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How do you prove a Markov chain is 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.
Is Markov chain reducible?
Definition A Markov chain is called irreducible if and only if all states belong to one communication class. A Markov chain is called reducible if and only if there are two or more communication classes. A finite Markov chain is irreducible if and only if its graph representation is a strongly connected graph.
How do you show a Markov chain?
Markov Chains: A discrete-time stochastic process X is said to be a Markov Chain if it has the Markov Property: Markov Property (version 1): For any s, i0,…,in−1 ∈ S and any n ≥ 1, P(Xn = s|X0 = i0,…,Xn−1 = in−1) = P(Xn = s|Xn−1 = in−1).
What happens when a Markov chain is aperiodic?
Consequently, if the Markov chain is irreducible, then all states have the same period. The proof is another easy exercise. There is a simple test to check whether an irreducible Markov chain is aperiodic: If there is a state i for which the 1 step transition probability p(i,i)>. 0, then the chain is aperiodic.
How to define accessibility of a Markov chain?
Another (equivalent) definition for accessibility of states is the following: Let the number of steps until the Markov chain reaches the state for the first time. We define if for all . Theorem 2.7 Let be such that . In this case is accessible from if and only if .
How is the random dynamic of a Markov chain represented?
The random dynamic of a finite state space Markov chain can easily be represented as a valuated oriented graph such that each node in the graph is a state and, for all pairs of states (ei, ej), there exists an edge going from ei to ej if p (ei,ej)>0. The value of the edge is then this same probability p (ei,ej).
Is the Markov chain with transition matrix irreducible?
As a consequence, the state space can be completely divided into disjoint equivalence classes with respect to the equivalence relation . The Markov chain with transition matrix is called irreducibleif the state space consists of only one equivalence class, i.e. for all .