Size-biased branching population measures and the multi-type x log x condition
Peter Olofsson
Source: Bernoulli Volume 15, Number 4
(2009), 1287-1304.
Abstract
We investigate the x log x condition for a general (Crump–Mode–Jagers) multi-type branching process with a general type space by constructing a size-biased population measure that relates to the ordinary population measure via an intrinsic martingale Wt. Sufficiency of the x log x condition for a non-degenerate limit of Wt is proved and conditions for necessity are investigated.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.bj/1262962236
Digital Object Identifier: doi:10.3150/09-BEJ211
Mathematical Reviews number (MathSciNet): MR2597593
Zentralblatt MATH identifier: 1202.60141
References
Alsmeyer, G. (1994). On the Markov renewal theorem. Stochastic Process. Appl. 50 37–56.
Mathematical Reviews (MathSciNet): MR1262329
Zentralblatt MATH: 0789.60066
Digital Object Identifier: doi:10.1016/0304-4149(94)90146-5
Athreya, K.B. (2000). Change of measures for Markov chains and the LlogL theorem for branching processes. Bernoulli 6 323–338.
Mathematical Reviews (MathSciNet): MR1748724
Digital Object Identifier: doi:10.2307/3318579
Project Euclid: euclid.bj/1081788031
Athreya, K.B. and Ney, P. (1972). Branching Processes. Berlin: Springer.
Mathematical Reviews (MathSciNet): MR373040
Biggins, J.D. and Kyprianou, A.E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
Mathematical Reviews (MathSciNet): MR2058149
Zentralblatt MATH: 1056.60082
Digital Object Identifier: doi:10.1239/aap/1086957585
Project Euclid: euclid.aap/1086957585
Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Belmont, CA: Duxbury Press.
Mathematical Reviews (MathSciNet): MR2722836
Geiger, J. (1999). Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Probab. 36 301–309.
Mathematical Reviews (MathSciNet): MR1724856
Zentralblatt MATH: 0942.60071
Digital Object Identifier: doi:10.1239/jap/1032374454
Project Euclid: euclid.jap/1032374454
Jagers, P. (1989). General branching processes as Markov fields. Stochastic Process. Appl. 32 183–212.
Mathematical Reviews (MathSciNet): MR1014449
Zentralblatt MATH: 0678.92009
Digital Object Identifier: doi:10.1016/0304-4149(89)90075-6
Jagers, P. (1992). Stabilities and instabilities in population dynamics. J. Appl. Probab. 29 770–780.
Mathematical Reviews (MathSciNet): MR1188534
Zentralblatt MATH: 0769.60079
Digital Object Identifier: doi:10.2307/3214711
Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
Mathematical Reviews (MathSciNet): MR742953
Zentralblatt MATH: 0535.60075
Digital Object Identifier: doi:10.2307/1427068
Joffe, A. and Waugh, W.A.O’N. (1982). Exact distributions of kin numbers in a Galton–Watson process. J. Appl. Probab. 19 767–775.
Mathematical Reviews (MathSciNet): MR675140
Zentralblatt MATH: 0496.60089
Digital Object Identifier: doi:10.2307/3213829
Kurtz, T.G., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten–Stigum theorem for multitype branching processes. In Classical and Modern Branching Processes (K.B. Athreya and P. Jagers, eds.). IMA Vol. Math. Appl. 84 181–186. New York: Springer.
Mathematical Reviews (MathSciNet): MR1601753
Zentralblatt MATH: 0867.60067
Digital Object Identifier: doi:10.1007/978-1-4612-1862-3_18
Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 420–446.
Mathematical Reviews (MathSciNet): MR2299923
Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
Nerman, O. (1981). On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrsch. Verw. Gebiete 57 365–395.
Mathematical Reviews (MathSciNet): MR629532
Olofsson, P. (1996). General branching processes with immigration. J. Appl. Probab. 33 940–948.
Mathematical Reviews (MathSciNet): MR1416216
Zentralblatt MATH: 0870.60081
Digital Object Identifier: doi:10.2307/3214975
Olofsson, P. (1998). The xlogx condition for general branching processes. J. Appl. Probab. 35 537–544.
Mathematical Reviews (MathSciNet): MR1659492
Zentralblatt MATH: 0926.60063
Digital Object Identifier: doi:10.1239/jap/1032265202
Project Euclid: euclid.jap/1032265202
Olofsson, P. and Shaw, C.A. (2002). Exact sampling formulas for multi-type Galton–Watson processes. J. Math. Biol. 45 279–293.
Mathematical Reviews (MathSciNet): MR1934415
Zentralblatt MATH: 1013.92019
Digital Object Identifier: doi:10.1007/s002850200148
Waymire, E. and Williams, S.C. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348 585–632.
Mathematical Reviews (MathSciNet): MR1322959
Zentralblatt MATH: 0857.60028
Digital Object Identifier: doi:10.1090/S0002-9947-96-01500-0
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