Accumulated hosts on the definition of stationary distributions for Markov chains

Abstract

There are two unequal definitions for the Markov chain stationary distribution: Definition 1 Let {x(n), n=0,1,2,…} be the Markov chain, E={0,1,2,…) is the state space. If for An and i∈E, there is P{X(n)=i}=P{X(o)=i}=Pi, then {Pi, i∈E} is called the stationary distribution of the Markov chain. Definition 2 Let {x(n).n=0,1,2,…} be the Markov chain, E={0,1,2,…} be the state space, Pij be the one-step transition probability, {πi,i∈ E} is a probability distribution. If {πi, i∈E} satisfies the equation system πi=sum from j=0 to ∞πj Pji , i=0,1,2,…, then {πi,i∈E} is called a stationary distribution of the Markov chain. This paper compares these two definitions through a series of theorems, so as to find out their similarities and differences.