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Unformatted text preview: is that if Tn ! 1 as
n ! 1, then we have succeeded in deﬁning the process for all time and we
are done. This will be the case in almost all the examples we consider. The
bad news is that limn!1 Tn < 1 can happen. In most models, it is senseless
to have the process make an inﬁnite amount of jumps in a ﬁnite amount of
time so we introduce a “cemetery state” to the state space and complete the
deﬁnition by letting T1 = limn!1 Tn and setting
X (t) = for all t T1 To show that explosions can occur we consider.
Example 4.5. Pure birth processes with power law rates. Suppose
q (i, i + 1) = ip and all the other q (i, j ) = 0. In this case the jump to n + 1
is made at time Tn = t1 + · · · + tn , where tn is exponential with rate np .
Etn = 1/np , so if p > 1
n
X
ETn =
1/mp
P1 m=1 This implies ET1 =
m=1 1/m < 1, so T1 < 1 with probability one.
When p = 1 which is the case for the Yule process
p ETn = (1/ ) n
X m=1 1/m ⇠ (log n)/ 123 4.2. COMPUTING THE TRANSITION PROBABILITY as n ! 1. This is, by itself, not enough to establish that Tn ! 1, but...
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 Spring '10
 DURRETT
 The Land

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