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Independent And Stationary Sequences Of Random Variables - Chapter 19

Chapter 19 EXAMPLES AND ADDENDA The separate sections of this chapter are not related to one another except in so far as they illustrate or extend the results of Chapter 18 . © 1 . The central limit theorem for homogeneous Markov chains Consider a homogeneous Markov chain with a finite number of states (labelled 1, 2, . . ., k) and transition matrix P = (p i ;) (see, for instance, Chapter III of [47] ) . If Xn is the state of the system at time n, we have the sequence of random variables X1 , X2 , . . ., Xn. | Chapter 19 EXAMPLES AND ADDENDA The separate sections of this chapter are not related to one another except in so far as they illustrate or extend the results of Chapter 18. 1. The central limit theorem for homogeneous Markov chains Consider a homogeneous Markov chain with a finite number of states labelled 1 2 k and transition matrix P pi7 see for instance Chapter III of 47 . If Xn is the state of the system at time n we have the sequence of random variables X1 X2 . Xn . 19.1.1 We denote by pff the probability of moving from state i to state j in n steps. If for some s 0 ptf 0 for all i j then Markov s theorem 47 states that the limits Pj lim Pi M- 00 exist for all i and j and do not depend on i and that for constants C p 0 p l max pj Cp . 19.1.2 ij The numbers pr p2 . pk form a stationary probability distribution in the sense that if P Xl j pj for all j then the variables Xn form a stationary sequence. It then follows from 19.1.2 that Xn is uniformly mixing since if 4 iX1 i1 X2 i2 . X iJ r in r s 366 EXAMPLES AND ADDENDA Chap. 19 P AB Pilpili2.pir_iirp n r.pix_lis P A pirin rPin rin r Pis - I is P A P B P A Pin rPin rin r i.pis_iis so that P AB P A P B P A p in -pin r P A Cpn. Let be any real function defined on the states of the chain. Application of Theorem 18.5.2 shows that the central limit theorem applies to the sequence f Xj whenever -Ef XJ 2 2 f Ef p J-E p JH PGl-E PG . J 1 If n ti1 n2 . nk is any other initial distribution we denote the corresponding probability and expectation by Pn and En. Theorem 19.1.1. Assume that 19.1.2 holds and that oVO. Then for any initial distribution 7t lim pj-tf 4 2 - f e-i 2du . n- oo t0 77 J i J J-oo Proof. The theorem is already proved for the case 7i p1 p2 . pk . Thus denoting the normalised sum as usual by Z and setting r log n e - 2 - En eitZn e 2 - E eitZ En eitZn - E ei z f it r E. exp -j I PQ-E PQ -1 .077 j 1 J f h r E exp -j J XJ-E pQ -1 cm j 1 it n I E -E exp py-E VQ o l .on J r 1 J k Z iPiL1-PjrjPjr 1jr 2---Pj .