Integral and asymptotic representations of geo-stable densities

The fact that geo-stable distributions do not have an explicit representation causes severe difficulties in empirical work. Here, we provide representations for geometric-stable probability densities in terms of fast converging integrals.     

On asymptotic distribution on the a-adic integers

We show that the values of a polynomial with a-adic coefficients at integer and rational prime arguments are asymptotically distributed on the a-adic integers and that the integer parts of certain sequences known to be uniformly distributed modulo one, are uniformly distributed on the a-adic integers.     

Error bounds for asymptotic expansions of the distribution of the MLE in a GMANOVA model

Summary In this paper we obtain asymptotic expansions for the distribution function and the density function of a linear combination of the MLE in a GMANOVA model, and for the density function of the MLE itself. We also obtain certain error bounds for the asymptotic expansions.     

Asymptotics of orthogonal polynomials with asymptotic Freud-like weights

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.     

Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P_n(cosθ;a,b). One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P_n(cosθ) in θ(0,π/2].     

The asymptotic property of approximation to |x| by Newman's rational operators

Let . The present note gives the asymptotoc formula of max . This revised version was published online in June 2006 with corrections to the Cover Date.     

Best simultaneous approximations: Norms, signatures, and asymptotic directions

Results concerning the distribution of directions of best simultaneous approximations in various norms are obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 730–737, May, 2000.     

The asymptotic distributions of the statistics of the skew elliptical variables

In this paper, the asymptotic properties of the quadratic forms and the T statistic of the skew elliptical variables are studied. Consistent estimators of some parameters are obtained. The robustness of the significance level of the one-sided t test within the family of the skew normal family is investigated.     

Bivariate CDF iterations and asymptotic independence

Asymptotic independence under weak convergence of sequences of bivariate random vectors is studied via the behavior of a certain class of bivariate cdf iterands that includes the extreme value iterand and a certain maximin iterand.     

Numerical radius for the asymptotic stability of delay differential equations

In this note, we make use of the numerical radius to investigate the asymptotic stability of linear systems of delay differential equations. We present examples to illustrate our assumption is weaker than the classical ones.     

A globally uniform asymptotic expansion of the hermite polynomials

In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn（√2n＋1α）to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme （SIAM J.Math.Anal.,（1994）,25：304-321）.A new estimate for the remainder is given.     

Power and exponential asymptotic forms of correlation functions

Using the Ornstein.Zernike equation, we obtain two asymptotic equations, one describing the exponential asymptotic behavior and the other describing the power asymptotic behavior of the total correlation function h(r). We show that the exponential asymptotic form is applicable only on a bounded distance interval l < r < L. The power asymptotic form is always applicable for r > L and reproduces the form of the interaction potential. In this case, as the density of a rarified gas decreases, L → l, the exponential asymptotic form vanishes, and only the power asymptotic form remains. Conversely, as the critical point is approached, L → ℞, and the applicability domain of the exponential asymptotic form increases without bound. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 454–464, September, 2008.     

关于函数S(x)及S_*(x)的渐近性质

对于任意实数x∈(1,∞),定义S(x)=m in{m∈N∶x m!};x∈[1,∞),S*(x)=m ax{m∈N∶m!x}.主要目的是研究这两个函数的渐近性质,并给出了它们的渐近公式.     

Estimating function with asymptotic bias and its estimator

We introduce the estimating function with asymptotic bias and investigate the asymptotic behavior of the estimator based on it by using their relationship. The estimator based on the estimating function with asymptotic bias has the asymptotic normality with asymptotic bias. We show that this theory has several interesting applications in practical statistics.     

B样条在一些渐近组合问题中的应用

本文考察了B样条函数及其导数的渐近性质,并给出了收敛阶;考察了经典Eulerian数和两类广义Eulerian数的渐近性质;给出了以Hermite多项式表示的细化Eulerian数的渐近形式.Carlitz等人利用中心极限定理得到Eulerian数渐近公式的逼近阶为43阶.利用样条方法,我们得到更为精确的逼近阶.将样条方法引入到组合数的渐近分析中,为离散对象的研究提供了一种新的分析方法.     

Some asymptotic expansions of moments of order statistics

Series expansions of moments of order statistics are obtained from expansions of the inverse of the distribution function. They are valid for certain types of distributions with regularly varying tails. We show that the expansions converge quickly when the sample size is moderate to large, and we obtain bounds on the rate of convergence. The special case of the Cauchy distribution is treated in more detail.     

An asymptotic expansion for the incomplete beta function

A new asymptotic expansion is derived for the incomplete beta function , which is suitable for large , small and . This expansion is of the form

where is the incomplete Gamma function ratio and . This form has some advantages over previous asymptotic expansions in this region in which depends on as well as on and .

     

Another approach to asymptotic expansions for Euler’s approximations of semigroups

In [2], optimal bounds for the remainder terms in asymptotic expansions for Euler’s approximations of semigroups were derived. The approach was based on applications of the Fourier-Laplace transforms, which allowed one to reduce the problem to estimation of error terms in the Law of Large Numbers. In this paper, we propose an alternative (direct) approach based on application of certain integro-differential identities (the so-called multiplicative representations of differences). Such identities were introduced by Bentkus [3] and applied (see Bentkus and Paulauskas [4]) to derive the optimal convergence rates in Chernoff-type lemmas and Euler’s approximations of semigroups. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 267–284, April–June, 2006.     

Error bounds for asymptotic approximations of the partition function

We consider Markovian queueing models with a finite number of states and a product form solution for its steady state probability distribution. Starting from the integral representation for the partition function in complex space we construct error bounds for its asymptotic expansion obtained by the saddle point method. The derivation of error bounds is based on an idea by Olver applicable to integral transforms with an exponentially decaying kernel. The bounds are expressed in terms of the supremum of a certain function and are asymptotic to the absolute value of the first neglected term in the expansion as the large parameter approaches infinity. The application of these error bounds is illustrated for two classes of queueing models: loss systems and single chain closed queueing networks.