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Convergence of Markov processes near saddle fixed points.

Turner, Amanda G. (2007) Convergence of Markov processes near saddle fixed points. The Annals of Probability, 35 (3). pp. 1141-1171.

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    Abstract

    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 ẋt=b(xt), 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 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.

    Item Type: Article
    Journal or Publication Title: The Annals of Probability
    Additional Information: RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics
    Uncontrolled Keywords: Limit theorem ; Markov jump process ; martingale inequality ; OK Corral gunfight model ; saddle fixed point ; ordinary differential equation.
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 2395
    Deposited By: ep_importer
    Deposited On: 01 Apr 2008 09:24
    Refereed?: Yes
    Published?: Published
    Last Modified: 09 Oct 2013 13:12
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/2395

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