The Annals of Probability

Convergence of Markov processes near saddle fixed points

Amanda G. Turner

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We consider sequences (XtN)t≥0 of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form t=b(xt), where $b(x)=\bigl(\smallmatrix{-\mu\ 0 \cr 0\ \lambda}\bigr)x+\tau(x)$ for some λ, μ>0 and τ(x)=O(|x|2). Here the processes are indexed so that the variance of the fluctuations of XtN 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 logN 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 XtN first hits one of the lines y=x or y=−x, then

Nμ/{(2(λ+μ))}|XTN| ⇒ |Z|μ/{(λ+μ)},

for some zero mean Gaussian random variable Z.

Article information

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

First available in Project Euclid: 10 May 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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