By A. K. Basu

This publication, appropriate for complex undergraduate, graduate and learn classes in statistics, utilized arithmetic, operation study, laptop technology, diverse branches of engineering, company and administration, economics and lifestyles sciences etc., is aimed among undemanding chance texts and complicated works on stochastic strategies. What distinguishes the textual content is the representation of the theorems via examples and functions.

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**Extra resources for Introduction to Stochastic Process**

**Example text**

E. State 1 is recurrent and by solidarity theorem the other two states 0 and 2 are also recurrent. Now jui = X n f ^ = l x 0 + 2+ l = 2 . State 1 is positive recurrent. Thus the states of the chain are periodic (each with period 2) and positive recurrent. 7, pfl ) dl/u= 2/ 2 =1 for all /. 2 Foster type theorems The following theorems, associated with Foster, give criteria for transient and recurrent chains in terms of solution of certain equations. C. is irreducible. 11 (Foster, 1953) Let the Markov chain be irreducible.

The solution is Poo'” = Similarly, po(Xn= 1) = p ^ = P0(X„= I) = + d -p -9 )'(l- — - (I - p - q)" - E - , ) (Poo(0) = 1), Discrete Time Markov Chain 31 p \l} = — ----- (1 - p - q ) n- 2 — , ^ 10 p +q y P +q and 11 = — — + (1 - p - q ) n— q—, p +q 1 ~q p + q _q —» p~ 1 SX 1 Pn = Thus, p +q p_ ’ P -q - p~ q _ \0 < p + q < 2 p as n —> «> if \ p < 1, q < 1 p q 1 p +q q This is the steady state distribution and also a stationary distribution. e. | I - p - q \ < 1. So we have a unique stationary distribution n determined by, 7I0 = 7" -, 7i\= —7— and p +q p +q lJ J , if 0< t' n does not depend on initial distribution.

C. never returns to j at any time n > 1 and that contradicts the fact that j is recurrent. So f f = 1. Since f f = 1, there exists n x such that p \f^ > 0. Now p \ff+n+n° ^ > p \fl)p ^ p ^ o) and hence 2 p[n) > 2 p ^ n+nQ) > p-"l}pyfo) 2 n=1 n=\ ■ ’ J n=\ (since j is recurrent). Hence, i is recurrent. Since i is recurrent and i ~^j(fij= 1) from the first part of proof it follows that f f =1. 4, (m) (n-m) P ii where 0 < n < n(n > 1). 7) Discrete Time Markov Chain 27 For £ > 0, take n and n so large that ( 2 .