k ) of the {0, 1}-valued stochastic process ξ(t) = 1 c (t) , t ∈ 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( ∩ [0, T ]) of within [0, T ] as T grows large. Second, we prove that the family of events {ξ(t) = 1} = {t / ∈ }, t ∈ R 1 , satisfies an almost-Markovtype 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 limitKeywords Poisson grain model on the line · Total length of clumps · Mixed cumulants · Higher-order covariances · Thermodynamic limit · Cramér-type large deviations relations · Berry-Esseen bound Mathematics Subject Classification (2000)Primary 60D05, 60G55 · Secondary 60G60, 62F05

The Method of Cumulants for the Normal Approximation

“…For functionals of Markov chains or stochastic processes with mixing, it is convenient to work with another set of quantities, called centered moments [112, Chapter 4], higher-order covariances [62] or Boolean cumulants [41, Section 10]. Mixing properties of the underlying stochastic process lead to good bounds on the centered moments, and then bounds on cumulants are deduced from a Boolean-to-classical cumulants formula.…”

“…Poisson cylinder processes are studied in [65], [66] and volumes of simplices in high-dimensional Poisson-Delaunay tessellations in [56]. The method is also applicable to the covered volume in the Boolean model, [51,61,62]. General functionals of random m-dependent fields are studied in [51,58,60,64].…”