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1 2p (1.31) It should not be surprising that the system stabilizes when p < 1/2. In
this case movements to the left have a higher probability than to the right, so
there is a drift back toward 0. On the other hand if steps to the right are more 55 1.10. INFINITE STATE SPACES* frequent than those to the left, then the chain will drift to the right and wander
o↵ to 1.
II. When p > 1/2 all states are transient. (1.23) implies that if x > 0, Px (T0 < 1) = ((1 p)/p)x . To ﬁgure out what happens in the borderline case p = 1/2, we use results
from Sections 1.8 and 1.9. Recall we have deﬁned Vy = min{n 0 : Xn = y }
and (1.17) tells us that if x > 0
Px (VN < V0 ) = x/N
If we keep x ﬁxed and let N ! 1, then Px (VN < V0 ) ! 0 and hence
Px (V0 < 1) = 1
In words, for any starting point x, the random walk will return to 0 with probability 1. To compute the mean return time, we note that if ⌧N = min{n : Xn 62
(0, N )}, then we have ⌧N V0 and by (1.26) we have E1 ⌧N = N 1. Letting
N ! 1 and combining the last tw...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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