Advances in Applied Probability

Large deviations for the empirical measure of heavy-tailed Markov renewal processes

Mauro Mariani and Lorenzo Zambotti

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A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.

Article information

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 648-671.

First available in Project Euclid: 19 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60K15: Markov renewal processes, semi-Markov processes

Large deviation empirical measure Markov renewal process heavy tail


Mariani, Mauro; Zambotti, Lorenzo. Large deviations for the empirical measure of heavy-tailed Markov renewal processes. Adv. in Appl. Probab. 48 (2016), no. 3, 648--671.

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  • Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. (New York) 51), 2nd edn. Springer, New York.
  • Basile, G. and Bovier, A. (2010). Convergence of a kinetic equation to a fractional diffusion equation. Markov Process. Relat. Fields 16, 15–44.
  • Bertini, L., Faggionato, A. and Gabrielli, D. (2015). Large deviations of the empirical flow for continuous time Markov chains. Ann. Inst. H. Poincaré Prob. Statist. 51, 867–900.
  • Brin, M. and Stuck, G. (2002). Introduction to Dynamical Systems. Cambridge University Press.
  • Condamin, S. et al. (2008). Probing microscopic origins of confined subdiffusion by first-passage observables. Proc. Nat. Acad. Sci. USA 105, 5675–5680.
  • Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.
  • Den Hollander, F. (2000). Large Deviations (Fields Inst. Monogr. 14). American Mathematical Society, Providence, RI.
  • Duffy, K. and Metcalfe, A. P. (2005). How to estimate the rate function of a cumulative process. J. Appl. Prob. 42, 1044–1052.
  • Duffy, K. and Rodgers-Lee, M. (2004). Some useful functions for functional large deviations. Stoch. Stoch. Reports 76, 267–279.
  • Duffy, K. R., Macci, C. and Torrisi, G. L. (2011). On the large deviations of a class of modulated additive processes. ESAIM Prob. Statist. 15, 83–109.
  • Ganesh, A., Macci, C. and Torrisi, G. L. (2005). Sample path large deviations principles for Poisson shot noise processes, and applications. Electron. J. Prob. 10, 1026–1043.
  • Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin.
  • Lefevere, R. and Zambotti, L. (2010). Hot scatterers and tracers for the transfer of heat in collisional dynamics. J. Statist. Phys. 139, 686–713.
  • Lefevere, R., Mariani, M. and Zambotti, L. (2011). Large deviations for renewal processes. Stoch. Process. Appl. 121, 2243–2271.
  • Lefevere, R., Mariani, M. and Zambotti, L. (2011). Large deviations of the current in stochastic collisional dynamics. J. Math. Phys. 52, 033302.
  • Maiboroda, R. E. and Markovich, N. M. (2004). Estimation of heavy-tailed probability density function with application to Web data. Comput. Statist. 19, 569–592.
  • Russell, R. (1997). The large deviations of random time-changes. Doctoral Thesis, Trinity College Dublin.