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.     

A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence

We consider a stationary time series {X_t} given byX_t=∑^∞_k=−∞ ψ_kZ_t−k, where {Z_t} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {X_t} is squared integrable andm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm ∑_|k|?m ψ²_k→0.     

Asymptotic Joint Normality of Counts of Uncorrelated Motifs in Recursive Trees

We study the fringe of random recursive trees, by analyzing the joint distribution of the counts of uncorrelated motifs. Our approach allows for finite and countably infinite collections. To be able to deal with the collection when it is infinitely countable, we use measure-theoretic themes. Each member of a collection of motifs occurs a certain number of times on the fringe. We show that these numbers, under appropriate normalization, have a limiting joint multivariate normal distribution. We give a complete characterization of the asymptotic covariance matrix. The methods of proof include contraction in a metric space of distribution functions to a fixed-point solution (limit distribution). We discuss two examples: the finite collection of all possible motifs of size four, and the infinite collection of rooted stars. We conclude with remarks to compare fringe-analysis with matching motifs everywhere in the tree.     

Central limit theorems for a class of irreducible multicolor urn models

We take a unified approach to central limit theorems for a class of irreducible multicolor urn models with constant replacement matrix. Depending on the eigenvalue, we consider appropriate linear combinations of the number of balls of different colors. Then under appropriate norming the multivariate distribution of the weak limits of these linear combinations is obtained and independence and dependence issues are investigated. Our approach consists of looking at the problem from the viewpoint of recursive equations.     

加权和线性过程的渐近正态性

本文建立了α-混合序列情形的加权和平稳线性过程的渐近正态性.获得的结论基于最少的权条件.所得结论将Abadir等[Econometric Theory,2014,30(1):252-284]中的结论推广至α-混合序列情形.     

Asymptotic Normality of Kernel Type Density Estimators for Random Fields

Kernel type density estimators are studied for random fields. It is proved that the estimators are asymptotically normal if the set of locations of observations become more and more dense in an increasing sequence of domains. It turns out that in our setting the covariance structure of the limiting normal distribution can be a combination of those of the continuous parameter and the discrete parameter cases. The proof is based on a new central limit theorem for α-mixing random fields. Simulation results support our theorems. Final version 29 October 2004     

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     

Analysis of a generalized Friedman’s urn with multiple drawings

We study a generalized Friedman’s urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched.     

递推数列极限的初等求法和收敛渐近性

借助实例介绍一些非线性递推数列，特别是分式线性递推数列极限的初等求法。就一般分式线性递推数列，明确其收敛渐近性，并通过相关推论展示其应用。     

On densities for solutions to stochastic fixed point equations

We consider systems of stochastic fixed point equations that arise in the asymptotic analysis of random recursive structures and algorithms such as Quicksort, large Pólya urn processes, and path lengths of random recursive trees and split trees. The main result states sufficient conditions on the fixed point equations that imply the existence of bounded, smooth, rapidly decreasing Lebesgue densities.     

一类适应随机变量序列的强极限定理

研究了一类适应随机变量序列的局部收敛性,推广了文献[1]中的结论.并在假定部分和序列为极限鞅时,得到了极限鞅的强极限定理.最后给出了*-mixing序列的强大数定律.     

Central and Functional Central Limit Theorems for a Class of Urn Models

We consider an approach based on tails to certain central limit and functional central limit theorems for a class of two color urn models. In particular, some of the results are derived from an associated Ornstein–Uhlenbeck process, and for another result we give an alternative proof based on martingale tails.      

An Asymptotic Behavior of the Remainder in the Central Limit Theorem for Moments of Sums of Independent Random Variables

The objective of the paper is to study the asymptotic behavior of the reminder in the central limit theorem for moments of sums of independent random variables.     

Asymptotic Normality for Non-linear Functionals of Non-causal Linear Processes with Summable Weights

Let be a non-causal linear process with weights a_js satisfying certain summability conditions, and the iid sequence of innovation {_i} having zero mean and finite second moment. For a large class of non-linear functional K which includes indicator functions and polynomials, the present paper develops the central limit theorem for the partial sums      

Cayley树上随机场的马尔可夫逼近与一类小偏差定理

通过引进样本相对熵率作为Cayley树上任意随机场与马尔可夫链场之间的偏差的一种度量, 建立了关于状态序偶频率的一类小偏差定理. 证明中应用了研究马尔可夫链强极限定理的一种新的分析方法.     

直线上时间随机环境下随机游动的渐近性质

在状态空间是可数情形下,本文给出了时间随机环境下随机游动的一个一般模型.随后,在环境是独立同分布情形下得到了直线上时间随机环境下紧邻随机游动的一个常返与暂留准则和强大数定律;最后讨论了其中心极限定理,它类似与简单随机游动的相应结果.     

非齐次隐马尔可夫模型随机变换的若干强极限定理

利用鞅方法讨论了非齐次隐马尔可夫模型变换的强极限定理,作为特殊情形,将随机选择的概念拓展到非齐次隐马尔可夫模型中,得到了关于有限非齐次隐马尔可夫模型随机选择与随机公平比的若干极限定理.     

Asymptotic behavior of a tagged particle in simple exclusion processes

We review in this article central limit theorems for a tagged particle in the simple exclusion process. In the first two sections we present a general method to prove central limit theorems for additive functional of Markov processes. These results are then applied to the case of a tagged particle in the exclusion process. Related questions, such as smoothness of the diffusion coefficient and finite dimensional approximations, are considered in the last section.     

Local Asymptotic Normality in Extreme Value Index Estimation

This paper deals with the estimation of the extreme value index in local extreme value models. We establish local asymptotic normality (LAN) under certain extreme value alternatives. It turns out that the central sequence occurring in the LAN expansion of the likelihood process is up to a rescaling procedure the Hill estimator. The central sequence plays a crucial role for the construction of asymptotic optimal statistical procedures. In particular, the Hill estimator is asymptotically minimax.     

时间随机环境下随机游动的渐近行为

本文给出了可数状态空间中时间随机环境下随机游动的一个统一的模型 .对于最常见的情况 ,即d维最近邻域随机环境下随机游动 ,如果环境是严平稳的 ,则在一定条件下 ,该随机游动满足强大数定律和中心极限定理 .特别地 ,当环境独立同分布时 ,我们可以得到更为具体的结果 ,该结果类似于经典的随机游动的相应结论 .