scholarly journals Random inscribed polytopes in projective geometries

2021 ◽  
Author(s):  
Florian Besau ◽  
Daniel Rosen ◽  
Christoph Thäle
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Normal Approximation ◽  
Independent Random Variables ◽  
Polyhedral Sets ◽  
Continuous Density ◽  
Projective Geometries ◽  
Mean Width ◽  
The Mean ◽  
Geometric Estimates

AbstractWe establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and a Berry–Esseen bound for functionals of independent random variables.

scholarly journals Variance asymptotics and central limit theorems for volumes of unions of random closed sets

2002 ◽  
Vol 34 (03) ◽  
pp. 520-539 ◽  
Author(s):  
Tomasz Schreiber
Keyword(s):  
Central Limit ◽  
Borel Measure ◽  
Strong Law ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Large Numbers ◽  
Multidimensional Ball ◽  
Mean Width ◽  
The Mean ◽  
Heavy Tailed

Let X, X 1, X 2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X 1 ∪ ∙ ∙ ∙ ∪ X n )) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝ d with centres distributed according to a spherically-symmetric heavy-tailed law.

A Central Limit Theorem for Cumulative Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

scholarly journals RANDOM MATRICES: UNIVERSAL PROPERTIES OF EIGENVECTORS

2012 ◽  
Vol 01 (01) ◽  
pp. 1150001 ◽  
Author(s):  
TERENCE TAO ◽  
VAN VU
Keyword(s):  
Random Matrices ◽  
Real Axis ◽  
Central Limit ◽  
Random Matrix ◽  
Joint Distribution ◽  
Limit Theorems ◽  
The Matrix ◽  
Universal Properties ◽  
The Mean ◽  
First Four Moments

The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of the entries of the matrix. In this paper, we extend the four moment theorem to also cover the coefficients of the eigenvectors of a Wigner random matrix. A similar result (with different hypotheses) has been proved recently by Knowles and Yin, using a different method. As an application, we prove some central limit theorems for these eigenvectors. In another application, we prove a universality result for the resolvent, up to the real axis. This implies universality of the inverse matrix.

scholarly journals On central limit theorems for branching processes with dependent immigration

2020 ◽  
pp. 7-15
Author(s):  
V. Golomoziy ◽  
S. Sharipov
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Branching Processes ◽  
Central Limit Theorems ◽  
Standard Normal Distribution ◽  
Immigration Process ◽  
Standard Normal ◽  
Mean And Variance ◽  
The Mean ◽  
Regularly Varying

In this paper we consider subcritical and supercritical discrete time branching processes with generation dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m−dependent. The first result states on weak convergence of the fluctuation subcritical branching processes with m−dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.

scholarly journals On an extremal problem in the central limit theorems for sums of independent random variables

1964 ◽  
Vol 4 (3) ◽  
pp. 343-352
Author(s):  
V. M. Zolotarev
Keyword(s):  
Extremal Problem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: В. М. Золотарев. Об одной экстремальной задаче в предельных теоремах для сумм независимых случайных величин V. M. Zolotariov. Apie vieną ribinių teoremų, nepriklausomų atsitiktinių dydžių sumoms, ekstremalinį uždavinį

Poisson's equation for the recurrent M/G/1 queue

1994 ◽  
Vol 26 (04) ◽  
pp. 1044-1062 ◽  
Author(s):  
Peter W. Glynn
Keyword(s):  
Central Limit ◽  
Waiting Time ◽  
Limit Theorems ◽  
Time Sequence ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Initial State ◽  
Additive Functionals ◽  
Asymptotic Expressions ◽  
The Mean

This paper shows how to calculate solutions to Poisson's equation for the waiting time sequence of the recurrent M/G/l queue. The solutions are used to construct martingales that permit us to study additive functionals associated with the waiting time sequence. These martingales provide asymptotic expressions, for the mean of additive functionals, that reflect dependence on the initial state of the process. In addition, we show how to explicitly calculate the scaling constants that appear in the central limit theorems for additive functionals of the waiting time sequence.

bm-INDEPENDENCE AND bm-CENTRAL LIMIT THEOREMS ASSOCIATED WITH SYMMETRIC CONES

2010 ◽  
Vol 13 (03) ◽  
pp. 461-488 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Beta Function ◽  
Positive Cone ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Symmetric Cones ◽  
Index Sets ◽  
The Given ◽  
Independence Of Algebras

We study the properties of the (noncommutative) bm-independence of algebras, indexed by partially ordered sets. The index sets are given by positive cones, in particular the symmetric cones, which include the positive-definite Hermitian matrices with complex or quaternionic entries. We formulate and prove the general versions of the bm-Central Limit Theorems for bm-independent random variables, indexed by lattices in such positive cones. The limit measures we obtain are symmetric and compactly supported on the real line. Their (even) moment sequences (gn)n≥0 satisfy the generalized recurrence for the Catalan numbers: [Formula: see text], where the coefficients γ(r) are computed by the Euler's beta-function of the first kind, related to the given positive cone. Example of a nonsymmetric cone, the Vinberg's cone, is also studied in this context.

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