Limit theorems for random normalized distortion

Cohort, Pierre

doi:10.1214/aoap/1075828049

Cited by 24 publications

(13 citation statements)

References 20 publications

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“…points sets (that is to say letting X n consist of i.i.d. 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.…”

Section: Functionals On Gibbsian Loss Networkmentioning

confidence: 96%

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: 96%

Limit theorems for geometric functionals of Gibbs point processes

Schreiber¹,

Yukich²

2013

Ann. Inst. H. Poincaré Probab. Statist.

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“…of N (0, I d ). This rather unexpected choice is motivated by the fact that this distribution provides the lowest in average random quantization error (see [2]). …”

Section: Competitive Learning Vector Quantization (Clv Q) This Procementioning

confidence: 99%

Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces

2010

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Abstract.We describe quantization designs which lead to asymptotically and order optimal functional quantizers for Gaussian processes in a Hilbert space setting. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions.Mathematics Subject Classification. 60G15, 60E99.

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“…By specialising the general results of [14] for the linear Bezier surfaces it is possible to express the approximation error as v(y, X ) − f (y) = 1 i =j 3 u i u j q y, C f (y, X ) , where q(y, C f (y, X )) is the half of the mean value for the second order directional derivative f (z)(x i − x j ), (x i − x j ) for z ∈ C f (y, X ) if the points y, x i , x j form a non-degenerate triangle. This mean value can be represented as A(ϕ(y, x i , x j ))(x i − x j ) 2 for a certain point ϕ from the corresponding triangle with vertices y, x i , x j , where A is the matrix-valued function such that 1 Denote the right-hand side by ψ(y, x 1 , x 2 , x 3 ). Note that the order of the vertices of C f (y, X ) is not essential because of the symmetry in this definition.…”

Section: Optimal Bezier Approximationsmentioning

confidence: 99%

Optimal approximation and quantisation

Molchanov

Tontchev

2007

Journal of Mathematical Analysis and Applications

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A number of interesting functionals F (X ) of a finite point set X in the Euclidean space can be represented as integrals of a function η that depends on the integration variable y and X restricted onto a certain set C(y, X ) that is determined by y and X and satisfies separation and uniform boundedness conditions. For instance, C(y, X ) can be the Voronoi cell generated by X that contains y. We single out the general properties of C and η that ensure that the normalised infimum of F (X ) over all sets X with cardinality n converges to a limit that can be identified using a particular form of η. The considered functionals include those that appear in quantisation problems for probability measures, and in finding the optimal approximation of a function by splines, tangent planes and triangulated surfaces. For instance, it is shown that the minimum L β -approximation error (normalised by n β ) of a sufficiently smooth bivariate convex function f using the convex triangulation converges to the L 1/(β+1) -norm of the determinant of the second derivative of f times a certain absolute constant.

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Limit theorems for random normalized distortion

Cited by 24 publications

References 20 publications

Limit theorems for geometric functionals of Gibbs point processes

Limit theorems for geometric functionals of Gibbs point processes

Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces

Optimal approximation and quantisation

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