Limiting Distributions in Markov Chains

One of the principal questions involving Markov chains is what are the long-time (asymptotic) properties of the chain? Before we can formulate the question precisely, we need to introduce some ideas. Suppose we have a Markov chain {Xn}n∈N on a finite or countable state space S with transition probabilities Pi,j = P (Xn+1 = j | Xn = i), i, j ∈ S. We let N denote the number of elements in S (note that N =∞ is allowed).