本文主要研究有关超群性质的两部分内容.首先,对于离散时间有限Markov链,若其可逆且具有单重特征值,则与其可交换的马氏链也可配称.本文证明了对于一般可逆的离散时间Markov链,与其可交换的马氏链仍是可配称的;然而对于一般具有单重特征值的离散时间Markov链,与其可交换的马氏链不再满足可配称性.其次,对于有限状态的不可约生灭马氏核,分别给出了对称和对角线为0情形下有关超群性质的可计算判定.在对称情形下,满足超群性质的生灭矩阵的阶数至多为2在对角线为0的情形下,利用主子式给出了满足超群性质的显式表达.

In this paper, we mainly study the following two parts about the hypergroup property.First,for a discrete Markov chain with finite states.If it is uniplicit,then for any commutative Markov chain,it must be symmetric.In the first part of this paper, the condition of "uniplicit" is relaxed,for general reversible Markov chain,the commutative Markov chain is still symmetric; Whill,for the general Markov chain,and all its eigenvalues are of multiplicity 1, the commutative Markov chain is no longer symmetric.Then, for the irreducible birth and death Markov kernel with finite states, some computable criteria for the hypergroup property in symmetric and diagonal cases are given.In the symmetric case,the order of birth and death matrix satisfying hypergroup property is at most 2; When the diagonal is 0,an explicit expression satisfying hypergroup property by using the minor is given.