Increasing returns economics and generalized Pólya processes

This article extends the traditional Pólya scheme consisting of one urn with two colors to the schemes where multiple independent and/or interdependent urns with multiple additions and/or withdraws and several independent and/or interdependent colors are considered. It also argues that with these schemes many complex economic systems subject to increasing returns can be formalized mathematically, for they allow for positive and negative feedbacks among many variables, “jumps,” “bad” behaved dynamics, dis continuities, and interrelation among systems. © 2013 Wiley Periodicals, Inc. Complexity 19: 21–37, 2013     

Rate of Convergence of the Discrete Pólya-1 Algorithm

The rate of convergence of the discrete Pólya-1 algorithm is studied. Examples are given to show that the rates derived are sharp.     

A generalized Pólya's urn with graph based interactions

Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power . We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if then there is a single point with non‐zero entries such that the proportion converges to v almost surely. A special case is when G is regular and . We show e.g. that if G is non‐bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,614–634, 2015     

Pósa's conjecture for graphs of order at least 2 × 108

In 1962 Pósa conjectured that every graph G on n vertices with minimum degree \begin{align*}\delta(G)\ge \frac{2}{3}n\end{align*} contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Pósa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than \begin{align*}\frac{2}{3}n\end{align*}. Still in 1996, Komlós, Sárközy, and Szemerédi proved Pósa's Conjecture, using the Regularity and Blow‐up Lemmas, for graphs of order n ≥ n₀, where n₀ is a very large constant. Here we show without using these lemmas that n₀:= 2 × 10⁸ is sufficient. We are motivated by the recent work of Levitt, Sárközy and Szemerédi, but our methods are based on techniques that were available in the 90's. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A Tighter Erdős‐Pósa Function for Long Cycles

We prove that there exists a bivariate function f with such that for every natural k and ?, every graph G has at least k vertex‐disjoint cycles of length at least ? or a set of at most vertices that meets all cycles of length at least ?. This improves a result by Birmelé et al. (Combinatorica, 27 (2007), 135–145), who proved the same result with .     

利用区间删失数据的分布函数估计及其收敛速度

讨论任意m点均匀分布(m≥3)的情况,用m点均匀分布的累积分布函数去逼近连续总体的分布函数,在适当的条件下,证明了用区间删失数据估计分布函数具有收敛速度D(n)-2/9.     

The CRT is the scaling limit of unordered binary trees

We prove that a uniform, rooted unordered binary tree (also known as rooted, binary Pólya tree) with n leaves has the Brownian continuum random tree as its scaling limit for the Gromov‐Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 467–501, 2011     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

分段线性插值收敛速度的一种估计

给出了分段线性插值收敛速度的一种估计.     

每个节点具有多服务台的Jackson网络流体逼近的收敛速度

研究一个每个节点具有多服务台的Jackson网络.在服务强度为1的条件下,研究了Jackson网络的泛函重对数率与其流体逼近的收敛速度,证明了如果该网络的外部到达过程,服务过程有泛函重对数率,且在流体变换下以指数速度收敛到其相应的流体模型,则其队长过程、负荷过程、忙期过程等也具有相应的性质.     

拉格朗日插值算子的L_p收敛速度

本文得到了以第二类Ｔｃｈｅｂｙｃｈｅｆ多项式的零点为插值结点组的拉格朗日插值于Ｌｐ（１＜ｐ＜２）下的收敛速度     

The Erdős–Pósa Property for Long Circuits

For an integer ? at least 3, we prove that if G is a graph containing no two vertex‐disjoint circuits of length at least ?, then there is a set X of at most vertices that intersects all circuits of length at least ?. Our result improves the bound due to Birmelé, Bondy, and Reed (The Erd?s–Pósa property for long circuits, Combinatorica 27 (2007), 135–145) who conjecture that ? vertices always suffice.     

Estimates for the Rate of Convergence of the Galerkin Method for Abstract Hyperbolic Equations

We study the rate of convergence of the semidiscrete Galerkin method for linear hyperbolic equations in a Hilbert space. We establish asymptotic estimates for the error arising as a result of the arbitrariness in the choice of subspaces in which the approximation problems are solved.     

Long cycles through prescribed vertices have the Erdős‐Pósa property

We prove that for every graph, any vertex subset S, and given integers : there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in time, where s denotes the cardinality of S.     

加快迭代收敛速度的两个命题

定义了单调收敛函数和交错收敛函数,并根据其收敛特点,提出并证明了加快其收敛速度的两个命题.算例表明其效果较好.     

On the structure of random plane-oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.     

未知方向密度估计的收敛率

设Ｘ为ｐ维随机向量,对于未知的投影方向θｏ（‖θｏ‖＝１）,本文利用θｏ的估计与核密度估计相结合的方法给出了θ＾Ｔ０Ｘ的密度（方向密度）的核型密度估计,获得了此估计的逐点渐近正态性,逐点精确强收敛率,一致精确强收敛率以及均方误差收敛率,所得结果与最优性与已知方向上的核密度估计完全一致。作为例子,对θｏ为Ｘ协方差阵的最大特征值所对应的特征方向,我们给出了θｏ的满足条件的估计极其方向密度估计。     

共轭混合型Jacobi级数的收敛速度(英文)

对于有界变差的连续函数，本文给出了其共轭混合型Jacobi级数的收敛速度．     

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model ,where yi's are responses, xi = (xi1, xi2,…,xip)' and ti ∈T are known and nonrandom design points, T is a compact set in the real line is an unknown parameter vector, g(·) is an unknown function and {Ei} isa linear process, i.e., random variables with zeromean and variance o2e. Drawing upon B-spline estimation of g(·) and least squares estimation of 0, we construct estimators of the autocovariances of {Ei}- The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {Ei} are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.