Approximation Theory in the Central Limit Theorem

Paulauskas, V.; Račkauskas, Alfredas

doi:10.1007/978-94-011-7798-6

Cited by 52 publications

(9 citation statements)

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“…This method is based on the Lindeberg strategy [32] of replacing non-gaussian random variables with gaussian ones. (For more modern discussions about Lindeberg's method, see [8,40].) Using this method, we are able to prove universality for general Wigner matrices under very mild assumptions.…”

Section: 2mentioning

confidence: 98%

Random matrices: Universality of local eigenvalue statistics

2011

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In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and k-point correlation and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

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Section: 2mentioning

confidence: 98%

Random matrices: Universality of local eigenvalue statistics

2011

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“…Note that if ξ,ξ are both real valued, then the hypothesis of matching moments to second order is automatic since the ξ,ξ are normalized to have mean zero and variance one. The leaing idea is to use Lindeberg replacement trick, originated in Lindeberg [1922] (see also Paulauskas & Raskauskas [2009], Chatterjee [2014] for more recent discussions). The arguments we will use follow the spirit of Tao & Vu [2011, 2014, where characteristic polynomials of random matrices were considered.…”

Section: Comparability Of Log-magnitudesmentioning

confidence: 99%

Local Universality of Zeroes of Random Polynomials

Tao

2014

Int Math Res Notices

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In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials f = n i=1 ciξiz i and f = n i=1 ciξiz i , where the ξi andξi are iid random variables that match moments to second order, the coefficients ci are deterministic, and the degree parameter n is large. Our results show, under some light conditions on the coefficients ci and the tails of ξi,ξi, that the correlation functions of the zeroes of f andf are approximately the same. As an application, we give some answers to the classical question "How many zeroes of a random polynomials are real ?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions f andf if their log magnitudes log |f |, log |f | are close in distribution, and if some non-concentration bounds are obeyed.

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“…where " weak −→ " means "weak convergence" of probability measures [1,9]. An important special case arises if one starts with a random ν-vectorz having independent componentsz = (z 1 , .…”

Section: Discretization Methodsmentioning

confidence: 99%

“…, is a sequence of discrete probability distributions of the type (97a-d). According to the central limit theorem [1,9] it is known that …”

Section: Discretization Methodsmentioning

confidence: 99%

Reliability Analysis of Technical Systems/Structures by means of Polyhedral Approximation of the Safe/Unsafe Domain

Marti

2007

GAMM-Mitteilungen

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Key words Reliability analysis, survival conditions, optimizational representation of the state function, polyhedral approximation of the safe/unsafe domain, probability inequalities, approximation by discretization methods Reliability analysis of technical structures and systems is based on an appropriate (limit) state function separating the safe and unsafe/states in the space of random parameters. Starting with the survival conditions, hence, the state equation and the condition for the admissibility of states, an optimizational representation of the state function can be given in terms of the minimum function of a certain minimization problem. Selecting a certain number of boundary points of the safe/unsafe domain, hence, on the limit state surface, the safe/unsafe domain is approximated by a convex polyhedron defined by the intersection of the half spaces in the parameter space generated by the tangent hyperplanes to the safe/unsafe domain at the selected boundary points on the limit state surface. The resulting approximative probability functions are then defined by means of probabilistic linear constraints in the parameter space, where, after an appropriate transformation, the probability distribution of the parameter vector can be assumed to be normal with zero mean vector and unit covariance matrix. Working with separate linear constraints, approximation formulas for the probability of survival of the structure are obtained immediately. More exact approximations are obtained by considering joint probability constraints, which, in a second approximation step, can be evaluated by using probability inequalities and/or discretization of the underlying probability distribution.

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Approximation Theory in the Central Limit Theorem

Cited by 52 publications

References 0 publications

Random matrices: Universality of local eigenvalue statistics

Random matrices: Universality of local eigenvalue statistics

Local Universality of Zeroes of Random Polynomials

Reliability Analysis of Technical Systems/Structures by means of Polyhedral Approximation of the Safe/Unsafe Domain

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