矩阵秩在判定齐次马尔可夫链遍历性中的应用

Application of Matrix Rank in Judging the Ergodicity of Homogeneous Markov Chains

利用常规方法判断齐次马尔可夫链的遍历性有时显得比较麻烦,文章引入矩阵秩,给出判断齐次马尔可夫链遍历性的一个新方法.设{Xk,k≥0}为具有n个状态的齐次马氏链,R(Z)=m,则有:当m=n时,齐次马氏链{Xk,k≥0}具有遍历性;当m<n时,齐次马氏链{Xk,k≥0}不具有遍历性.通过2个实例说明,应用此方法来判断齐次马氏链的遍历性更简便、快捷,可以有效地提高效率.

Applying general methods sometimes makes it difficult to judge the ergodicity of homogeneous Markov chains.As a new way of judgment,the matrix rank is introduced.Assume {Xk,k≥0} is a homogeneous Markov chain with n states,R（Z） = m,then when m = n,{Xk,k≥0}has its ergodicity;when m 〈 n,ergodicity doesn′t exist in {Xk,k≥0}.It is simpler and more convenient for us to use the conclusion to judge ergodicity of homogeneous Markov chains.The effectiveness and accuracy of the method are obviously shown by two examples.