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Register a new userLimit theorems for random simplices in high dimensions
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Let $r=r(n)$ be a sequence of integers such that $rleq n$ and let $X_1,ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $rsim alpha n$ for some $0 < alpha < 1$).
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In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guedon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thale: High-dimensional limit theorems for random vectors in $ell_p^n$-balls, Commun. Contemp. Math. (2019)].
We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $log Z_n$.
Central limit theorems for the log-volume of a class of random convex bodies in $mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $ntoinfty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $ell_p$-ball. In particular, this includes the cone and the uniform probability measure.
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems known so far about limiting distributions of random Betti numbers. In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: ErdH{o}s-Renyi random clique complexes, random Vietoris-Rips complexes, and random v{C}ech complexes. These results may be of practical interest in topological data analysis.
We correct the proofs of the main theorems in our paper Limit theorems for Betti numbers of random simplicial complexes.
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