Double stochastic matrix positive recurrent?

[diddy_kaufen]

Homework Statement

When is a Markov chain with double stochastic matrix positive recurrent?

Homework Equations

Double stochastic matrix is when the sum of the column vectors, and not just the row vectors, is 1.

The Attempt at a Solution

I know I have to show that the expected value of the hitting time is $E(h_N)<\infty$ as $N\rightarrow\infty$. But I don't know how to approach this since it is an infinite space. Hints or guidance greatly appreciated.

edit, I think I got it, will edit later today

 

[mighty2000]

First, let's define some terms. A Markov chain is a stochastic process where the future state depends only on the current state, and not on any previous states. A double stochastic matrix is a square matrix where the sum of the elements in each column is equal to 1. Positive recurrence refers to a state being visited infinitely often with probability 1.

To show that a Markov chain with a double stochastic matrix is positive recurrent, we need to show that the expected value of the hitting time, denoted as E(h_N), is finite as N approaches infinity. The hitting time, h_N, is the number of steps it takes for the Markov chain to reach a certain state, N, starting from some initial state.

To approach this problem, we can use the concept of irreducibility. A Markov chain is said to be irreducible if it is possible to go from any state to any other state in a finite number of steps. If a Markov chain is irreducible, then all states in the chain are in the same communicating class, meaning that they can communicate with each other. In other words, there is a non-zero probability of transitioning from any state to any other state.

Now, let's assume that our Markov chain is irreducible. This means that for any state N, there is a non-zero probability of transitioning from any other state to state N in a finite number of steps. This also means that the expected hitting time, E(h_N), must be finite, as there is always a non-zero probability of reaching state N in a finite number of steps.

Next, let's consider the double stochastic matrix. Since the sum of the elements in each column is equal to 1, this means that the probability of transitioning from any state to any other state is equal to 1. This ensures that the Markov chain is irreducible, as there is always a non-zero probability of transitioning to any state from any other state.

Therefore, we can conclude that a Markov chain with a double stochastic matrix is positive recurrent, as the expected hitting time, E(h_N), is finite due to the irreducibility of the chain and the equal probabilities of transitioning from any state to any other state.