This is not good, as a lot of the intuition for recurrence comes from assuming that it will. Let be a real-valued function defined on the states of a Markov chain . Definition 1 . 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. 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.
Intuitively it means the distribution after N steps is close to the distribution after N+1 steps.
To make precise the conditions under which it has this interpretation, we first need the definition of an aperiodic Markov chain. This method, called the Metropolis algorithm, is applicable to a wide range of Bayesian inference problems.
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. 4.4.1 Property of Markov chains. 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. It can be shown that if the Markov chain is irreducible and aperiodic then. If belongs to a class of zero states or is inessential, then as . 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. In any irreducible, aperiodic Markov chain the limits π j = lim n → ∞ p ij n exist and are independent of the initial distribution. Thus the rows of Pn are more and more similar to the row vector π as n becomes large. Jean-Michel Réveillac, in Optimization Tools for Logistics, 2015. If a Markov chain is irreducible, aperiodic, and positive recurrent, then, for every i,j ∈ S, lim n→∞ Pn ij = πj. 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. 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. An aperiodic Markov chain with positive recurrent states.
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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.
Recall that Pn ij = P(Xn = j|X0 = i), and note that the limit is independent of the initial state.
If and belong to the same class of positive states, then the limit (3) exists, and in the aperiodic case the limit (4) exists.
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