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Does a Markov chain converge?
Do all Markov chains converge in the long run to a single stationary distribution like in our example? No. It turns out only a special type of Markov chains called ergodic Markov chains will converge like this to a single distribution.
How do you know if a Markov chain converges?
We will prove that if the Markov chain is irreducible and aperiodic, then there exists a stationary distribution, the stationary distribution is unique, and the Markov chain will converge to the stationary distribution (note the Perron-Frobenius theorem).
What is a periodic Markov chain?
A state in a Markov chain is periodic if the chain can return to the state only at multiples of some integer larger than 1. Periodic behavior complicates the study of the limiting behavior of the chain.
How do you find the equilibrium of a Markov chain?
Answer A Markov chain with transition matrix P has an equilibrium distribution if p(n) = p0Pn →π as n → ∞, independently of the initial distribution p(0). π ∗ is a stationary distribution if π∗P =π∗. If π is an equilibrium distribution then it it is a stationary distribution, but not conversely.
Does the stationary property of Markov chain always exist?
In general a Markov chain can have more than one equivalence class. We will show that for an irreducible Markov chain, a stationary distri- bution exists if and only if all states are positive recurrent, and in this case the stationary distribution is unique.
Do all Markov chains have steady state probabilities?
Do all Markov chains have the property that eventually the distribution settles to the “same steady” state regardless of the initial state? This Markov chain doesn’t converge to a unique steady state. This Markov chain doesn’t converge at all!
How can you tell if a graph is aperiodic?
In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph.
Are recurrent States periodic?
If a state is periodic, it is positive recurrent.
What is Markov analysis?
Markov analysis is a method used to forecast the value of a variable whose predicted value is influenced only by its current state, and not by any prior activity. Markov analysis is often used for predicting behaviors and decisions within large groups of people.
What are the assumptions of Markov analysis?
Markov assumptions: (1) the probabilities of moving from a state to all others sum to one, (2) the probabilities apply to all system participants, and (3) the probabilities are constant over time.
What is null recurrent Markov chain?
If all states in an irreducible Markov chain are null recurrent, then we say that the Markov chain is null recurrent. If all states in an irreducible Markov chain are transient, then we say that the Markov chain is transient.
Why is the stationary distribution unique?
Assuming irreducibility, the stationary distribution is always unique if it exists, and its existence can be implied by positive recurrence of all states. The stationary distribution has the interpretation of the limiting distribution when the chain is ergodic.