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On the reverse process of a critical multitype Galton-Watson processes without variances. (English) Zbl 0532.60078
The reverse process of a Galton-Watson branching process was introduced by W. W. Esty [J. Appl. Probab. 12, 574-580 (1975; Zbl 0313.60056)], who proved, among other things, a gamma limit law in the critical one-type case. The present author [J. Multivariate Anal. 12, 161-177 (1982; Zbl 0492.60081)] extended this result to the multitype case when the variances of the offspring distribution exists. In the present paper he extends his results to the case where these variances do not exist but the tails of the offspring distributions satisfy a regular variation condition.
Reviewer: D.R.Grey

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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References:
[1] Esty, W.W, The reverse Galton-Watson process, J. appl. probab., 12, 574-580, (1975) · Zbl 0313.60056
[2] Goldstein, M.I; Hoppe, F.M, Critical multitype branching processes with infinite variance, J. math. anal. appl., 65, 675-686, (1978) · Zbl 0408.60082
[3] Goldstein, M.I; Hoppe, F.M, Necessary conditions for normed convergence of critical bienaymé-Galton-Watson processes without variance, J. multivar. anal., 8, 55-62, (1978) · Zbl 0382.60092
[4] Hoppe, F.M, The critical bienaymé-Galton-Watson process, J. stochastic process. appl., 5, 57-66, (1977) · Zbl 0356.60042
[5] Nakagawa, T, Reverse processes and some limit theorems of multitype Galton-Watson processes, J. multivar. anal., 12, 161-177, (1982) · Zbl 0492.60081
[6] Vatutin, V.A, Limit theorems for critical multitype Markov branching processes with infinite second moments, Mat. sb., 103, 253-264, (1977), [Russian] · Zbl 0396.60072
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