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Unformatted text preview: he results above for branching processes we see that if we denote the mean
P
number of children by µ = k kak , then
If µ < 1, 0 is positive recurrent
If µ = 1, 0 is null recurrent
If µ > 1, 0 is transient To bring out the parallels between the three examples, note that when µ > 1 or
p > 1/2 the process drifts away from 0 and is transient. When µ < 1 or p < 1/2
the process drifts toward 0 and is positive recurrent. When µ = 1 or p = 1/2,
there is no drift. The process eventually hits 0 but not in ﬁnite expected time,
so 0 is null recurrent. 1.11 Chapter Summary A Markov chain with transition probability p is deﬁned by the property that
given the present state the rest of the past is irrelevant for predicting the future:
P (Xn+1 = y Xn = x, Xn 1 = xn 1 , . . . , X0 = x0 ) = p(x, y ) 60 CHAPTER 1. MARKOV CHAINS The m step transition probability
pm (i, j ) = P (Xn+m = y Xn = x)
is the mth power of the matrix p.
Recurrence and transience
The ﬁrst thing we need to determine about a Markov chain is which states
are recurrent and which are transient. To do this we...
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 Spring '10
 DURRETT
 The Land

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