A Markov chain can have one or a number of properties that give it specific functions, which are often used to manage a concrete case.. 4.4.1.1 Absorbing chain. Let be a real-valued function defined on the states of a Markov chain . Recall that Pn ij = P(Xn = j|X0 = i), and note that the limit is independent of the initial state. An irreducible Markov chain is said to be aperiodic if for some n ≥ 0 and some state j, P {X n = j | X 0 = j} > 0 and P {X n + 1 = j | X 0 = j} > 0. 4.4.1 Property of Markov chains.

If there is any state in the Markov Chain that can return to itself in 1 step, then the whole chain is aperiodic provided that you've already determined that there exists some subset of the chain that is closed and irreducible. This method, called the Metropolis algorithm, is applicable to a wide range of Bayesian inference problems. It can be shown that if the Markov chain is irreducible and aperiodic then. While a recurrent state has the property that the Markov chain is expected to return to the state an infinite number of times, the Markov chain is not necessarily expected to return even once within a finite number of steps.

The Markov chain Monte Carlo sampling strategy sets up an irreducible, aperiodic Markov chain for which the stationary distribution equals the posterior distribution of interest. Thus the rows of Pn are more and more similar to the row vector π as n becomes large. There's some more subtlety to what makes a chain aperiodic if there is no self-loop, but it may be excessive for your purposes. If and belong to the same class of positive states, then the limit (3) exists, and in the aperiodic case the limit (4) exists. Proposition 12.3: In any irreducible, periodic Markov chain the limits π j = lim n → ∞ p ij n exist and are independent of the initial distribution. This is not good, as a lot of the intuition for recurrence comes from assuming that it will. In any irreducible, aperiodic Markov chain the limits π j = lim n → ∞ p ij n exist and are independent of the initial distribution. A chain can be absorbing when one of its states, called the absorbing state, is such it is impossible to leave once it has been entered. If belongs to a class of zero states or is inessential, then as . Definition 1 . If a Markov chain is irreducible, aperiodic, and positive recurrent, then, for every i,j ∈ S, lim n→∞ Pn ij = πj. Intuitively it means the distribution after N steps is close to the distribution after N+1 steps. Jean-Michel Réveillac, in Optimization Tools for Logistics, 2015.

To make precise the conditions under which it has this interpretation, we first need the definition of an aperiodic Markov chain.

An aperiodic Markov chain with positive recurrent states. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share …


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