Let (Ω,A,μ,T) be a measure preserving dynamical system. The speed of convergence in probability in the ergodic theorem for a generic function on Ω is arbitrarily slow.

Let ${(X_t,Y_t)}_{t\in 𝕋}$ be a discrete or continuous-time Markov process with state space $𝕏\times {ℝ}^d$ where $𝕏$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. ${(X_t,Y_t)}_{t\in 𝕋}$ is assumed to be a Markov additive process. In particular, this implies that the first component ${\left(X_t\right)}_{t\in 𝕋}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process ${\left(Y_t\right)}_{t\in 𝕋}$ is shown to satisfy the...