The Uniqueness、Perturbation Bound And Algorithm Of The Stationary Probability Matrix For Transition Probability Tensors

Nonnegative tensors are widely used in engineering and science.Transition probability tensors,as a special kind of nonnegative tensor,are derived from mathematical models describing high-order Markov chains and higherorder multivariate Markov chains.The stationary probability matrix of the transition probability tensor for higher-order multivariate Markov chains reflects that the probability distribution of the stochastic process in each state will not change after a long time.It plays an important role in biological gene sequence analysis,multilinear PageRank,management planning and other practical problems.In particular,the uniqueness condition of the stationary probability matrix of transition probability tensors has become one of the hot topics in tensor theory and its application.In 2019,a uniqueness condition of the stationary probability matrix of transition probability tensors for high-order multivariate Markov chains was given by Li et al.However,this condition is relatively strong and difficult to verify.In this dissertation,we focus on research the uniqueness condition of the stationary probability matrix of transition probability tensors for high-order multivariate Markov chains,and on this basis,further research the perturbation bound of the stationary probability matrix and algorithm for solving stationary probability matrix,which mainly includes the following four aspects.Firstly,some uniqueness conditions of the stationary probability matrix of transition probability tensors are given by using inequality scaling techniques,optimal set method,contract mapping principle,adding parameter method,and so on,the uniqueness of the stationary probability matrix of transition probability tensors with the special structure are also studied,and the uniqueness condition and concrete form of the stationary probability matrix of the 4-order 2 dimensional irreducible transition probability tensor are obtained,then,the new results are compared with the existing ones through practical application examples and theoretical proof.Secondly,under the uniqueness conditions of the stationary probability matrix,perturbation analysis of the stationary probability matrix is carried out,several perturbation bounds of the stationary probability matrix are given,and then the obtained perturbation bounds are compared.Numerical examples show that the proposed perturbation bounds are meaningful.Thirdly,by applying the uniqueness conditions of the stationary probability matrix,the new convergence conditions and error bounds for the existing algorithm that calculating the stationary probability matrix are obtained,we also propose a new algorithm for solving stationary probability matrix and analyze the convergence of new algorithm.Finally,the above research ideas and methods are applied to general nonnegative tensors,and some new uniqueness conditions for positive Z-eigenvectors of nonnegative tensors are gained.