Creat An Invariant Measure For Markov Chains

Now the Time series analysis is widely used, this requires a stationary distribu-tion, which means that there must be an invariant measure. This paper will elaborate the existing condition of two state spaces and how to creat the invariant measure.The first one is that the state space is general state space. If the Markov chain Φ is ψ-irreducible and there is an accessible atom, we can creat a subinvariant measure using the taboo probability of the atom, specially, the subinvariant still is invariant under the condition that the Markov chain is reccurent. Without the accessible atom, we need to creat the subinvariant measure by using the splitting skill. If the Markov chain is not ψ-irreducible, but there is a subinvariant measure and a positive subset A of B+(X), we also can creat the minimal measure. Under the condition that A is reccurent, the minimal measure still is invariant.The second one is that the state space is a topological state space, we don’t need to know whether the Markov chain Φ is ψ-irreducible, we just want to know that the transition probability of Φis continuous, which means that the Markov chain is weak Feller. If the transition probability sequence{Pk{x,·),k∈Z+} is tight, the limit of the transition prabability sequence is the only invariant measure respecting to vague convergence. We enlarge the probability space on whichΦ is defined and creat a bivariate chain ψ. If ψis Harris reccurect, similar to the first one, we can creat the invariant measure by using the taboo probability Uh of ψfor h is a con-tinuous function which define on X and0<h(x)≤1.