What does it mean for a Markov chain to converge?

Convergence to equilibrium means that, as the time progresses, the Markov chain ‘forgets’ about its initial distribution λ. In particular, if λ = δ(i), the Dirac delta concentrated at i, the chain ‘forgets’ about initial state i.

What is steady state distribution in Markov chain?

We create a Maple procedure called steadyStateVector that takes as input the transition matrix of a Markov chain and returns the steady state vector, which contains the long-term probabilities of the system being in each state. The input transition matrix may be in symbolic or numeric form.

Can a periodic Markov chain converge?

When P is irreducible (but not necessarily aperiodic), then π still exists and is unique, but the Markov chain does not necessarily converge to π from every starting state. This has the unique stationary distribution π = (1/2,1/2), but does not converge from either of the two initial states.

How do you know if a Markov chain converges?

By elementary arguments (page 2) we know that starting from any initial distribu- tion q, if the iteration q,qP,qP2,… converges, then it must converge to this unique stationary distribution. However, it remains to be shown that if the Markov chain determined by P is regular, then the iteration always converges.

How do you find the unique steady state vector?

Here is how to compute the steady-state vector of A .

1. Find any eigenvector v of A with eigenvalue 1 by solving ( A − I n ) v = 0.
2. Divide v by the sum of the entries of v to obtain a vector w whose entries sum to 1.
3. This vector automatically has positive entries. It is the unique steady-state vector.

How do you find the steady state of a matrix?

In order to find the steady-state vector s=(s1s2) you need to solve a simple matrix equation (T−I)s=0. and, thus, you have a linear system {−s1/2+s2=0;s1/2−s2=0.

What is the equilibrium vector of the Markov chain?

EQUILIBRIUM VECTOR OF A MARKOV CHAIN If a Markov chain with transition matrix P is regular, then there is a unique vector V such that, for any probability vector v and for large values of n, Vector V is called the equilibrium vector or the fixed vector of the Markov chain.