F.M. Hoppe. Supercritical multitype branching processes. Ann. Prob., 4:393--401, 1976. Home/Search Document Not in Database Summary Related Articles Check |
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Large Deviations For Supercritical Multi-Type Branching Processes - Jones (2001) (Correct)
....left and right eigenvectors corresponding to , normed so that 1 u = 1 and t u = 1. It is well known that under Assumption A0 there exist Seneta Heyde norming constants # n=0 such that c n Z n Lt in probability, where 0 L and P(L[i] 0) q[i] P(Z n [i, 0 for some n) [Hop76]. If EZ 1 [i, j] log Z 1 [i, j] for all i, j, then we can take c n = n and EL = u. By use of a multivariate Sevastyanov transformation, it is possible to restrict our attention to processes for which q = 0. The minimum population size of such processes is either bounded, or grows ....
....continuous, and we have F [i] x) 0 for x q if and only if # i, 0. 3.1. Karlin McGregor function and an upper bound Let #[i] be the Laplace transform of L[i] then # is analytic on (0, since L 0, and satisfies the multivariate Poincare equation #(s) f(#(s) for s [0, #) From Hoppe [Hop76] Theorem 2.3, we have that up to a scale factor, # is the unique strictly decreasing convex solution to this equation with #(0 ) 1. Put log # log 0, so that = 1 #, and define the (multi type) Karlin McGregor function for f to be F (s) s F (#(s) for s #) Then F is ....
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F.M. Hoppe. Supercritical multitype branching processes. Ann. Prob., 4:393--401, 1976.
Branching Processes - Athreya, Vidyashankar (1999) (113 citations) (Correct)
....2 As in single type case even if (3.9) fails to hold there always exist Senata constants c n such that such that u DeltaZ n cn converges to a nontrivial limit W and cn 1 cn converges to ae and Zn v DeltaZ n converges to u on the set of nonextinction. This result was established by Hoppe [42]. Thus the relative proportions of the various types stabilizes to a deterministic distribution and the growth rates of all types are identical to the exponential rate of ae n . In the multitype supercritical case the population vector Z n is such that u Delta Z n grows at a geometric rate ....
Hoppe,F.M.(1976): Supercritical multitype branching processes, Annals of Probability, No. 3, 393-401.
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