16.1 Elements of markov sequences  (Page 20/13)

Elements of markov sequences

Markov sequences (Markov chains) are often studied at a very elementary level, utilizing algebraic tools such as matrix analysis. In this section, we show that the fundamentalMarkov property is an expression of conditional independence of “past” and “future," given the “present.” The essential Chapman-Kolmogorov equation is seen as aconsequence of this conditional independence. In the usual time-homogeneous case with finite state space, the Chapman-Kolmogorov equation leads to the algebraicformulation that is widely studied at a variety of levels of mathematical sophistication. With the background laid, we only sketch some of the more common results. This shouldprovide a probabilistic perspective for a more complete study of the algebraic analysis.

Markov sequences

We wish to model a system characterized by a sequence of states taken on at discrete instants which we call transition times . At each transition time, there is either a change to a new state or a renewal of the state immediately before thetransition. Each state is maintained unchanged during the period or stage between transitions. At any transition time, the move to the next state is characterized by a conditional transition probability distribution. We suppose that the system is memoryless , in the sense that the transition probabilities are dependent upon the current state (and perhaps the period number), butnot upon the manner in which that state was reached. The past influences the future only through the present . This is the Markov property , which we model in terms of conditional independence.

For period i , the state is represented by a value of a random variable X _i , whose value is one of the members of a set E , known as the state space . We consider only a finite state space and identify the states by integers from 1 to M . We thus have a sequence