On the Effects of Small Deviations in the Transition Matrix of a Finite Markov Chain

Let P={Pij; i, j=l, 2, ... , r} be the transition matrix of a stationary, regular Markov chain C with r states and a={ai; i=l, 2, ... , r} be its limiting vector which represents the stationary distribution of the chain. Suppose that there is another regular Markov chain C' with the transition matrix pI ={p;j} and the limiting vector a' ={a;}. If pI is close to P, we can expect that a' is also close to a. Then how close are they? If P and pI are exactly known, then we can answer the question by calculating both a and a'. However the question is difficult if pr is not exactly known and the only thing being known is that P' is close to P in some measure. Such a situation arises whenever we infer the transition matrix of a Markov chain. In such a case, we can only get an approximate value of the transition matrix, and we are concerned with bounds within which the real limiting vector exists. This problem is more difficult than it may first appear. Because, each entry ai of the limiting vector a is written as a quotient of two determinants of matrices, and generally it is not easy to determine bounds of variation of a determinant caused by small changes of its entries.