scholarly journals Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics

2020 ◽  
Vol 57 (3) ◽  
pp. 928-955
Author(s):  
Viktor Beneš ◽  
Christoph Hofer-Temmel ◽  
Günter Last ◽  
Jakub Večeřa
Keyword(s):  
Limit Theorem ◽  
Asymptotic Properties ◽  
Boolean Model ◽  
Uniqueness Result ◽  
Finite Range ◽  
Activity Parameter ◽  
Negative Interaction ◽  
U Statistics ◽  
Disagreement Percolation ◽  
Particle Process

AbstractWe study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation, we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A by-product of our approach is a new uniqueness result for Gibbs particle processes.

Almost sure central limit theorem for the products of U-statistics

Metrika ◽  
2009 ◽  
Vol 73 (1) ◽  
pp. 61-76 ◽  
Author(s):  
Zuoxiang Peng ◽  
Zhongquan Tan ◽  
Saralees Nadarajah
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
U Statistics

scholarly journals Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

2017 ◽  
Vol 54 (2) ◽  
pp. 569-587 ◽  
Author(s):  
Ollivier Hyrien ◽  
Kosto V. Mitov ◽  
Nikolay M. Yanev
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Cell Populations ◽  
Subcritical Case ◽  
Immigration Process ◽  
Large Numbers

Abstract We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1996 ◽  
Vol 28 (02) ◽  
pp. 333-334
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
Individual Grains

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

The Estimation of mean shape and mean particle number in overlapping particle systems in the plane

1995 ◽  
Vol 27 (01) ◽  
pp. 102-119 ◽  
Author(s):  
Wolfgang Weil
Keyword(s):  
Particle Number ◽  
Convex Set ◽  
Boolean Model ◽  
Particle Systems ◽  
Typical Particle ◽  
The Mean ◽  
Convexity Assumptions ◽  
Particle Process ◽  
Simply Connected ◽  
Number Of Particles

A stationary (but not necessarily isotropic) Boolean model Y in the plane is considered as a model for overlapping particle systems. The primary grain (i.e. the typical particle) is assumed to be simply connected, but no convexity assumptions are made. A new method is presented to estimate the intensity y of the underlying Poisson process (i.e. the mean number of particles per unit area) from measurements on the union set Y. The method is based mainly on the concept of convexification of a non-convex set, it also produces an unbiased estimator for a (suitably defined) mean body of Y, which in turn makes it possible to estimate the mean grain of the particle process.

Learning Block Structures in U-statistic Based Matrices

Biometrika ◽  
2020 ◽  
Author(s):  
Weiping Zhang ◽  
Baisuo Jin ◽  
Zhidong Bai
Keyword(s):  
Matrix Theory ◽  
Block Structure ◽  
Group Structure ◽  
Asymptotic Properties ◽  
Finite Sample ◽  
Eigenvalues And Eigenvectors ◽  
Sample Matrix ◽  
U Statistics ◽  
Large Matrix ◽  
Block Structures

Abstract We introduce a conceptually simple, efficient and easily implemented approach for learning the block structure in a large matrix. Using the properties of U-statistics and large dimensional random matrix theory, the group structure of many variables can be directly identified based on the eigenvalues and eigenvectors of the scaled sample matrix. We also established the asymptotic properties of the proposed approach under mild conditions. The finite-sample performance of the approach is examined by extensive simulations and data examples.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1999 ◽  
Vol 31 (02) ◽  
pp. 283-314 ◽  
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
The Individual ◽  
Individual Grains

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝ d generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

scholarly journals SPECTRAL PROPERTIES OF CHAOTIC PROCESSES

2006 ◽  
Vol 06 (03) ◽  
pp. 355-371
Author(s):  
BERNARD BERCU ◽  
CLÉMENTINE PRIEUR
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Spectral Properties ◽  
Law Of Large Numbers ◽  
Fourier Transforms ◽  
Asymptotic Properties ◽  
Initial State ◽  
Strong Law ◽  
Expanding Map ◽  
Large Numbers

We investigate the spectral asymptotic properties of the stationary dynamical system ξt= φ(Tt(X0)). This process is given by the iterations of a piecewise expanding map T of the interval [0,1], invariant for an ergodic probability μ. The initial state X0is distributed over [0,1] according to μ and φ is a function taking values in ℝ. We establish a strong law of large numbers and a central limit theorem for the integrated periodogram as well as for Fourier transforms associated with (ξt: t ∈ ℕ). Several examples of expanding maps T are also provided.

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