Limit theorems for additive functionals of a Markov chain

Abstract

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $\pi$. Let $\Psi$ a function on the state space of the chain, with $\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.Comment: Accepted for the publication in Annals of Applied Probabilit

Limit theorems for some continuous-time random walks

“…The results presented in this section extend those of [18] to the case of two-dimensional Markov chains. They can be proved using quite similar arguments.…”

Nonlinear Diffusion with Fractional Laplacian Operators

“…Interest in studying the nonlinear model we propose is two-fold: on the one hand, experts in the mathematics of diffusion want to understand the combination of fractional operators with porous medium type propagation. On the other hand, models of this kind arise in statistical mechanics when modeling for instance heat conduction with anomalous properties and one introduces jump processes into the modeling [46], see also [48,47]. It is mentioned in heat control by [6].…”

“…This is a model of so-called anomalous diffusion, a much studied topic in physics, probability and finance, see for instance [1,47,48,55,71,72] and their references. The equation is solved with the aid of well-known Functional Analysis tools; for instance, it is proved that it generates a semigroup of ordered contractions in L 1 (R n ).…”

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part II: Degenerate Collision Frequency

“…The case with degenerate collision frequency we are considering has been studied in [2]. It does not correspond to a particles dynamics physical situation but rather to the modelization of some chains of oscillators, see [13,22].…”