Linear functionals for critical multitype galton-watson branching processes View Full Text


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Article Info

DATE

2000-04

AUTHORS

V. A. Vatutin

ABSTRACT

We consider a critical K-type Galton-Watson branching process {Z(t)=(Z1(t),…,ZK(t)): t=0,1,…}. It is well known that, under rather general assumptions on the characteristics of the branching process, for any real vector the distribution of the sequence of sums, properly scaled and given thatZ(t)≠0 converges to a limit law as t→∞. In addition, the scaling function is of order t if the variances of the number of direct descendants of particles of all types are finite. But the limiting distribution has a unit atom at zero if the vectorw is orthogonal to the left eigenvector of the mean matrix of the process corresponding to its Perron root. If the variances of the number of direct descendants of particles of all types are finite, then to get a nontrivial limiting distribution for suchw (under the condition of nonextinction) one should always scaleZ(t)w by a function proportional to. In the case where the variances of the number of direct descendants of some types are infinite, the order of a scaling function providing existence of a nontrivial limit essentially depends onw. In the present note, we take the next step, namely, for a large class of processes with K≥3 types of particles and infinite variances of the number of direct descendants, we show that one can find two vectorsw1 andw2 orthogonal to the mentioned left eigenvector, such that the processesZ(t)w1 andZ(t)w2 conditioned on nonextinction up to moment t have different orders of growth in t as t→∞. More... »

PAGES

1502-1509

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URI

http://scigraph.springernature.com/pub.10.1007/bf02673726

DOI

http://dx.doi.org/10.1007/bf02673726

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