一类半马氏过程的常返性与正常返性

The recurrent and positive recurrent of a class of Semi-Markov Process

对于逗留时为正整数值的半马氏过程程｛ξ（ｔ）｝，本文研究了｛ξ（ｔ）｝、相应过程｛ξ（ｎ）｝、嵌入链｛ξ_ｎ｝的常返性和正常返性之间的关系．定理 ２．１证明了三过程的状态常返性是等价的．定理 ２．２证明了ξ（ｔ）、ξ（ｎ）的状态正常返性是等价的，ξ（ｔ）的状态正常返是嵌入链状态正常返的充分条件．定理２．３给出了ξ（ｎ）的状态正常返的充分条件．该条件在状态空间有限时也是必要条件．

As far as the semi-Markov process where the holding time is positive integral is concerned,we have studied the recurrent and positive recurrent relationships among {ξ(t) },corresponding process { ξ(n) } and imbedded chain {ξ_n}. Theorem 2. 1 prove that the state current of the three process is equal in value. Theorem 2. 2 prove that the state positive recurrent of {ξ(t) } and {ξ(n) ) is equal in value. The state positive current of {ξ(t) } is sufficient for the state positive recurrent of imbedded chain. Theorem 2. 3 provide a sufficient condition for the state positive corrent of {ξ(n) }. The condition is also necessary on the condition of definite states.