Annals of Applied Probability

Limit theorems for additive functionals of a Markov chain

Milton Jara, Tomasz Komorowski, and Stefano Olla

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Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with α-tails with respect to π, α∈(0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N1/αnNΨ(Xn) to an α-stable law. A “martingale approximation” approach and a “coupling” approach give two different sets of conditions. We extend these results to continuous time Markov jump processes Xt, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N−1/α0NtV(Xs) ds to a stable process. The result is applied to show that an appropriately scaled limit of solutions of a linear Boltzman equation is a solution of the fractional diffusion equation.

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Ann. Appl. Probab., Volume 19, Number 6 (2009), 2270-2300.

First available in Project Euclid: 25 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 76P05: Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]

Stable laws self-similar Lévy processes limit theorems linear Boltzmann equation fractional heat equation anomalous heat transport


Jara, Milton; Komorowski, Tomasz; Olla, Stefano. Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19 (2009), no. 6, 2270--2300. doi:10.1214/09-AAP610.

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  • [1] Bal, G., Papanicolaou, G. and Ryzhik, L. (2002). Radiative transport limit for the random Schrödinger equation. Nonlinearity 15 513–529.
  • [2] Basile, G., Olla, S. and Spohn, H. (2009). Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. Anal. DOI: 10.1007/s00205-008-0205-6. To appear.
  • [3] Brown, B. M. and Eagleson, G. K. (1971). Martingale convergence to infinitely divisible laws with finite variances. Trans. Amer. Math. Soc. 162 449–453.
  • [4] Csáki, E. and Csörgö, M. (1995). On additive functionals of Markov chains. J. Theoret. Probab. 8 905–919.
  • [5] De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 787–855.
  • [6] Derriennic, Y. and Lin, M. (2003). The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 73–76.
  • [7] Doeblin, W. (1938). Sur deux problèmes de M. Kolmogoroff concernant les chaînes dénombrables. Bull. Soc. Math. France 66 210–220.
  • [8] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
  • [9] Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829–846.
  • [10] Erdös, L. and Yau, H.-T. (2000). Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Comm. Pure Appl. Math. 53 667–735.
  • [11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [12] Fannjiang, A. C. (2005). White-noise and geometrical optics limits of Wigner–Moyal equation for wave beams in turbulent media. Comm. Math. Phys. 254 289–322.
  • [13] Fouque, J.-P., Garnier, J., Papanicolaou, G. and Sølna, K. (2007). Wave Propagation and Time Reversal in Randomly Layered Media. Stochastic Modelling and Applied Probability 56. Springer, New York.
  • [14] Goldstein, S. (1995). Antisymmetric functionals of reversible Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 31 177–190.
  • [15] Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk 188 739–741.
  • [16] Jara, M., Komorowski, T. and Olla, S. (2009). Limit theorems for additive functionals of a Markov Chain. Available at arXiv:0809.0177v1.
  • [17] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
  • [18] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [19] Lax, P. D. (2002). Functional Analysis. Wiley, New York.
  • [20] Lepri, S., Livi, R. and Politi, A. (2003). Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377 1–80.
  • [21] Lukkarinen, J. and Spohn, H. (2007). Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal. 183 93–162.
  • [22] Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724.
  • [23] Mellet, A., Mischler, S. and Mouhot, C. (2009). Fractional diffusion limit for collisional kinetic equations. Available at arXiv:0809.2455.
  • [24] Nagaev, S. V. (1957). Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 378–406.
  • [25] Olla, S. (2001). Central limit theorems for tagged particles and for diffusions in random environment. In Milieux Aléatoires. Panoramas et Synthèses 12 23–25. Soc. Math. France, Paris.
  • [26] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [27] Spohn, H. (1977). Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17 385–412.