## The Annals of Probability

- Ann. Probab.
- Volume 35, Number 3 (2007), 1141-1171.

### Convergence of Markov processes near saddle fixed points

#### Abstract

We consider sequences (*X*_{t}^{N})_{t≥0} of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form *ẋ*_{t}=*b*(*x*_{t}), where for some *λ*, *μ*>0 and *τ*(*x*)=*O*(|*x*|^{2}). Here the processes are indexed so that the variance of the fluctuations of *X*_{t}^{N} is inversely proportional to *N*. The simplest example arises from the OK Corral gunfight model which was formulated by Williams and McIlroy [*Bull. London Math. Soc.* **30** (1998) 166–170] and studied by Kingman [*Bull. London Math. Soc.* **31** (1999) 601–606]. These processes exhibit their most interesting behavior at times of order log*N* so it is necessary to establish a fluid limit that is valid for large times. We find that this limit is inherently random and obtain its distribution. Using this, it is possible to derive scaling limits for the points where these processes hit straight lines through the origin, and the minimum distance from the origin that they can attain. The power of *N* that gives the appropriate scaling is surprising. For example if *T* is the time that *X*_{t}^{N} first hits one of the lines *y*=*x* or *y*=−*x*, then

*N*^{μ/{(2(λ+μ))}}|*X*_{T}^{N}| ⇒ |*Z*|^{μ/{(λ+μ)}},

for some zero mean Gaussian random variable *Z*.

#### Article information

**Source**

Ann. Probab., Volume 35, Number 3 (2007), 1141-1171.

**Dates**

First available in Project Euclid: 10 May 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1178804325

**Digital Object Identifier**

doi:10.1214/009117906000000836

**Mathematical Reviews number (MathSciNet)**

MR2319718

**Zentralblatt MATH identifier**

1134.60019

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 37C25: Fixed points, periodic points, fixed-point index theory 60G46: Martingales and classical analysis 60J75: Jump processes

**Keywords**

Limit theorem Markov jump process martingale inequality OK Corral gunfight model saddle fixed point ordinary differential equation

#### Citation

Turner, Amanda G. Convergence of Markov processes near saddle fixed points. Ann. Probab. 35 (2007), no. 3, 1141--1171. doi:10.1214/009117906000000836. https://projecteuclid.org/euclid.aop/1178804325