Stat150_Spring08_Markov_classification

Stat150_Spring08_Markov_classification - Statistics 150:...

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Unformatted text preview: Statistics 150: Spring 2007 February 3, 2007 0-0 1 Classification of states Definition State i is called persistent (or recurrent ) if P ( X n = i for some n 1 | X = i ) = 1 , which is to say that the probability of eventual return to i , having started from i , is 1. If this probability is strictly less than 1 , the state is called transient . 1 Let f ij ( n ) = P ( X 1 6 = j,X 2 6 = j,...,X n- 1 6 = j,X n = j | X = i ) be the probability that the first visit to state j , starting from i , takes place at the n th step. Define f ij = X n =1 f ij ( n ) to be the probability that the chain ever visits j , starting from i . Of course, j is persistent if and only if f ij = 1 . 2 We seek a criterion for persistence in term of the n-step transition probabilities. We define the generating functions P ij = X n =0 s n p ij ( n ) , F ij ( s ) = X n =0 s n f ij ( n ) with the convention that p ij (0) = ij , and f ij (0) = 0 for all i and j . Clearly f ij = F ij (1) . 3 Proposition 1.1. 1. P ii ( s ) = 1 + F ii ( s ) P ii ( s ) ; 2. P ij ( s ) = F ij ( s ) P jj ( s ) if i 6 = j . Proof. We have p ij ( m ) = m X r =1 f ij ( r ) p jj ( m- r ) , m = 1 , 2 ,.......
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This note was uploaded on 05/07/2008 for the course STAT 150 taught by Professor Evans during the Spring '08 term at University of California, Berkeley.

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Stat150_Spring08_Markov_classification - Statistics 150:...

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