Second Order Markov Chains with Finite Vectors Known Stationary

We consider the stationary probability vectors of a second order Markov chain, and discuss the geometrical structure of the set of these vectors with finite ones known stationary. By elementary methods, we deduce that if three vectors on a line segment in the standard simplex are stationary probability vectors of a second order Markov chain, then the set of stationary probability vectors of the chain contains all vectors on the segment. Based on properties of quadratic stochastic operators and affine subspaces, we give two generalizations of the above result for any plane sections of the standard simplex. The generalizations claim that the stationarity of all vectors of any given plane section of the simplex is equivalent to the stationarity of finite vectors in some appropriate positions of the section. Furthermore, the obtained results are applied to discuss the minimum number of such known stationary vectors as interior points for faces.

Automated summary

This paper examines the stationary probability vectors of a second order Markov chain and investigates the geometrical structure of the set of these vectors with finite ones known stationary. Using elementary methods, it is deduced that if three vectors on a line segment in the standard simplex are stationary probability vectors of a second order Markov chain, then the set of stationary probability vectors of the chain contains all vectors on the segment. Two generalizations of this result are provided for any plane sections of the standard simplex based on properties of quadratic stochastic operators and affine subspaces. These generalizations claim that the stationarity of all vectors of any given plane section of the simplex is equivalent to the stationarity of finite vectors in some appropriate positions of the section. Furthermore, the obtained results are applied to discuss the minimum number of such known stationary vectors as interior points for faces.