Mean arrival times of sets for Markov chains (Q5899759)

Let E be the state space of a homogeneous Markov chain (M.C.) \(L=\{\xi_ n\}_ 0^{\infty}\). Suppose that all states in E are commuting and \(E=\cup^{\infty}_{0}E_ i\), \(E_ i\cap E_ j=\emptyset\), \(\forall i\neq j\). Based on the division of E, which arises in studying stochastic systems, and using a technique of limit-passing over, this paper gives sufficient conditions for the mean hitting time of subsets \(A\cup B(\subset E)\), where L is finite, \(E_ x\tau_{A\cup B}<\infty\), \(A\subset E\), \(B\subset E\), \(A\cap B=\emptyset\); \(B\cap E_ i\neq \emptyset\), for \(i\in N_ B\) (some subset of E); \(E_ i\setminus A\neq \emptyset\), \(\forall i.\) A recurrent M.C. \(L(\bar x)=\{{\tilde \xi}_ n\}_ 0^{\infty}\) is introduced for every \(\bar x\in E(A)\triangleq \Pi (E_ i\setminus A)\), with transition probability matrix \(P(\bar x)=(p(\bar x,j,k))\), where \(\bar x=(x_ 0,x_ 1,...)\) and \(p(\bar x,j,k)=P_{x_ j}(\xi_ 1\in E_ k)\). The unique positive solution of the equation \(\pi(\bar x)=P(\bar x)\pi(\bar x)\) and the mean number of hitting times at j by \(L(\bar x)\) before hitting time of i under the condition of \({\tilde\xi}_ 0=j\), \(_ ip_{jj}(\bar x)\), are considered to form sufficient conditions for \(E_ x\tau_{A\cup B}<\infty\).