N n0 n0 eqt 48 and check by dierentiating that 1 1 x d

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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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