2
votes
0answers
27 views

Geometric ergodicity and mixing - stationary case

I have this theorem: The Markov Chain {$X_n$} is stationary and geometrically ergodic if and only if {$X_n$} is stationary and absolutely regular with $\beta_n=O(\gamma^n)$ for some $\gamma\in ...
0
votes
1answer
18 views

A problem on Markov process

Suppose, $\Pi_{\theta}$ be the transition probability function of a Markov chain. For any function $f$ define $$\Pi_{\theta}f_{\theta}(x) = \int f(y,\theta)\Pi_{\theta}(x,dy).$$ Is there any ...
1
vote
1answer
44 views

Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
1
vote
0answers
61 views

Rate of convergence of $n$-fold convolution over $\mathbb{Z}_p$.

Suppose that $\mu$ is a probabilist measure on $\mathbb{Z}_p$ such that $d_{TV}(\mu,\mu_U)\leqslant \delta < 1$. What are the best upper bound on $$ d_{TV}(\underbrace{\mu*\mu*\ldots*\mu}_{n ...
2
votes
0answers
70 views

simple proof of the $L^2$ weak law for discrete-time ergodic Markov processes

Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ ...
1
vote
0answers
31 views

literature to learn more on ergodic harris recurrent chains with an atom

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's ...
2
votes
2answers
59 views

Convergence of an Ergodic process

I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
3
votes
0answers
133 views

Generalization of Dobrushin's Ergodic Decomposition for continuous Markov Chains

Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It ...
0
votes
0answers
215 views

How to calculate the limit kernel of a non-ergodic Markov Chain?

This question is about finding the limit kernel $P^\infty$ of a non-ergodic Markov Chain. The problem We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ ...
0
votes
1answer
115 views

Manipulating ergodic Markov chains in order to make them non-ergodic

Consider a Markov chain, for simplicity let us consider time discrete chains. The problem We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ (having ...