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How do you find the equilibrium distribution 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.
What is the equilibrium distribution?
A physical, economic or social system which has settled down to a stable statistical behaviour has a stationary, or equilibrium, distribution which specifies the limiting proportions spent in the designated states of the system.
What is a stationary discrete time Markov chain?
Definition. A discrete-time Markov chain is a sequence of random variables. with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states: if both conditional probabilities are well defined, that is, if.
What is invariant probability?
A row vector v is a probability vector if all the components are non-negative and sum to 1. A probability vector π is an invariant probability distribution for stochastic matrix P if πP = π. In other words, an invariant probablity distribution of P is a left eigenvector of P with eigenvalue 1.
Do all Markov chains have steady state?
No. Some Markov chains reach a state of equilibrium but some do not. Some Markov chains transitions do not settle down to a fixed or equilibrium pattern. Therefore we’d like to have a way to identify Markov chains that do reach a state of equilibrium.
When does a Markov chain reach an equilibrium position?
Thus, once a Markov chain has reached a distribution π Tsuch that π P = πT, it will stay there. If πTP = πT, we say that the distribution πT is an equilibrium distribution. Equilibriummeans a level position: there is no more change in the distri- bution of X. t as we wander through the Markov chain.
When do we say that the distribution πt is an equilibrium distribution?
If πTP = πT, we say that the distribution πT is an equilibrium distribution. Equilibriummeans a level position: there is no more change in the distri- bution of X. t as we wander through the Markov chain.
What makes an irreducible Markov chain unique?
An irreducible Markov chain (finite or countably infinite) has a unique equilibrium distribution if and only if all states are positive recurrent.
Is the invariant vector unique in a Markov chain?
For a Markov chain with N < ∞ states, the set I of invariant probability vectors is a non-empty simplex in R N whose extreme points correspond to the recurrent classes of the chain. Thus, the vector is unique iff there is exactly one recurrent class; the transient states (if any) play absolutely no role (as in Jens’s example).