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Journal articles on the topic 'Multitype Galton-Watson process'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Sagitov, Serik, and Maria Conceição Serra. "Multitype Bienaymé–Galton–Watson processes escaping extinction." Advances in Applied Probability 41, no. 01 (2009): 225–46. http://dx.doi.org/10.1017/s0001867800003207.

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In the framework of a multitype Bienaymé–Galton–Watson (BGW) process, the event that the daughter's type differs from the mother's type can be viewed as a mutation event. Assuming that mutations are rare, we study a situation where all types except one produce on average less than one offspring. We establish a neat asymptotic structure for the BGW process escaping extinction due to a sequence of mutations toward the supercritical type. Our asymptotic analysis is performed by letting mutation probabilities tend to 0. The limit process, conditional on escaping extinction, is another BGW process

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2

Sagitov, Serik, and Maria Conceição Serra. "Multitype Bienaymé–Galton–Watson processes escaping extinction." Advances in Applied Probability 41, no. 1 (2009): 225–46. http://dx.doi.org/10.1239/aap/1240319583.

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In the framework of a multitype Bienaymé–Galton–Watson (BGW) process, the event that the daughter's type differs from the mother's type can be viewed as a mutation event. Assuming that mutations are rare, we study a situation where all types except one produce on average less than one offspring. We establish a neat asymptotic structure for the BGW process escaping extinction due to a sequence of mutations toward the supercritical type. Our asymptotic analysis is performed by letting mutation probabilities tend to 0. The limit process, conditional on escaping extinction, is another BGW process

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3

Biggins, J. D., and A. E. Kyprianou. "Measure change in multitype branching." Advances in Applied Probability 36, no. 02 (2004): 544–81. http://dx.doi.org/10.1017/s0001867800013604.

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The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions

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4

Biggins, J. D., and A. E. Kyprianou. "Measure change in multitype branching." Advances in Applied Probability 36, no. 2 (2004): 544–81. http://dx.doi.org/10.1239/aap/1086957585.

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The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions

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5

Spåtaru, Aurel. "A maximum sequence in a critical multitype branching process." Journal of Applied Probability 28, no. 4 (1991): 893–97. http://dx.doi.org/10.2307/3214692.

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Let (Zn) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn) consider the right eigenvector u = (u1, · ··, up) > 0, and set . It is shown that lim inf, lim sup whenever Z0 = i, where .

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6

Spåtaru, Aurel. "A maximum sequence in a critical multitype branching process." Journal of Applied Probability 28, no. 04 (1991): 893–97. http://dx.doi.org/10.1017/s0021900200042807.

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Let (Zn ) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn ) consider the right eigenvector u = (u 1, · ··, up ) &gt; 0, and set . It is shown that lim inf, lim sup whenever Z 0 = i, where .

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7

Doku-Amponsah, Kwabena. "LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS." Far East Journal of Mathematical Sciences (FJMS) 102, no. 10 (2017): 2307–19. http://dx.doi.org/10.17654/ms102102307.

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8

Dyakonova, E. E. "On a multitype Galton-Watson process with state-dependent immigration." Journal of Mathematical Sciences 99, no. 3 (2000): 1244–49. http://dx.doi.org/10.1007/bf02674083.

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9

Cerf, Raphaël, and Joseba Dalmau. "Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector." ESAIM: Probability and Statistics 23 (2019): 797–802. http://dx.doi.org/10.1051/ps/2019007.

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Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

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10

Prehn, Uwe, and Ines Langer. "The Expected Source Time of a Subcritical Multitype Galton-Watson Process." Mathematische Nachrichten 138, no. 1 (1988): 83–92. http://dx.doi.org/10.1002/mana.19881380106.

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11

Pénisson, Sophie. "Beyond the Q-process: various ways of conditioning the multitype Galton-Watson process." Latin American Journal of Probability and Mathematical Statistics 13, no. 1 (2016): 223. http://dx.doi.org/10.30757/alea.v13-09.

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12

Joffe, A., and W. A. O'n Waugh. "The kin number problem in a multitype Galton–Watson population." Journal of Applied Probability 22, no. 1 (1985): 37–47. http://dx.doi.org/10.2307/3213746.

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The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), J

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13

Joffe, A., and W. A. O'n Waugh. "The kin number problem in a multitype Galton–Watson population." Journal of Applied Probability 22, no. 01 (1985): 37–47. http://dx.doi.org/10.1017/s0021900200028990.

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The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), J

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14

Klebaner, Fima C. "Linear growth in near-critical population-size-dependent multitype Galton–Watson processes." Journal of Applied Probability 26, no. 3 (1989): 431–45. http://dx.doi.org/10.2307/3214402.

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We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combina

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15

Klebaner, Fima C. "Linear growth in near-critical population-size-dependent multitype Galton–Watson processes." Journal of Applied Probability 26, no. 03 (1989): 431–45. http://dx.doi.org/10.1017/s0021900200038043.

