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 Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 1, Pages 30–51 (Mi tvp323)

Deviations from typical type proportions in critical multitype Galton–Watson processes

V. A. Vatutina, K. Fleischmannb

a Steklov Mathematical Institute, Russian Academy of Sciences
b Weierstrass Institute for Applied Analysis and Stochastics, Germany

Abstract: Consider a critical $K$-type Galton–Watson process $\{\mathbf{Z}(t): t=0,1,\ldots \}$ and a real vector $\mathbf{w}=(w_{1},\ldots ,w_{K})^{\top}$. It is well known that under rather general assumptions, $\langle \mathbf{Z} (t),\mathbf{w}\rangle :=\sum_{k}Z_{k}(t)w_{k}$ conditioned on nonextinction and appropriately scaled has a limit in law as $t\uparrow \infty$ [V. A. Vatutin, Math. USSR Sb., 32 (1977), pp. 215–225]. However, the limit degenerates to $0$ if the vector $\mathbf{w}$ deviates seriously from "typical" type proportions, i.e., if $\mathbf{w}$ is orthogonal to the left eigenvectors related to the maximal eigenvalue of the mean value matrix. We show that in this case (under reasonable additional assumptions on the offspring laws) there exists a better normalization which leads to a nondegenerate limit. Opposed to the finite variance case, which was already resolved in [K. Athreya and P. Ney, Ann. Probab., 2 (1974), pp. 339–343] and [I. S. Badalbaev and A. Mukhitdinov, Theory Probab. Appl., 34 (1989), pp. 690–694], the limit law (for instance, its “index”) may seriously depend on $\mathbf{w}$.

Keywords: marked particle, typical type proportions, nondegenerate limit, nonextinction, deviations, asymptotic expansion.

DOI: https://doi.org/10.4213/tvp323

Full text: PDF file (841 kB)

English version:
Theory of Probability and its Applications, 2001, 45:1, 23–40

Bibliographic databases:

Citation: V. A. Vatutin, K. Fleischmann, “Deviations from typical type proportions in critical multitype Galton–Watson processes”, Teor. Veroyatnost. i Primenen., 45:1 (2000), 30–51; Theory Probab. Appl., 45:1 (2001), 23–40

Citation in format AMSBIB
\Bibitem{VatFle00} \by V.~A.~Vatutin, K.~Fleischmann \paper Deviations from typical type proportions in critical multitype Galton--Watson processes \jour Teor. Veroyatnost. i Primenen. \yr 2000 \vol 45 \issue 1 \pages 30--51 \mathnet{http://mi.mathnet.ru/tvp323} \crossref{https://doi.org/10.4213/tvp323} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1810973} \zmath{https://zbmath.org/?q=an:0988.60084} \transl \jour Theory Probab. Appl. \yr 2001 \vol 45 \issue 1 \pages 23--40 \crossref{https://doi.org/10.1137/S0040585X97978038} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000167428900002} 

• http://mi.mathnet.ru/eng/tvp323
• https://doi.org/10.4213/tvp323
• http://mi.mathnet.ru/eng/tvp/v45/i1/p30

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This publication is cited in the following articles:
1. Reder C., “Transient behaviour of a Galton–Watson process with a large number of types”, Journal of Applied Probability, 40:4 (2003), 1007–1030
2. Penisson S., “Beyond the Q-Process: Various Ways of Conditioning the Multitype Galton-Watson Process”, ALEA-Latin Am. J. Probab. Math. Stat., 13:1 (2016), 223–237
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