On asymptotics of large Haar distributed unitary matrices

Let U _n be an n × n Haar unitary matrix. In this paper, the asymptotic normality and independence of Tr U _n , Tr U _n ² ,..., Tr U _n ^k are shown by using elementary methods. More generally, it is shown that the renormalized truncated Haar unitaries converge to a Gaussian random matrix in distribution. This revised version was published online in August 2006 with corrections to the Cover Date.     

Second order freeness and fluctuations of random matrices: II. Unitary random matrices

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.     

Tail asymptotics of random sum and maximum of log-normal risks

In this paper we derive the asymptotic behaviour of the survival function of both random sum and random maximum of log-normal risks. As for the case of finite sum and maximum investigated in Asmussen and Rojas-Nandayapa (2008) also for the more general setup of random sums and random maximum the principle of a single big jump holds. We investigate both the log-normal sequences and some related dependence structures motivated by stationary Gaussian sequences.     

A limit theorem for the eigenvalues of product of two random matrices

The existence of limit spectral distribution of the product of two independent random matrices is proved when the number of variables tends to infinity. One of the above matrices is the Wishart matrix and the other is a symmetric nonnegative definite matrix.     

Continuum limits for classical sequential growth models

A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,…,n ? 1} and putting i below j if there is a path i = i₁…i_k = j in the graph with i₁ < … < i_k. Rideout and Sorkin 14 provide computational evidence that suitably normalized sequences of random graph orders have a “continuum limit.” We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large‐scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,…,n ‐ 1} such that the random partial order on {0,…,n ‐ 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension ≥ 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

Gumbel分布最大吸引场的次序统计量的精致渐近性

得到一类Gumbel分布最大吸引场的随机容量样本的次序统计量的精致渐近性,揭示了收敛速度、权函数、边界函数及极限状态之间的联系.这类吸引场真包含了全体(γ),γ>0分布族.     

独立情形下自正则和的精确渐近性

研究均值为零非退化的独立同分布的随机变量序列正则和收敛性,在适当条件下,获得了自正则和精确渐近性的一般结果.     

Random coefficient differential equation models for bacterial growth

In the mathematical modeling of population growth, and in particular of bacterial growth, parameters are either measured directly or determined by curve fitting. These parameters have large variability that depends on the experimental method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this work the parameters that appear in the Monod kinetics growth model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.     

On the growth factor in Gaussian elimination for generalized Higham matrices

A Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite. We prove that, for such A, the growth factor in Gaussian elimination is less than 3. Moreover, a slightly larger bound holds true for a broader class of complex matrices A=B+iC, where B and C are Hermitian and positive definite. Copyright © 2002 John Wiley & Sons, Ltd.     

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

Bounding the growth factor in Gaussian elimination for Buckley's class of complex symmetric matrices

A Buckley matrix is an n × n complex symmetric matrix A = I _n + iC, where C is real symmetric positive definite. We prove that, for such A the growth factor in Gaussian elimination is not greater than $${1 + \sqrt{17} \over 4} \simeq 1.28078\ldots$$ Copyright © 2000 John Wiley & Sons, Ltd.     

Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source

We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann-Hilbert boundary value problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 34–57, April, 2009.     

Corrected versions of cross-validation criteria for selecting multivariate regression and growth curve models

This paper is concerned with cross-validation (CV) criteria for choice of models, which can be regarded as approximately unbiased estimators for two types of risk functions. One is AIC type of risk or equivalently the expected Kullback-Leibler distance between the distributions of observations under a candidate model and the true model. The other is based on the expected mean squared error of prediction. In this paper we study asymptotic properties of CV criteria for selecting multivariate regression models and growth curve models under the assumption that a candidate model includes the true model. Based on the results, we propose their corrected versions which are more nearly unbiased for their risks. Through numerical experiments, some tendency of the CV criteria will be also pointed.     

Random fuzzy shock models and bivariate random fuzzy exponential distribution

In this paper, a random fuzzy shock model and a random fuzzy fatal shock model are proposed. Then bivariate random fuzzy exponential distribution is derived from the random fuzzy fatal shock model. Furthermore, some properties of the bivariate random fuzzy exponential distribution are proposed. Finally, an example is given to show the application of the bivariate random fuzzy exponential distribution.     

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

ABSTRACT

In this paper, for centred homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.