Lemma 43 is a stationary distribution if and only if q

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Unformatted text preview: ts into two at rate , so q (i, i + 1) = i. To find the transition probability of the Yule process we will guess and verify that pt (1, j ) = e t (1 ) for j tj 1 e i.e., a geometric distribution with success probability e To explain the mean we note that d EX (t) = E X (t) dt j implies 1 t (4.13) and hence mean e t . E1 X (t) = e t . To check (4.13), we will use the forward equation (4.9) to conclude that if 1 then p0 (1, j ) = j pt (1, j ) + (j 1)pt (1, j 1) (4.14) t where pt (1, 0) = 0. The use of the forward equation here is dictated by the fact that we are only writing down formulas for pt (i, j ) when i = 1. To check the proposed formula for j = 1 we note that p0 (1, 1) = t e t = pt (1, 1) Things are not so simple for j > 1: p0 (1, j ) = t t e +e t (1 (j ) e tj 1 1)(1 e ) tj 2 (e t ) 127 4.3. LIMITING BEHAVIOR Recopying the first term on the right and using e t = (1 the second, we can rewrite the right-hand side of the above as e t (1 +e ) t (j 1)(1 tj 2 (j t (1 e t )+ in 1) tj 1 e e e )...
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