scholarly journals Pólya Urns Via the Contraction Method

2014 ◽  
Vol 23 (6) ◽  
pp. 1148-1186 ◽  
Author(s):  
MARGARETE KNAPE ◽  
RALPH NEININGER
Keyword(s):  
Asymptotic Behaviour ◽  
Discrete Time ◽  
Normal Limit ◽  
Rooted Trees ◽  
Limit Laws ◽  
Contraction Method ◽  
Limit Distributions

We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

scholarly journals On rates of convergence in the probabilistic analysis of algorithms

2021 ◽  
Author(s):  
◽  
Jasmin Straub
Keyword(s):  
Normal Limit ◽  
Analysis Of Algorithms ◽  
Rates Of Convergence ◽  
Random Trees ◽  
Divide And Conquer ◽  
Complexity Measures ◽  
Limit Laws ◽  
Contraction Method ◽  
Normal Limits ◽  
Recursive Structures

Within the last thirty years, the contraction method has become an important tool for the distributional analysis of random recursive structures. While it was mainly developed to show weak convergence, the contraction approach can additionally be used to obtain bounds on the rate of convergence in an appropriate metric. Based on ideas of the contraction method, we develop a general framework to bound rates of convergence for sequences of random variables as they mainly arise in the analysis of random trees and divide-and-conquer algorithms. The rates of convergence are bounded in the Zolotarev distances. In essence, we present three different versions of convergence theorems: a general version, an improved version for normal limit laws (providing significantly better bounds in some examples with normal limits) and a third version with a relaxed independence condition. Moreover, concrete applications are given which include parameters of random trees, quantities of stochastic geometry as well as complexity measures of recursive algorithms under either a random input or some randomization within the algorithm.

Limit distributions of the maximum size of a tree in a random recursive forest

2002 ◽  
Vol 12 (1) ◽  
Author(s):  
Yu.L. Pavlov ◽  
E.A. Loseva
Keyword(s):  
Probability Distribution ◽  
Asymptotic Behaviour ◽  
Maximum Size ◽  
Rooted Trees ◽  
Limit Distributions ◽  
Uniform Probability ◽  
Uniform Probability Distribution

AbstractWe consider the set of forests consisting of N recursive non-rooted trees with n vertices where the uniform probability distribution is defined. We give the complete description of the asymptotic behaviour of the maximum size of a tree in a forest as n → ∞.

Some ARMA models for dependent sequences of poisson counts

1988 ◽  
Vol 20 (4) ◽  
pp. 822-835 ◽  
Author(s):  
Ed Mckenzie
Keyword(s):  
Asymptotic Behaviour ◽  
Discrete Time ◽  
Joint Distribution ◽  
Autoregressive Process ◽  
Two Dimensional ◽  
Arma Models ◽  
Time Reversibility ◽  
Vector Autoregressive ◽  
Arma Processes ◽  
Vector Autoregressive Process

A family of models for discrete-time processes with Poisson marginal distributions is developed and investigated. They have the same correlation structure as the linear ARMA processes. The joint distribution of n consecutive observations in such a process is derived and its properties discussed. In particular, time-reversibility and asymptotic behaviour are considered in detail. A vector autoregressive process is constructed and the behaviour of its components, which are Poisson ARMA processes, is considered. In particular, the two-dimensional case is discussed in detail.

Probability metrics and recursive algorithms

1995 ◽  
Vol 27 (3) ◽  
pp. 770-799 ◽  
Author(s):  
S. T. Rachev ◽  
L. Rüschendorf
Keyword(s):  
Asymptotic Behaviour ◽  
Recursive Algorithms ◽  
Probability Metrics ◽  
Contraction Method ◽  
Probability Metric ◽  
Wide Range

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.

scholarly journals LIMIT LAWS FOR DISTORTED CRITICAL RETURN TIME PROCESSES IN INFINITE ERGODIC THEORY

2007 ◽  
Vol 07 (01) ◽  
pp. 103-121 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
MEHDI SLASSI
Keyword(s):  
Asymptotic Behaviour ◽  
Uniform Distribution ◽  
Ergodic Theory ◽  
Ergodic Measure ◽  
Return Time ◽  
Limit Laws ◽  
Measure Spaces ◽  
Distribution Laws ◽  
Infinite Ergodic Theory ◽  
Measure Preserving Transformations

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling–Kac condition. We identify two critical cases for which we prove uniform distribution laws.

Limit theorems for threshold-stopped random variables with applications to optimal stopping

1990 ◽  
Vol 22 (2) ◽  
pp. 396-411 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Normalizing Constants

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

scholarly journals Asymptotic Behaviour of Continuous Time and State Branching Processes

2000 ◽  
Vol 68 (1) ◽  
pp. 68-84 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Processes ◽  
Limit Laws ◽  
Classical Setting

AbstractWe prove some limit theorems for contiunous time and state branching processes. The non-degenerate limit laws are obtained in critical and non-critical cases by conditioning or introducing immigration processes. The limit laws in non-critical cases are characterized in terms of the cononical measure of the cumulant semigroup. The proofs are based on estimates of the cumulant semigroup derived from the forward and backward equations, which are easier than the proffs in the classical setting.

Stochastic Analysis of ‘Simultaneous Merge–Sort'

1997 ◽  
Vol 29 (03) ◽  
pp. 669-694
Author(s):  
M. Cramer
Keyword(s):  
Asymptotic Behaviour ◽  
Stochastic Analysis ◽  
Challenging Problem ◽  
Contraction Method ◽  
Merge Sort ◽  
Kolmogorov Metric

The asymptotic behaviour of the recursion is investigated; Yk describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2k –1 and Mk can be interpreted as the number of comparisons of ‘Simultaneous Merge–Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of Mk must converge towards a constant if it converges in distribution at all.

Sign in / Sign up

Export Citation Format

Share Document