scholarly journals On death processes and urn models

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Continuous Time ◽  
Urn Models ◽  
Sampling Without Replacement ◽  
International Audience

International audience We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models.

scholarly journals Limit laws for a class of diminishing urn models.

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Random Variables ◽  
Urn Models ◽  
Limit Laws ◽  
Sampling Without Replacement ◽  
International Audience ◽  
Special Instance ◽  
Special Case

International audience In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by $M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}$ and $c∈\mathbb{N} _0$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case $a=c=d=1$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, $a=d=1$ and $c=0$, and corresponding generalizations, $a,d∈\mathbb{N}$ and $c=0$.

Ehrenfest urn models

1965 ◽  
Vol 2 (02) ◽  
pp. 352-376 ◽  
Author(s):  
Samuel Karlin ◽  
James McGregor
Keyword(s):  
Exponential Distribution ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Urn Models ◽  
Time Intervals ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Random Times

In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in stateiif there areiballs in urn I, N −iballs in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distributionWhen an event occurs a ball is chosen at random (each of theNballs has probability 1/Nto be chosen), removed from its urn, and then placed in urn I with probabilityp, in urn II with probabilityq= 1 −p, (0 <p< 1).

scholarly journals The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is Nondegenerate (Extended Abstract)

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Patrick Bindjeme ◽  
james Allen fill
Keyword(s):  
Large Class ◽  
Continuous Time ◽  
Random Variable ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean

International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

scholarly journals Note on the weighted internal path length of b-ary trees

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Path Length ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Time Parameter ◽  
Recursive Sequences ◽  
International Audience

Analysis of Algorithms International audience In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

scholarly journals A functional limit law for the profile of plane-oriented recursive trees.

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Henning Sulzbach
Keyword(s):  
Continuous Time ◽  
Functional Limit ◽  
Convergence Theorems ◽  
International Audience ◽  
Recursive Trees ◽  
Martingale Convergence

International audience We give a functional limit law for the normalized profile of random plane-oriented recursive trees. The proof uses martingale convergence theorems in discrete and continuous-time. This complements results of Hwang (2007).

scholarly journals Limiting Distributions for a Class Of Diminishing Urn Models

2012 ◽  
Vol 44 (1) ◽  
pp. 87-116 ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Urn Models ◽  
Moment Sequence ◽  
Limiting Distributions ◽  
Sampling Without Replacement ◽  
The Moment

In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix and c = pa with . We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

scholarly journals Classification of large Pólya-Eggenberger urns with regard to their asymptotics

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Nicolas Pouyanne
Keyword(s):  
Algebraic Approach ◽  
Urn Models ◽  
Search Trees ◽  
International Audience

International audience This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having "large'' eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn's replacement matrix and examples of each case are treated. We study the cases of so-called cyclic urns in any dimension and $m$-ary search trees for $m \geq 27$.

scholarly journals Boundary effect in competition processes

2019 ◽  
Vol 56 (3) ◽  
pp. 750-768
Author(s):  
Vadim Shcherbakov ◽  
Stanislav Volkov
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Boundary Effect ◽  
Competitive Interaction ◽  
Continuous Time Markov Chain ◽  
Urn Models ◽  
Birth Processes ◽  
Special Case

AbstractThis paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

scholarly journals Limiting Distributions for a Class Of Diminishing Urn Models

2012 ◽  
Vol 44 (01) ◽  
pp. 87-116 ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Urn Models ◽  
Moment Sequence ◽  
Limiting Distributions ◽  
Sampling Without Replacement ◽  
The Moment

In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix and c = pa with . We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

scholarly journals Individuals at the origin in the critical catalytic branching random walk

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Valentin Topchii ◽  
Vladimir Vatutin
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Continuous Time ◽  
Branching Random Walk ◽  
Conditional Limit Theorem ◽  
Catalytic Branching ◽  
International Audience ◽  
Number Of Individuals ◽  
Conditional Limit

International audience A continuous time branching random walk on the lattice $\mathbb{Z}$ is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we prove a conditional limit theorem for the number of individuals at the origin.

Sign in / Sign up

Export Citation Format

Share Document