有限状态马氏过程的自相关函数的单调性(英文)

On the Monotonicity of Autocorrelation Function for Finite State Markov Processes

使用生成元(转移速率矩阵)的若当分解,给出了以自相关函数描述的可逆性和不可逆性之间的区别。对于有限状态的马氏过程,证明了不可逆性蕴涵着存在状态空间上的实值观测函数,使得它的自相关函数在[0,∞)上不单调,且在某个时间间隔内,它是负值的;而可逆性则蕴涵着状态空间上所有的实值观测函数的自相关函数在[0,∞)上单调,且为正值的。

By using the Jordan decomposition of the generator （transition rate matrix ）, the difference in terms of the autocorrelation function between the reversibility and the irreversibility is shown. For a finite state Markov process with continuous time, its irreversibility implies that there exists an observable function on its state space such that its autocorrelation function is nonmonotonic over [0,∞ ） and negative at certain time interval; its reversibility implies that all the autocorrelation functions are monotonic and positive over [0, ∞ ）.