In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov processes.We show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and ...In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov processes.We show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t,x,y)(with respect to m)which is bounded in(x,y)for every t>0,then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution.We also present several classes of Markov processes satisfying the above conditions.展开更多
The causal states of computational mechanics define the minimal sufficient memory for a given discrete stationary stochastic process. Their entropy is an important complexity measure called statistical complexity （or...The causal states of computational mechanics define the minimal sufficient memory for a given discrete stationary stochastic process. Their entropy is an important complexity measure called statistical complexity （or true measure complexity）. They induce the s-machine, which is a hidden Markov model （HMM） generating the process. But it is not the minimal one, although generative HMMs also have a natural predictive interpretation. This paper gives a mathematical proof of the idea that the s-machine is the minimal HMM with an additional （partial） determinism condition. Minimal internal state entropy of a generative HMM is in analogy to statistical complexity called generative complexity. This paper also shows that generative complexity depends on the process in a nice way. It is, as a function of the process, lower semi-continuous （w.r.t. weak-, topology）, concave, and behaves nice under ergodic decomposition of the process.展开更多
基金supported by National Natural Science Foundation of China(GrantNo.11171010)Beijing Natural Science Foundation(Grant No.1112001)
文摘In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov processes.We show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t,x,y)(with respect to m)which is bounded in(x,y)for every t>0,then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution.We also present several classes of Markov processes satisfying the above conditions.
文摘The causal states of computational mechanics define the minimal sufficient memory for a given discrete stationary stochastic process. Their entropy is an important complexity measure called statistical complexity （or true measure complexity）. They induce the s-machine, which is a hidden Markov model （HMM） generating the process. But it is not the minimal one, although generative HMMs also have a natural predictive interpretation. This paper gives a mathematical proof of the idea that the s-machine is the minimal HMM with an additional （partial） determinism condition. Minimal internal state entropy of a generative HMM is in analogy to statistical complexity called generative complexity. This paper also shows that generative complexity depends on the process in a nice way. It is, as a function of the process, lower semi-continuous （w.r.t. weak-, topology）, concave, and behaves nice under ergodic decomposition of the process.
