The central limit theorem for stationary Markov chains under invariant splittings

Central Limit Theorems for Conditional Markov Chains

This paper studies Central Limit Theorems for real-valued functionals of Conditional Markov Chains. Using a classical result by Dobrushin (1956) for non-stationary Markov chains, a conditional Central Limit Theorem for fixed sequences of observations is established. The asymptotic variance can be estimated by resampling the latent states conditional on the observations. If the conditional means...

Limit theorems for stationary Markov processes with L2-spectral gap

Let (Xt, Yt)t∈T be a discrete or continuous-time Markov process with state space X × R where X is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (Xt, Yt)t∈T is assumed to be a Markov additive process. In particular, this implies that the first component (Xt)t∈T is also a Markov process. Markov random walks or additive f...

Central and Local Limit Theories in Markov Dependent Random Variables

Variance Estimation in the Central Limit Theorem for Markov Chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent ...

Isometric Cocycles Related to Beam Splittings

We provide in the context of quantum Markov chains due to Accardi coming up from iterated beam splittings a limit theorem concerning with a refinement of the discrete time. The limit object refers to a continuous time and should be called quantum Markov process. Closely related to our construction of such quantum Markov processes are isometric cocycles. We derive, upto technical conditions, a c...