Table of Contents
What is an ergodic matrix?
Ergodic Markov Chains. Defn: A Markov chain is called an ergodic or irreducible Markov chain if it is possible to eventually get from every state to every other state with positive probability. Ex: The wandering mathematician in previous example is an ergodic Markov chain.
What does it mean for a Markov chain to be reversible?
equivalently, π(x)P(x, y) = π(y)P(y, x) for all x, y ∈ X. (1) A Markov chain whose stationary distribution π and transition probability matrix P satisfy (1) is called reversible.
Is ergodic irreducible?
Ergodic Markov chains are also called irreducible. A Markov chain is called a regular chain if some power of the transition matrix has only positive elements. However, in others the Regular Markov chain concept doesn’t exist. And Ergodic replaces Regular in all literate.
How can you tell if a Markov chain is reversible?
A Markov chain with invariant measure π is reversible if and only if πiPij = πjPji, for all states i and j. Another useful fact is that once reversibility is checked, invariance is automatic.
Is the reverse of a Markov chain a Markov chain?
Our first result is that the reversed process is still a Markov chain, but not time homogeneous in general.
What is an ergodic state in a Markov chain?
A Markov chain is said to be ergodic if there exists a positive integer such that for all pairs of states in the Markov chain, if it is started at time 0 in state then for all , the probability of being in state at time is greater than .
What is a recurrent state in Markov chain?
A recurrent state has the property that a Markov chain starting at this state returns to this state infinitely often, with probability 1. A transient state has the property that a Markov chain starting at this state returns to this state only finitely often, with probability 1.
What is aperiodic Markov chain?
A class is said to be periodic if its states are periodic. Similarly, a class is said to be aperiodic if its states are aperiodic. Finally, a Markov chain is said to be aperiodic if all of its states are aperiodic. If i↔j, then d(i)=d(j).
Is an ergodic state is recurrent?
Positive recurrent, aperiodic states are called ergodic states.