scholarly journals THE DEGREE PROFILE AND GINI INDEX OF RANDOM CATERPILLAR TREES

2018 ◽  
Vol 33 (4) ◽  
pp. 511-527
Author(s):  
Panpan Zhang ◽  
Dipak K. Dey
Keyword(s):  
Asymptotic Distribution ◽  
Numerical Experiments ◽  
Gini Index ◽  
Joint Distribution ◽  
Preferential Attachment ◽  
Multinomial Distribution ◽  
Dispersion Matrix ◽  
Urn Model ◽  
New Type ◽  
Generalized Pólya Urn

AbstractIn this paper, we investigate the degree profile and Gini index of random caterpillar trees (RCTs). We consider RCTs which evolve in two different manners: uniform and nonuniform. The degrees of the vertices on the central path (i.e., the degree profile) of a uniform RCT follows a multinomial distribution. For nonuniform RCTs, we focus on those growing in the fashion of preferential attachment. We develop methods based on stochastic recurrences to compute the exact expectations and the dispersion matrix of the degree variables. A generalized Pólya urn model is exploited to determine the exact joint distribution of these degree variables. We apply the methods from combinatorics to prove that the asymptotic distribution is Dirichlet. In addition, we propose a new type of Gini index to quantitatively distinguish the evolutionary characteristics of the two classes of RCTs. We present the results via several numerical experiments.

scholarly journals On a memory game and preferential attachment graphs

2016 ◽  
Vol 48 (2) ◽  
pp. 585-609 ◽  
Author(s):  
Hüseyin Acan ◽  
Paweł Hitczenko
Keyword(s):  
Preferential Attachment ◽  
Random Variable ◽  
Urn Model ◽  
Limiting Distributions ◽  
Asymptotically Normal ◽  
Generalized Pólya Urn ◽  
Limit Result ◽  
Few Vertices ◽  
Memory Game ◽  
Main Technical Tool

Abstract In their recent paper Velleman and Warrington (2013) analyzed the expected values of some of the parameters in a memory game; namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods, we obtain the joint asymptotic normality of the degree counts in the preferential attachment graphs. Furthermore, we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz et al. (2014) on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. In order to prove that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multitype generalized Pólya urn model.

scholarly journals Plane recursive trees, Stirling permutations and an urn model

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Joint Distribution ◽  
Urn Model ◽  
Pólya Urn ◽  
International Audience ◽  
Polya Urn ◽  
Generalized Pólya Urn ◽  
Recursive Trees

International audience We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Pólya urn.

scholarly journals A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

2021 ◽  
Author(s):  
Sven-Erik Ekström ◽  
Paris Vassalos
Keyword(s):  
Distribution Function ◽  
Asymptotic Distribution ◽  
Numerical Experiments ◽  
Toeplitz Matrices ◽  
Future Research ◽  
Research Directions ◽  
Numerical Examples ◽  
Working Hypothesis ◽  
Wide Range ◽  
Future Research Directions

AbstractIt is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

Characterization of the Generalized Pólya Process and its Applications

2014 ◽  
Vol 46 (4) ◽  
pp. 1148-1171 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Joint Distribution ◽  
Arrival Times ◽  
Stochastic Intensity ◽  
Polya Process ◽  
New Type ◽  
Compound Process ◽  
Pólya Process

In this paper some important properties of the generalized Pólya process are derived and their applications are discussed. The generalized Pólya process is defined based on the stochastic intensity. By interpreting the defined stochastic intensity of the generalized Pólya process, the restarting property of the process is discussed. Based on the restarting property of the process, the joint distribution of the number of events is derived and the conditional joint distribution of the arrival times is also obtained. In addition, some properties of the compound process defined for the generalized Pólya process are derived. Furthermore, a new type of repair is defined based on the process and its application to the area of reliability is discussed. Several examples illustrating the applications of the obtained properties to various areas are suggested.

scholarly journals AILU FOR HELMHOLTZ PROBLEMS: A NEW PRECONDITIONER BASED ON THE ANALYTIC PARABOLIC FACTORIZATION

2001 ◽  
Vol 09 (04) ◽  
pp. 1499-1506 ◽  
Author(s):  
MARTIN J. GANDER ◽  
FRÉDÉRIC NATAF
Keyword(s):  
Numerical Experiments ◽  
New Approach ◽  
New Type

We investigate a new type of preconditioner which is based on the analytic factorization of the operator into two parabolic factors. Approximate analytic factorizations lead to new block ILU preconditioners. We analyze the preconditioner at the continuous level where it is possible to optimize its performance. Numerical experiments illustrate the effectiveness of the new approach.

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (2) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

We prove strong convergence of the proportions Un/Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.

scholarly journals Asymptotic distribution of sum and maximum for Gaussian processes

1999 ◽  
Vol 36 (4) ◽  
pp. 1031-1044 ◽  
Author(s):  
Hwai-Chung Ho ◽  
William P. McCormick
Keyword(s):  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Joint Distribution ◽  
Covariance Function ◽  
Random Variables ◽  
Regularity Conditions ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Standard Normal

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

The Asymptotic Distribution of the S–Gini Index

2002 ◽  
Vol 44 (4) ◽  
pp. 439-446 ◽  
Author(s):  
Ričardas Zitikis ◽  
Joseph L. Gastwirth
Keyword(s):  
Asymptotic Distribution ◽  
Gini Index

The G/M/m queue with finite waiting room

1975 ◽  
Vol 12 (4) ◽  
pp. 779-792 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Asymptotic Distribution ◽  
Waiting Time ◽  
Joint Distribution ◽  
Transient Solution ◽  
Waiting Room ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
The Moment ◽  
Number Of Customers

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

scholarly journals On n/p-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices

2018 ◽  
Vol 33 ◽  
pp. 24-40 ◽  
Author(s):  
Jolanta Pielaszkiewicz ◽  
Dietrich Von Rosen ◽  
Martin Singull
Keyword(s):  
Normal Distribution ◽  
Covariance Matrix ◽  
Asymptotic Distribution ◽  
Joint Distribution ◽  
Recursive Formula ◽  
Multivariate Normal ◽  
Wishart Matrices ◽  
The Matrix ◽  
Matrix Normal ◽  
Matrix Normal Distribution

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

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