hw 6 - X is as follows Pick one of the particles uniformly...

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205B homework, week 6; due Thursday March 5 1. Let ( X n ) be an irreducible Markov chain on states I = { 0 , 1 , 2 ,... } . Let g : I R be such that (a) E i g ( X 1 ) g ( i ) for all i , with strict inequality for some i . (b) sup i E i | g ( X 1 ) - g ( i ) | < . Prove that ( X n ) is not positive-recurrent. Give an example to show it may be null-recurrent. 2. Let ( X n : n 0) be a non -homogeneous Markov chain on states { 1 , 2 ,...,K } . Let T be its tail σ -field. Prove that there exists a parti- tion ( B 1 ,...,B m ) , m K of Ω such that T = σ ( B 1 ,...,B m ) up to null sets. [Hint. Consider E ( Z | X n ) for tail-measurable Z .] 3. Let P ( i,j ) be a Markov transition matrix on { 0 , 1 , 2 ,...,K } such that 0 and K are absorbing, { 1 , 2 ,...,K - 1 } forms a strongly connected component and X j jP ( i,j ) = i for each 0 i K. Fix B 2. Define a Markov process ( X n ; n 0) on state-space { 1 , 2 ,...,K } B as follows. A state ( x (1) ,...,x ( B )) represents the positions of B particles, particle b being in position x ( b ). Initially all particles are at position i 0 , for some 1 i 0 K - 1. A step of the process
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Unformatted text preview: X is as follows. Pick one of the particles uniformly at random, and let it perform a move according to P ( · , · ). If the move takes the particle to a position which is not 0, that concludes the step of X . Otherwise the particle tries to move to 0, in which case it is immediately replaced at the position of another particle, picked uniformly at random from the other B-1 particles. Call this latter move a 0-jump. Ultimately the process will reach the absorbing state with all particles in position K . Let N be the random total number of 0-jumps made. Prove E ± B-1 B ² N = i K . What can you deduce about EN ? [Hint: Let A n be the average position of the B particles after n steps. Find a martingale related to A n .] 6...
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