Optimal Poisson quantisation

Molchanov, Ilya; Tontchev, Nikolay

doi:10.1016/j.spl.2007.02.004

Cited by 2 publications

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“…random variables) was first investigated by Zador [32] and later by Graf and Luschgy [14] and Cohort [7]. Letting X n be i. Molchanov and Tontchev [20] have pointed out the desirability for quantization via Poisson point sets and our purpose here is to establish asymptotics of the quantization error on Gibbsian input. This is done as follows.…”

Section: Functionals On Gibbsian Loss Networkmentioning

confidence: 97%

See 1 more Smart Citation

Limit theorems for geometric functionals of Gibbs point processes

Schreiber¹,

Yukich²

2013

Ann. Inst. H. Poincaré Probab. Statist.

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Given a Gibbs point process P Ψ on R d having a weak enough potential Ψ, we consider the random measures µ λ := P x∈P Ψ ∩Q λ ξ(x, P Ψ ∩ Q λ )δ x/λ 1/d , where Q λ := [−λ 1/d /2, λ 1/d /2] d is the volume λ cube and where ξ(·, ·) is a translation invariant stabilizing functional. Subject to Ψ satisfying a localization property and translation invariance, we establish weak laws of large numbers for λ −1 µ λ (f ), f a bounded test function on R d , and weak convergence of λ −1/2 µ λ (f ), suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input. American Mathematical Society 2000 subject classifications. Primary 60F05, 60G55, Secondary 60D051 where X ⊂ R d is locally finite and where the function ξ, defined on all pairs (x, X ), with x ∈ X , represents the interaction of x with respect to X . When X is a random n point set in R d (i.e. a finite spatial point process), the asymptotic analysis of the suitably scaled sums (1.1) as n → ∞ can often be handled by M -dependent methods, ergodic theory, or mixing methods. However there are situations where these classical methods are either not directly applicable, do not give explicit asymptotics in terms of underlying geometry and point densities, or do not easily yield explicit rates of convergence. Stabilization methods originating in [23] and further developed in [3,24,26], provide another approach for handling sums of spatially dependent terms.There are several similar definitions of stabilization, but the essence is captured by the notion of stabilization of the functional ξ with respect to a rate τ > 0 homogeneous Poisson point processfor all z ∈ R d . Let B r (x) denote the Euclidean ball centered at x with radius r ∈ R + := [0, ∞).Letting 0 denote the origin of R d , we say that a translation invariant ξ is stabilizing on P = P τ if there exists an a.s. finite random variable R := R ξ (τ ) (a 'radius of stabilization') such thatConsider the point measures3) where δ x denotes the unit Dirac point mass at x whereas Q λ := [−λ 1/d /2, λ 1/d /2] d is the λ-volume cube. Let B(Q 1 ) denote the class of all bounded f : Q 1 → R and for all random point measures µ on R d let f, µ := f dµ and letμ := µ − E [µ]. Stabilization of translation invariant ξ on P, as defined in (1.2), together with stabilization of ξ on P ∩ Q λ , λ ≥ 1, when combined with appropriate moment conditions on ξ, yields for all f ∈ B(Q 1 ) the law of large numbers [22, 25] lim λ→∞ λ −1 f, µ λ = τ E [ξ(0, P)] Q1 f (x)dx in L 1 and in L 2 , (1.4) and, if the stabilization radii on P and P ∩ Q λ , λ ≥ 1, decay exponentially fast, then [3, 21] lim λ→∞ λ −1 Var[ f, µ λ ] ...

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Section: Functionals On Gibbsian Loss Networkmentioning

confidence: 97%

“…Molchanov and Tontchev [20] have pointed out the desirability for quantization via Poisson point sets and our purpose here is to establish asymptotics of the quantization error on Gibbsian input. This is done as follows.…”

Section: Functionals On Gibbsian Loss Networkmentioning

confidence: 99%