scholarly journals Point process stabilization methods and dimension estimation

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Point Process ◽  
Central Limit ◽  
Point Processes ◽  
Dimension Estimation ◽  
Stabilization Methods ◽  
International Audience

International audience We provide an overview of stabilization methods for point processes and apply these methods to deduce a central limit theorem for statistical estimators of dimension.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

scholarly journals Gaussian fluctuations of Young diagrams under the Plancherel measure

2007 ◽  
Vol 463 (2080) ◽  
pp. 1069-1080 ◽  
Author(s):  
Leonid V Bogachev ◽  
Zhonggen Su
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Gaussian Process ◽  
Central Limit ◽  
Point Processes ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Finite Dimensional ◽  
The Central Limit Theorem ◽  
Local Correlations

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions λ ⊢ n ∈ (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like , the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin–Lebowitz–Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45°, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.

A new graph related to the directions of nearest neighbours in a point process

2003 ◽  
Vol 35 (1) ◽  
pp. 47-55 ◽  
Author(s):  
S. N. Chiu ◽  
I. S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Point Process ◽  
Process Simulation ◽  
Point Processes ◽  
Nearest Neighbour ◽  
Simulation Studies ◽  
Nearest Neighbours ◽  
Typical Point

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

scholarly journals Random Inscribing Polytopes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Ross M. Richardson ◽  
Van H. Vu ◽  
Lei Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Convex Bodies ◽  
Higher Moments ◽  
Random Polytopes ◽  
International Audience

International audience For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.

scholarly journals The degree distribution in unlabelled $2$-connected graph families

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Veronika Kraus
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Connected Graph ◽  
Degree Distribution ◽  
Random Variable ◽  
Expected Value ◽  
Asymptotically Linear ◽  
International Audience ◽  
Graph Families

International audience We study the random variable $X_n^k$, counting the number of vertices of degree $k$ in a randomly chosen $2$-connected graph of given families. We prove a central limit theorem for $X_n^k$ with expected value $\mathbb{E}X_n^k \sim \mu_kn$ and variance $\mathbb{V}X_n^k \sim \sigma_k^2n$, both asymptotically linear in $n$, for both rooted and unrooted unlabelled $2$-connected outerplanar or series-parallel graphs.

scholarly journals A functional central limit theorem for spatial birth and death processes

2008 ◽  
Vol 40 (3) ◽  
pp. 759-797 ◽  
Author(s):  
Xin Qi
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Point Processes ◽  
Functional Central Limit Theorem ◽  
Death Process ◽  
Birth Rates ◽  
Birth And Death Processes ◽  
Spatial Point Processes ◽  
Functional Central Limit

We give a functional central limit theorem for spatial birth and death processes based on the representation of such processes as solutions of stochastic equations. For any bounded and integrable function in Euclidean space, we define a family of processes which is obtained by integrals of this function with respect to the centered and scaled spatial birth and death process with constant death rate. We prove that this family converges weakly to a Gaussian process as the scale parameter goes to infinity. We do not need the birth rates to have a finite range of interaction. Instead, we require that the birth rates have a range of interaction that decays polynomially. In order to show the convergence of the finite-dimensional distributions of the above processes, we extend Penrose's multivariate spatial central limit theorem. An example of the asymptotic normalities of the time-invariance estimators for the birth rates of spatial point processes is given.

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