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In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.
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In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the central point theorem, and Tverberg's theorem on partitions of a point set.
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In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the central point theorem, and Tverberg’s theorem on partitions of a point set.
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The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets t...
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The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.
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In this paper we consider families of compact convex sets in Ｒ~d such that any subfamily of sizc at most d has a nonempty interscction. We prove some analogucs of the central point theorem and Tverberg's theorem for such families.
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This paper deals with weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a so...
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This paper deals with weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a sort of L-2-bracketing. We also consider l(infinity)-valued martingale difference arrays, and present natural generalizations of Jain-Marcus's and Ossiander's central limit theorems. As an application, the asymptotic behavior of log-likelihood ratio random fields in general statistical experiments with abstract parameters is derived. [References: 17]
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The Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance, and many other fields. In this paper we obtain a fun...
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The Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance, and many other fields. In this paper we obtain a functional central limit theorem for the nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.
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Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most delta. Summing powers of the volume of ...
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Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most delta. Summing powers of the volume of all k-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips complex is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. Univariate and multivariate central limit theorems are proven. Analogous results for the Cech complex are given. (C) 2020 Elsevier Inc. All rights reserved.
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Let K be a smooth convex set with volume one in R-d. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by K, is called a random polytope. We prove that severa...
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Let K be a smooth convex set with volume one in R-d. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by K, is called a random polytope. We prove that several key functionals of K-n satisfy the central limit theorem as n tends to infinity. (c) 2005 Elsevier Inc. All rights reserved.
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One of the main differences between the central limit theorem and the Poisson law of small numbers is that the former possesses the large sample property (LSP), i.e., the error of normal approximation to the sum of n independent i...
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One of the main differences between the central limit theorem and the Poisson law of small numbers is that the former possesses the large sample property (LSP), i.e., the error of normal approximation to the sum of n independent identically distributed (i.i.d.) random variables converges to 0 as n -> infinity. Since 1980s, considerable effort has been devoted to recovering the LSP for the law of small numbers in discrete random variable approximation. In this paper, we aim to establish the LSP for the superposition of i.i.d. finite point processes. (C) 2020 Elsevier B.V. All rights reserved.
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