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Large deviations for empirical measures of self-interacting Markov chains

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  • Budhiraja, Amarjit
  • Waterbury, Adam
  • Zoubouloglou, Pavlos

Abstract

Let Δo be a finite set and, for each probability measure m on Δo, let G(m) be a transition kernel on Δo. Consider the sequence {Xn} of Δo-valued random variables such that, given X0,…,Xn, the conditional distribution of Xn+1 is G(Ln+1)(Xn,⋅), where Ln+1=1n+1∑i=0nδXi. Under conditions on G we establish a large deviation principle for the sequence {Ln}. As one application of this result we obtain large deviation asymptotics for the Aldous et al. (1988) approximation scheme for quasi-stationary distributions of finite state Markov chains. The conditions on G cover other models as well, including certain models with edge or vertex reinforcement.

Suggested Citation

  • Budhiraja, Amarjit & Waterbury, Adam & Zoubouloglou, Pavlos, 2025. "Large deviations for empirical measures of self-interacting Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 186(C).
    • Handle: RePEc:eee:spapps:v:186:y:2025:i:c:s030441492500081x
    DOI: 10.1016/j.spa.2025.104640
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    References listed on IDEAS

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    1. Dupuis, Paul & Spiliopoulos, Konstantinos, 2012. "Large deviations for multiscale diffusion via weak convergence methods," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1947-1987.
    2. Franchini, Simone, 2017. "Large deviations for generalized Polya urns with arbitrary urn function," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3372-3411.
    3. Veretennikov, A. Yu., 2000. "On large deviations for SDEs with small diffusion and averaging," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 69-79, September.
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