What makes a stationary distribution unique?

Stationary distribution may refer to: 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.

What is Markov chain stationary distribution?

A stationary distribution of a Markov chain is a probability distribution that remains unchanged in the Markov chain as time progresses. Typically, it is represented as a row vector π whose entries are probabilities summing to 1, and given transition matrix P, it satisfies.

What makes a Markov chain stationary?

The stationary distribution of a Markov chain describes the distribution of Xt after a sufficiently long time that the distribution of Xt does not change any longer. Then, π is the stationary distribution if it has the property πT=πTP.

How do you prove that a stationary distribution is unique?

When there is only one equivalence class we say the Markov chain is irreducible. 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.

Under what conditions can you find a stationary distribution π?

Stationary distribution of a Markov Chain To put this notion in equation form, let π be a column vector of probabilities on the states that a Markov chain can visit. Then, π is the stationary distribution if it has the property πT=πTP.

What is invariant distribution Markov chain?

A probability distribution π = (πx ⩾ 0 : x ∈ X) such that ∑x∈X πx = 1 is said to be stationary distribution or invariant distribution for the Markov chain X if π = πP, that is πy = ∑x∈X πx pxy for all y ∈ X.

What does the stationary distribution represent?

The stationary distribution of a Markov chain describes the distribution of Xt after a sufficiently long time that the distribution of Xt does not change any longer.

What is the limiting distribution of this Markov chain?

The probability distribution π = [π0, π1, π2, ⋯] is called the limiting distribution of the Markov chain Xn if πj = lim n → ∞P(Xn = j | X0 = i) for all i, j ∈ S, and we have ∑ j ∈ Sπj = 1.

What is a second order Markov chain?

An n-th order Markov chain is one where the information of all the past states is predicated by the n-past states, i.e., for a discrete n-th order Markov chain, . If , this is a second order Markov chain. Usually, for a first-order Markov chain, the prefix ‘first-order’ is often omitted.

What is a generalized Markov chain?

Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in Bayesian statistics, thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory and speech processing.

What does Markov chain mean?

A Markov chain is “a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event”.