What does it mean for a Markov chain to converge?
Table of Contents
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 .
- Find any eigenvector v of A with eigenvalue 1 by solving ( A − I n ) v = 0.
- Divide v by the sum of the entries of v to obtain a vector w whose entries sum to 1.
- 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.