Contents
What is a irreducible Markov chain?
A Markov chain in which every state can be reached from every other state is called an irreducible Markov chain. If a Markov chain is not irreducible, but absorbable, the sequences of microscopic states may be trapped into some independent closed states and never escape from such undesirable states.
Are irreducible Markov chains aperiodic?
If we have an irreducible Markov chain, this means that the chain is aperiodic. Since the number 1 is co-prime to every integer, any state with a self-transition is aperiodic. If there is a self-transition in the chain (pii>0 for some i), then the chain is aperiodic.
When is a Markov chain said to be irreducible?
A Markov chain is said to be irreducible if it is possible to get to any state from any state. The following explains this definition more formally. A state j is said to be accessible from a state i (written i → j) if a system started in state i has a non-zero probability of transitioning into state j at some point.
Is the Markov chain and its transition matrix aperiodic?
The Markov chain and its transition matrix are called aperiodic if all states of are aperiodic. We will now show that the periods and coincide if the states belong to the same equivalence class of communicating states. For this purpose we introduce the notation if .
What is the probability of a Markov process changing?
Each number represents the probability of the Markov process changing from one state to another state, with the direction indicated by the arrow. For example, if the Markov process is in state A, then the probability it changes to state E is 0.4, while the probability it remains in state A is 0.6.
When does a Markov chain have a stationary distribution?
A Markov chain ( Xt) t≥0 has stationary distribution π (⋅) if for all j and for all t ≥ 0, ∑ i π(i)Pij(t) = π(j). The existence of a stationary distribution for the chain is equivalent to that chain being positive recurrent. gcd{t > 0: Pii(t) > 0} = 1.