Independent And Stationary Sequences Of Random Variables - Chapter 19

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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...

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Nội dung Text: 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 , . . . . (19 .1 .1)
We denote by p~~ ) the probability of moving from state i to state j in n steps.
If for some s > 0, p( ;) > 0 for all i, j, then Markov's theorem [47] states that
the limits
pj = lim p(n )
n-+ oo


exist for all i and j and do not depend on i, and that, for constants C, p
(0
~( A3 . . . i n
ti ) ~c ( k . . .in )
k
~ u(Al . . .(n+1)) = li(A2A . .(inn +1)) - Y" (All . . . n) 7
k

and more generally,
+s) (n+s)) (19.4.9)
y(A~, .. . = Y(A
Finally, the general case is obtained from (19 .4.9) and the equation

(19.4.10)
Ik
(k # 11)

and the stationarity is proved .
(2) To prove the uniform mixing condition, it is sufficient to prove that
(,~ 1 . . .k
I~(AII . . .ikn k+n . . .k+n+s _ ~~ 1 . . .k ~,( k+n . . .k+n+s
AJk+n . . .Jk+n+s) Y"(AiI . . .ik) ' (AJk+n . . .Jk+n+s)I
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