Can a Markov chain have no stationary distribution?
If p ≥ q, then there is no stationary distribution. Theorem 6.2. 1 A Markov chain over a finite state space S has a stationary distribution. Note that the stationary distribution mentioned in the above theorem is not necessarily unique.
Do all Markov chains have stationary distribution?
Not all Morkov chains have a stationary distribution but for some classes of probability transition matrix (those defining ergodic Markov chains), a stationary distribution is guaranteed to exist.
Does the stationary property of Markov chain always exist?
Stationary distribution may refer to: A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution.
How do you prove a Markov chain has a stationary distribution?
A distribution π is called a stationary distribution of a Markov chain P if πP = π. Thus, a stationary distribution is one for which advancing it along the Markov chain does not change the distribution: if the distribution of Xt is a stationary distribution π, then the distribution of Xt+1 will also be π.
Is Markov process stationary?
1: A stochastic process Y is stationary if the moments are not affected by a time shift, i.e., A theorem that applies only for Markov processes: A Markov process is stationary if and only if i) P1(y, t) does not depend on t; and ii) P1|1(y2,t2 | y1,t1) depends only on the difference t2 − t1.
Does periodic Markov chain have limiting distribution?
It turns out that in this case the Markov chain has a well-defined limiting behavior if it is aperiodic (states have period 1). Here is the idea: If π=[π1,π2,⋯] is a limiting distribution for a Markov chain, then we have π=limn→∞π(n)=limn→∞[π(0)Pn].
How do you know if a Markov chain has a unique stationary distribution?
If the entire state space of a Markov chain is irreducible, we can find a unique stationary distribution. When the entire state space of a Markov chain is not irreducible, we have to use the decomposition theorem, and find stationary distribution for every persistent group of states.
When Markov chain is stationary?
The stationary distribution of a Markov Chain with transition matrix P is some vector, ψ, such that ψP = ψ. In other words, over the long run, no matter what the starting state was, the proportion of time the chain spends in state j is approximately ψj for all j.
Is stationary distribution limiting distribution?
The limiting distribution of a regular Markov chain is a stationary distribution. If the limiting distribution of a Markov chain is a stationary distribution, then the stationary distribution is unique.
What is the difference between limiting and stationary distribution?
limiting distribution is independent of the initial state while stationary distribution is dependent on the initial state distribution. limiting distribution is asymptotic distribution while stationary distribution a special initial state distribution.
Is random walk stationary distribution?
The random walk is a stationary stochastic process. If we find a probability distribution π which satisfies the detailed balance condition, π(i)pij = π(j)pji , for all i,j ∈ V, then it is the stationary distribution. In particular, if G is d-regular, π(i) = d 2m = 1 n , for all i ∈ V and π is the uniform distribution.
What is the stationary distribution for a Markov chain?
For Markov Chain with several communication classes (example C1, C2) there exist stationary distributions (linear combination of π 1 and π 2). But how about this one, states {1,2,3} can communicate with each other while state {4} can access to other states. One persistent state and one transient state.
What is the difference between ergodic and absorbing Markov chains?
Ergodic Markov chains have a unique stationary distribution, and absorbing Markov chains have stationary distributions with nonzero elements only in absorbing states. The stationary distribution gives information about the stability of a random process and, in certain cases, describes the limiting behavior of the Markov chain.
What happens when there are multiple eigenvectors in a Markov chain?
When there are multiple eigenvectors associated to an eigenvalue of 1, each such eigenvector gives rise to an associated stationary distribution. However, this can only occur when the Markov chain is reducible, i.e. has multiple communicating classes.
Can a Markov chain have multiple communicating classes?
However, this can only occur when the Markov chain is reducible, i.e. has multiple communicating classes. In genetics, one method for identifying dominant traits is to pair a specimen with a known hybrid. Their offspring is once again paired with a known hybrid, and so on.