对M/T-SPH/1排队平稳指标的进一步研究—数值计算与渐近分析

讨论M/T-SPH/1排队平稳队长分布的数值计算,以及平稳队长和逗留时间分布各阶矩的数值计算及渐近分析.其中T-SPH表示可数状态吸收生灭链吸收时间的分布.在分布PGF和LST的基础上,首先给出了计算平稳队长分布,平稳队长以及逗留时间分布各阶矩的数值结果的递推公式.其次还讨论了平稳队长及平稳逗留时间分布各阶矩的尾部渐近特征.结果表明当参数取不同值时,两个指标尾部具有三种不同类型的衰减方式.最后还用数值例子检验了方法的有效性.     

The Asymmetric Leader Election Algorithm: Another Approach

The asymmetric leader election algorithm has obtained quite a bit of attention lately. In this paper we want to analyze the following asymptotic properties of the number of rounds: Limiting distribution function, all moments in a simple automatic way, asymptotics for p → 0, p → 1 (where p denotes the “killing” probability). This also leads to a few interesting new identities. We use two paradigms: First, in some urn model, we have asymptotic independence of urns behaviour as far as random variables related to urns with a fixed number of balls are concerned. Next, we use a technique easily leading to the asymptotics of the moments of extremevalue related distribution functions.      

Stability theorems and the second-order asymptotics in threshold phenomena for boundary functionals of random walks

In Section 1, we prove stability theorems for a series of boundary functionals of random walks. In Section 2, we suggest a new simpler proof of the theorem on threshold phenomena for the distribution of the maximum of the consecutive sums of random variables. In Section 3, we find the second-order asymptotics for this distribution under the assumption that the third moments of the random variables exist.     

The Wiener Index of simply generated random trees

Asymptotics are obtained for the mean, variance, and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton–Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 337–358, 2003     

Limit Laws for Symmetric k-Tensors of Regularly Varying Measures

We consider the asymptotics of certain symmetric k-tensors, the vector analogue of sample moments for i.i.d. random variables. The limiting distribution is operator stable as an element of the vector space of real symmetric k-tensors.     

Binomial Moments of the Distance Distribution and the Probability of Undetected Error

In [1] K. A. S. Abdel-Ghaffar derives a lower bound on the probability of undetected error for unrestricted codes. The proof relies implicitly on the binomial moments of the distance distribution of the code. We use the fact that these moments count the size of subcodes of the code to give a very simple proof of the bound in Abdel by showing that it is essentially equivalent to the Singleton bound. This proof reveals connections of the probability of undetected error to the rank generating function of the code and to related polynomials (Whitney function, Tutte polynomial, and higher weight enumerators). We also discuss some improvements of this bound.Finally, we analyze asymptotics. We show that an upper bound on the undetected error exponent that corresponds to the bound of Abdel improves known bounds on this function.     

On the Joint Path Length Distribution in Random Binary Trees

During the 10th Seminar on Analysis of Algorithms , MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary tree. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painlevé transcendent . This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n ^5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution's tails and moments. We will also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the Wentzel, Kramers, Brillouin (WKB) method.     

Some properties of a two-parameter family of poisson distributions of orderk

We study the problem of simulating the process of detecting a weighted sum of independent Poisson random variables. We investigate the properties of the resulting compound Poisson distribution: the analytic form of the probability function and recursion formulas for computing it, moments and semi-invariants, asymptotics of the distribution, and recursion relations for the derivatives with respect to the parameters. We give the results of model computations showing the set structure of the distribution. One figure. Bibliography: 8 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 46–54.     

Asymptotics of the moments of the number of cycles of a random A-permutation

We consider random permutations uniformly distributed on the set of all permutations of degree n whose cycle lengths belong to a fixed set A (the so-called A-permutations). In the present paper, we establish an asymptotics of the moments of the total number of cycles and of the number of cycles of given length of this random permutation as n → ∞.     

On the Paper "Asymptotics for the Moments of Singular Distributions"

In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments c_n(α) ≡ c_n = ∫¹₀tⁿdα(t), N = 0, 1, 2, ..., of some singular distributions α (with support [0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).     

The height and range of watermelons without wall

We determine the weak limit of the distribution of the random variables “height” and “range” on the set of p$p$-watermelons without wall restriction as the number of steps tends to infinity. Additionally, we provide asymptotics for the moments of the random variable “height”.     

The Order of the Error Term for Moments of the Log Likelihood Ratio Unit Root Test in an Autoregressive Process

This paper investigates the asymptotics of the log likelihood ratio test for a unit root in an autoregressive (AR) process of general order. The main result is that the expectation and variance (in fact, all moments) of the test statistic may, to the order of T-1, where T is the number of observations, be approximated by the expectation and variance of the corresponding test in an AR(1) process. This result has obvious implications for the asymptotics of unit root tests for panels. An explicit formula for the approximation error of a test in an AR(2) process is also given.     

A Computational Approach for Full Nonparametric Bayesian Inference Under Dirichlet Process Mixture Models

Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.     

Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model

We derive the asymptotics of the OLS estimator for a purely autoregressive spatial model. Only low-level conditions are used. As the sample size increases, the spatial matrix is assumed to approach a square-integrable function on the square (0,1)². The asymptotic distribution is a ratio of two infinite linear combinations of χ² variables. The formula involves eigenvalues of an integral operator associated with the function approached by the spatial matrices. Under the conditions imposed identification conditions for the maximum likelihood method and method of moments fail. A corrective two-step procedure using the OLS estimator is proposed.     

Lower order terms in the full moment conjecture for the Riemann zeta function

We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide some numerical evidence in favor of the conjecture.     

Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics

A Fokker-Planck type equation for interacting particles with exclusion principle is analyzed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in L¹ are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate.     

Analysis in distribution of two randomized algorithms for finding the maximum in a broadcast communication model

The limit laws of three cost measures are derived of two algorithms for finding the maximum in a single-channel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the “asymptotic transfers,” which roughly bridges the asymptotics of the “conquering cost of the subproblems” and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms.     

Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.     

Moments of the generalized hyperbolic distribution

In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily programmable for numerical evaluation of moments. For low order moments we also give an alternative derivation of the moments of the generalized hyperbolic distribution. The expressions given for these moments may be used to obtain moments for special cases such as the hyperbolic and normal inverse Gaussian distributions. Moments for limiting cases such as the skew hyperbolic t and variance gamma distributions can be found using the same approach.     

Asymptotic agreement of moments and higher order contraction in the Burgers equation

The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole-Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given by the inverse Cole-Hopf transformation of a summation of n heat kernels. The k-th order moments of the exact and the approximate solution are contracting with order in Lp-norm as t→∞. This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically.