Open Access
May 2007 Convergence of Markov processes near saddle fixed points
Amanda G. Turner
Ann. Probab. 35(3): 1141-1171 (May 2007). DOI: 10.1214/009117906000000836


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 $b(x)=\bigl(\begin{smallmatrix}-\mu\ 0 \cr 0\ \lambda\end{smallmatrix}\bigr)x+\tau(x)$ 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 $$N^{μ/{(2(λ+μ))}}|X_T^N| ⇒ |Z|^{μ/{(λ+μ)}},$$ for some zero mean Gaussian random variable Z.


Download Citation

Amanda G. Turner. "Convergence of Markov processes near saddle fixed points." Ann. Probab. 35 (3) 1141 - 1171, May 2007.


Published: May 2007
First available in Project Euclid: 10 May 2007

zbMATH: 1134.60019
MathSciNet: MR2319718
Digital Object Identifier: 10.1214/009117906000000836

Primary: 60F05
Secondary: 37C25 , 60G46 , 60J75

Keywords: limit theorem , Markov jump process , martingale inequality , OK Corral gunfight model , ordinary differential equation , saddle fixed point

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • May 2007
Back to Top