# Solution to the OK Corral Model via Decoupling of Friedman's Urn

@article{Kingman2003SolutionTT, title={Solution to the OK Corral Model via Decoupling of Friedman's Urn}, author={J. F. C. Kingman and Stanislav Volkov}, journal={Journal of Theoretical Probability}, year={2003}, volume={16}, pages={267-276} }

We consider the OK Corral model formulated by Williams and McIlroy(11) and later studied by Kingman.(7) In this paper we refine some of Kingman's results, by showing the connection between this model and Friedman's urn, and using Rubin's construction to decouple the urn. Also we obtain the exact expression for the probability of survival of exactly S gunmen given an initially fair configuration.

## 31 Citations

A uniform asymptotic expansion for weighted sums of exponentials

- Mathematics
- 2011

Some exactly solvable models of urn process theory

- Mathematics
- 2006

We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a…

On the Estimation of Parameter of Weighted Sums of Exponential Distribution

- Mathematics
- 2014

The random variable , with and being independent exponentially distributed random variables with mean one, is considered. Van Leeuwaarden and Temme (2011) attempted to determine good approximation…

Limiting Distributions for a Class Of Diminishing Urn Models

- MathematicsAdvances in Applied Probability
- 2012

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…

On death processes and urn models

- Mathematics, Computer Science
- 2011

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,…

On Sampling without replacement and OK-Corral urn models

- Mathematics
- 2010

In this work we discuss two urn models with general weight sequences $(A,B)$ associated to them, $A=(\alpha_n)_{n\in\N}$ and $B=(\beta_m)_{m\in\N}$, generalizing two well known P\'olya-Eggenberger…

Border aggregation model

- MathematicsThe Annals of Applied Probability
- 2018

Start with a graph with a subset of vertices called {\it the border}. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the…

Boundary effect in competition processes

- MathematicsJ. Appl. Probab.
- 2019

It is proved that, with probability one, eventually one of the components of the Markov chain tends to infinity, while the remaining one oscillates between values $0$ and $1$.

Linear competition processes and generalized Pólya urns with removals

- MathematicsStochastic Processes and their Applications
- 2021

DAVID STENLUND: Hitting times in urn models and occupation times in one-dimensional diffusion models

- Mathematics
- 2020

The main subject of this thesis is certain functionals of Markov processes. The thesis can be said to consist of three parts. The first part concerns hitting times in urn models, which are Markov…

## References

SHOWING 1-10 OF 17 REFERENCES

Attracting edge property for a class of reinforced random walks

- Mathematics
- 2003

Using martingale techniques and comparison with the generalized Urn scheme, it is shown that the edge reinforced random walk on a graph of bounded degree, with the weight function W(k)=kρ,ρ>1,…

The Ok Corral and the Power of the Law (A Curious Poisson‐Kernel Formula for a Parabolic Equation)

- Mathematics
- 1998

Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or…

Probability and Measure. (2nd. ed

- 1985

Sums of Independent Random Variables

- Mathematics
- 1975

I. Probability Distributions and Characteristic Functions.- 1. Random variables and probability distributions.- 2. Characteristic functions.- 3. Inversion formulae.- 4. The convergence of sequences…

Sums of independent random variables. (2nd

- 1975

Sums of independent random variables. (2nd. ed

- 1975

Embedding of Urn Schemes into Continuous Time Markov Branching Processes and Related Limit Theorems

- Mathematics
- 1968

Martingales in the Ok Corral

- Psychology
- 1999

In the model of the OK Corral formulated by Williams and McIlroy [2]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless…

Reinforced Random Walk

- Mathematics
- 2005

This thesis aim is to present results on a stochastic model called reinforced random walk. This process was conceived in the late 1980’s by Coppersmith and Diaconis and can be regarded as a…