Critical Multitype Branching Systems: Extinction Results
Jose Alfredo Lopez Mimbela (Centro de Investigacion en Matematicas Mexico)
Abstract
We consider a critical branching particle system in $\mathbb{R}^d$, composed of individuals of a finite number of types $i\in\{1,\ldots,K\}$. Each individual of type i moves independently according to a symmetric $\alpha_i$-stable motion. We assume that the particle lifetimes and offspring distributions are type-dependent. Under the usual independence assumptions in branching systems, we prove extinction theorems in the following cases: (1) all the particle lifetimes have finite mean, or (2) there is a type whose lifetime distribution has heavy tail, and the other lifetimes have finite mean. We get a more complex dynamics by assuming in case (2) that the most mobile particle type corresponds to a finite-mean lifetime: in this case, local extinction of the population is determined by an interaction of the parameters (offspring variability, mobility, longevity) of the long-living type and those of the most mobile type. The proofs are based on a precise analysis of the occupation times of a related Markov renewal process, which is of independent interest.
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Pages: 1356-1380
Publication Date: August 9, 2011
DOI: 10.1214/EJP.v16-908
References
- D.B.H. Cline, T. Hsing. Large deviation probabilities for sums of random variables with heavy or subexponential tails (1998). Math. Review number not available.
- D. Drasin, E. Seneta. A generalization of slowly varying functions. Proc. Amer. Math. Soc. 96 (1986), no. 3, 470--472. Math. Review MR0822442
- K. Fleischmann, V.A. Vatutin. An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system. Probab. Theory Related Fields 116 (2000), no. 4, 545--572. Math. Review MR1757599
- L.G. Gorostiza, S. Roelly, A. Wakolbinger. Persistence of critical multitype particle and measure branching processes. Probab. Theory Related Fields 92 (1992), no. 3, 313--335. Math. Review MR1165515
- M.I. Goldstein. Critical age-dependent branching processes: Single and multitype. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 74--88. Math. Review MR0278402
- L.G. Gorostiza, A. Wakolbinger. Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19 (1991), no. 1, 266--288. Math. Review MR1085336
- O. Kallenberg. Stability of critical cluster fields. Math. Nachr. 77 (1977), 7--43. Math. Review MR0443078
- J.A. L'opez-Mimbela, A. Wakolbinger. Clumping in multitype-branching trees. Adv. Appl. Prob. 28 (1996), 1034--1050. Math. Review MR1418245
- K. Matthes, J. Kerstan, J. Mecke. Infinitely divisible point processes. Translated from the German by B. Simon. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Chichester-New York-Brisbane, 1978. Math. Review MR0517931
- S.V. Nagaev. On the asymptotic behavior of one-sided large deviation probabilities. Teor. Veroyatnost. i Primenen. 26 (1981), no. 2, 369--372. Math. Review MR0616627
- G.R. Shorack, J.A. Wellner. Empirical Processes with Applications to Statistics. John Wiley & Sons, New York, (1986). Math. Review MR0838963
- V.A. Vatutin. Limit theorems for critical Markov branching processes with several types of particles and infinite second moments. Matem. Sb., 103 (1977), 253--264. (In Russian). Math. Review MR0443115
- V.A. Vatutin. A limit theorem for a critical Bellman--Harris branching process with several types of particles and infinite second moments. Theory Probab. Appl. 23 (1978), 4, 776--688. Math. Review MR0516277
- V.A. Vatutin. Discrete limit distributions of the number of particles in a Bellman--Harris branching process with several types of particles. Theory Probab. Appl. 24 (1979), 509--520. Math. Review MR0541363
- V.A. Vatutin, A. Wakolbinger. Spatial branching populations with long individual lifetimes. Theory Probab. Appl. 43 (1999), 620--632. Math. Review MR1692425
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