Question

A transition probability matrix P is said to be doubly stochastic if sum over each column equals 1, that is

$\sum_j P_{ij} =1$, for all j

If a doubly stochastic Markov chain has $n$ state, aperiodic, and positive reccurent. Calculate the limiting probabilities for every state!

Answer 1

In a doubly stochastic Markov chain with n aperiodic and positive recurrent states, the limiting probabilities are the same for every state and are calculated as 1/n.

Step-by-step explanation:

The question pertains to a doubly stochastic Markov chain with n states that is aperiodic and positive recurrent. In a doubly stochastic Markov chain, the limiting probabilities, also known as stationary probabilities, are the same for every state.

Due to the concept of double stochasticity (where the sum over each column and the sum over each row of the transition probability matrix equals 1), it is established that each state has a limiting probability of 1/n. This follows from the fact that a doubly stochastic matrix has an Eigenvector of the form (1/n, 1/n, ..., 1/n)' belonging to the Eigenvalue 1; this Eigenvector represents the stationary distribution, assuming all states are positive recurrent and the chain is aperiodic.