What makes a transition matrix irreducible?

What makes a transition matrix irreducible?

First, the chain must be irreducible, which means that any state j can be reached from any state i in a finite number of steps. The chain must be positive recurrent, meaning that, on average, the chain starting in state i returns to state i in a finite number of steps for all i ∈ S .

How do you know if a Markov chain is irreducible?

Definition A Markov chain is called irreducible if and only if all states belong to one communication class. A Markov chain is called reducible if and only if there are two or more communication classes. A finite Markov chain is irreducible if and only if its graph representation is a strongly connected graph.

How do you know if a transition matrix is aperiodic?

Since the number 1 is co-prime to every integer, any state with a self-transition is aperiodic. If there is a self-transition in the chain (pii>0 for some i), then the chain is aperiodic.

How do you know if a matrix is irreducible?

A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size).

What is irreducible stochastic process?

Irreducible is said to mean that the stochastic process can “go from any state to any state”.

Is an irreducible Markov chain closed?

Definition: an irreducible closed set C is a closed set such that x → y for all choices x, y ∈ C. An irreducible Markov chain is one where x → y for all x, y ∈ Σ. Theorem: In an irreducible closed set, either all states are transient or all states are recurrent.

What is the transition probability?

the probability of moving from one state of a system into another state. If a Markov chain is in state i, the transition probability, pij, is the probability of going into state j at the next time step.

What is irreducible state?

The transition probability matrices given below are for 4-state. A Markov chain is irreducible if all its states intercommunicate, i.e. if it is possible to pass between each pair of states in a finite number of steps with positive probability.

What is meant by irreducible factor?

Irreducible quadratic factors are quadratic factors that when set equal to zero only have complex roots. As a result they cannot be reduced into factors containing only real numbers, hence the name irreducible.

What is the limiting distribution for the transition probability matrix xn?

Then, Xn is a Markov chain on the states 0, 1, …, 6 with transition probability matrix The matrix is doubly stochastic, and it is regular ( P2 has only strictly positive entries), hence the limiting distribution is π = (1 7, …, 1 7).

How to tell if a matrix is irreducible or reducible?

If all entries are nonzero, then the matrix is irreducible. Otherwise, it’s reducible. If P is too large, compute P n with n as large as you can. Same test, slightly less accurate. Irreducibility means : you can go from any state to any other state in a finite number of steps.

What is the state transition probability matrix of a Markov chain?

The state transition probability matrix of a Markov chain gives the probabilities of transitioning from one state to another in a single time unit. It will be useful to extend this concept to longer time intervals. Definition 9.3: The n -step transition probability for a Markov chain is (9.4)P (n) i, j = Pr (X k + 1 = j|X k = i).

What is a doubly stochastic transition probability matrix?

A transition probability matrix is called doubly stochastic if the columns sum to one as well as the rows. Formally, P = || Pij || is doubly stochastic if P ij ≥ 0 and ∑ kP ik = ∑ kP kj = 1 for all i, j Consider a doubly stochastic transition probability matrix on the N states 0, 1, …, N − 1.