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We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combina

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16

Khaįrullin, R. H. "On estimating parameters of a multitype Galton-Watson process by ϕ-branching processes". Siberian Mathematical Journal 33, № 4 (1992): 703–13. http://dx.doi.org/10.1007/bf00971136.

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17

Hong, Jyy-I. "Coalescence on critical and subcritical multitype branching processes." Journal of Applied Probability 53, no. 3 (2016): 802–17. http://dx.doi.org/10.1017/jpr.2016.41.

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AbstractConsider a d-type (d<∞) Galton–Watson branching process, conditioned on the event that there are at least k≥2 individuals in the nth generation, pick k individuals at random from the nth generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and t

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18

Gadag, V. G., and M. B. Rajarshi. "Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process." Journal of Applied Probability 26, no. 1 (1989): 1–8. http://dx.doi.org/10.2307/3214311.

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In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to th

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19

Gadag, V. G., and M. B. Rajarshi. "Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process." Journal of Applied Probability 26, no. 01 (1989): 1–8. http://dx.doi.org/10.1017/s0021900200041747.

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Abstract:

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to th

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20

Gadag, V. G., and M. B. Rajarshi. "On multitype processes based on progeny length of particles of a supercritical Galton-Watson process." Journal of Applied Probability 24, no. 1 (1987): 14–24. http://dx.doi.org/10.2307/3214055.

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Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

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21

Gadag, V. G., and M. B. Rajarshi. "On multitype processes based on progeny length of particles of a supercritical Galton-Watson process." Journal of Applied Probability 24, no. 01 (1987): 14–24. http://dx.doi.org/10.1017/s0021900200030576.

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Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

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22

Seneta, E., and R. L. Tweedie. "Moments for stationary and quasi-stationary distributions of markov chains." Journal of Applied Probability 22, no. 1 (1985): 148–55. http://dx.doi.org/10.2307/3213754.

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A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

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23

Seneta, E., and R. L. Tweedie. "Moments for stationary and quasi-stationary distributions of markov chains." Journal of Applied Probability 22, no. 01 (1985): 148–55. http://dx.doi.org/10.1017/s0021900200029077.

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A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

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24

Bhattacharya, Ayan, Krishanu Maulik, Zbigniew Palmowski, and Parthanil Roy. "Extremes of multitype branching random walks: heaviest tail wins." Advances in Applied Probability 51, no. 2 (2019): 514–40. http://dx.doi.org/10.1017/apr.2019.20.

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AbstractWe consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index $\alpha$ , while the other types of particles have lighter tails than the particles of type Q. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poiss

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25

Reder, Christine. "Transient behaviour of a Galton–Watson process with a large number of types." Journal of Applied Probability 40, no. 04 (2003): 1007–30. http://dx.doi.org/10.1017/s002190020002026x.

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Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different m

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26

Reder, Christine. "Transient behaviour of a Galton–Watson process with a large number of types." Journal of Applied Probability 40, no. 4 (2003): 1007–30. http://dx.doi.org/10.1239/jap/1067436097.

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Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different m

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27

Dalmau, Joseba. "Distribution of the quasispecies for a Galton–Watson process on the sharp peak landscape." Journal of Applied Probability 53, no. 2 (2016): 606–13. http://dx.doi.org/10.1017/jpr.2016.25.

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Abstract We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe-a - 1)(ak/k!)∑i≥ 1ik/σi, where σ is the selective advantage of the master sequence and a is the product of the length of the chains with the mutatio

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28

Carvalho, Maria Lucília. "A joint estimator for the eigenvalues of the reproduction mean matrix of a multitype Galton-Watson process." Linear Algebra and its Applications 264 (October 1997): 189–203. http://dx.doi.org/10.1016/s0024-3795(97)82946-1.

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29

Wang, Hua Ming. "On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice." Acta Mathematica Sinica, English Series 30, no. 12 (2014): 2161–72. http://dx.doi.org/10.1007/s10114-014-3650-1.

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30

DUNN, A. M., R. S. TERRY, and D. E. TANEYHILL. "Within-host transmission strategies of transovarial, feminizing parasites of Gammarus duebeni." Parasitology 117, no. 1 (1998): 21–30. http://dx.doi.org/10.1017/s0031182098002753.

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The amphipod Gammarus duebeni harbours several species of vertically transmitted, feminizing microsporidian parasites. G. duebeni were collected from 3 localities in the UK. Animals from Budle Bay, Northumberland, were infected with Octosporea effeminans, and those from Millport, Isle of Cumbrae and Fenham Flats, Northumberland were infected with microsporidia of the genus Nosema. We derived expected distributions of parasites per host embryonic cell by modelling parasite transmission as a multitype, Galton–Watson branching process. Parasite prevalence (proportion of females infected) was sign

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31

VATUTIN, V. A., and N. M. YANEV. "Multitype critical Galton-Watson branching process with final types." Discrete Mathematics and Applications 1, no. 3 (1991). http://dx.doi.org/10.1515/dma.1991.1.3.321.

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