Mean Strong Periodicity of Markov Chains in Random Environments

For Markov chains in random environments, Cogburn[1,2] firstly introduced weakly ergodic concept that “starting time” is original point, and gave some conditions ensuring that the chains are weakly ergodic; Li Ying-qiu[3,4] introduced the definitions of uniformly weakly ergodicity and strongly ergodicity that “starting time” is any point and gave some conditions ensuring that the chains are uniformly weakly ergodicity and strongly ergodicity. The definitions of mean strongly ergodic that “starting time” is any point are introduced. Some conditions ensuring that the chains are mean strongly ergodic.