Markov Systems and Their Additive Functionals

Transformations Of_ Markov Processes Connected with Additive Functionals

8 Limit Theorems for Additive Functionals of a Markov Chain

Consider a Markov chain {X n } n≥0 with an ergodic probability measure π. Let Ψ 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 N 1/α N n Ψ(X n) to a α-stable law. A " martingale approximation " approach and " coupling " approach give two different sets of condition...

Non-markovian Limits of Additive Functionals of Markov Processes

Abstract. In this paper we consider an additive functional of an observable V (x) of a Markov jump process. We assume that the law of the expected jump time t(x) under the invariant probability measure π of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled limit of the functional is a Mittag-Leffler proces, provided that Ψ(x) := V (x)t(x) is square integ...

Deviation Bounds for Additive Functionals of Markov Processes

where X is a μ stationary and ergodic Markov process and V is some μ integrable function. These bounds are obtained under various moments assumptions for V , and various regularity assumptions for μ. Regularity means here that μ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc...).

Scaling Limits of Additive Functionals of Interacting Particle Systems

Using the renormalization method introduced in [17], we prove what we call the local Boltzmann-Gibbs principle for conservative, stationary interacting particle systems in dimension d = 1. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by-pro...