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7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove that the Markov Chain is reversible.

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P0,1 = 1 and otherwise Pi,i+1 = p <
0.5, Pi,i−1 = 1 − p > 0.5. In this case, since the chain can only make a transition (change
of state) of magnitude ±1, we immediately conclude that for each state i ≥ 0, “the rate
from i to i + 1 equals the rate from i + 1 to i”. This is by the same elementary reasoning,

as argued for why the rate out of statei equals the rate into state i, for each state i, for any function/path and has nothing to do with Markov chains: every time there is a change of state from i to i +1 there must be (soon after) a change of state from i +1 to i because that is the only way the process can, yet again, go from i to i +1; there is a one-to-one correspondence. But the rate from i to i+1 equals the rate from i +1 to i is equivalent to (in words) the time-reversibility equations, since here a pair i, j can only be of the form i,i +1 or i,i - 1. Thus the time-reversibility equations are n n1. Since Σηπη = 1 must hold, we get n20 - < 1, the geometric series converges and we can solve explicitly for the and since stationary distribution: )‰(1-p)-11-prim, n>1 (1+ which simplifies to 1- 2p 2(1-p) ㅡ Fn =

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7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove...
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