The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introd...The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introduces the concepts of the multitype canonical Markov branching chain in random environment （CMBCRE） and multitype Markov branching chain in random environment （MBCRE） and proved that CMBCRE must be MBCRE, and any MBCRE must be equivalent to another CMBCRE in distribution. The main results of this article are the construction of CMBCRE and some of its probability properties.展开更多
The concepts of random Markov matrix, Markov branching chain in randomenvironment (MBCRE) and Laplace functional of Markov branching chain in random environment (LFMBCRE)are introduced. The properties of LFMBCRE and t...The concepts of random Markov matrix, Markov branching chain in randomenvironment (MBCRE) and Laplace functional of Markov branching chain in random environment (LFMBCRE)are introduced. The properties of LFMBCRE and the explicit formulas of momentsof MBCRE are given.展开更多
In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTR...In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...展开更多
First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional ...First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification ofβ-MBCRE according to the different standards.展开更多
In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions assoc...In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.展开更多
A branching model {Z n} n≥0is considered where the offspring distribution of the population’s evolution is not only dependent on the population size, but also controlled by a Markovian environmental process {ξ n} n...A branching model {Z n} n≥0is considered where the offspring distribution of the population’s evolution is not only dependent on the population size, but also controlled by a Markovian environmental process {ξ n} n≥0. For this model, asymptotic behaviour is studied such as limn→∞Z n and limn→∞Z n/m n in the case that the mean m k, θof the offspring distribution converges to m>1 as the population size k grows to ∞. In the case that {ξ n} n≥0is an irreducible positive recurrent Markov chain, certain extinction (i.e. P(Z n=0 for some n)=1) and noncertain extinction (i.e. P(Z n=0 for some n)<1) are studied.展开更多
Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming s...Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming sequence.We prove that under a moment condition of order p∈(1,2),W-W_(n)=o(e^(-na))a.s.for some a>0 that we find explicitly;assuming the logarithmic moment condition holds,we haveW-W_(n)=o(n^(-α))a.s..In order to obtain these results,we provide the L^(p)-convergence of(W_(n));similar conclusions hold for a bisexual branching process in a varying environment.展开更多
基金Project supported by the National Natural Science Foundation of China and the Foundation of Wuhan University
文摘The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introduces the concepts of the multitype canonical Markov branching chain in random environment （CMBCRE） and multitype Markov branching chain in random environment （MBCRE） and proved that CMBCRE must be MBCRE, and any MBCRE must be equivalent to another CMBCRE in distribution. The main results of this article are the construction of CMBCRE and some of its probability properties.
文摘The concepts of random Markov matrix, Markov branching chain in randomenvironment (MBCRE) and Laplace functional of Markov branching chain in random environment (LFMBCRE)are introduced. The properties of LFMBCRE and the explicit formulas of momentsof MBCRE are given.
基金Supported by the National Natural Science Foundation of China (10771185 and 10871200)
文摘In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...
基金supported by the National Natural Science Foundation of China(Grant No:10371092)the Foundation of Wuhan University.
文摘First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification ofβ-MBCRE according to the different standards.
文摘In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.
文摘A branching model {Z n} n≥0is considered where the offspring distribution of the population’s evolution is not only dependent on the population size, but also controlled by a Markovian environmental process {ξ n} n≥0. For this model, asymptotic behaviour is studied such as limn→∞Z n and limn→∞Z n/m n in the case that the mean m k, θof the offspring distribution converges to m>1 as the population size k grows to ∞. In the case that {ξ n} n≥0is an irreducible positive recurrent Markov chain, certain extinction (i.e. P(Z n=0 for some n)=1) and noncertain extinction (i.e. P(Z n=0 for some n)<1) are studied.
基金supported by the Fundamental Research Funds for the Central University (Grant No.19JNLH09)Innovation Team Project in Guangdong Province,P.R.China (Grant No.2016WCXTD004)+1 种基金supported by the National Natural Science Foundation of China (Grants no.11731012,12271062)Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science&Technology)。
文摘Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming sequence.We prove that under a moment condition of order p∈(1,2),W-W_(n)=o(e^(-na))a.s.for some a>0 that we find explicitly;assuming the logarithmic moment condition holds,we haveW-W_(n)=o(n^(-α))a.s..In order to obtain these results,we provide the L^(p)-convergence of(W_(n));similar conclusions hold for a bisexual branching process in a varying environment.
基金Supported by the Natural Science Foundation of Anhui Province(kj2013Z331)the Humanities and Social Sciences Foundation for the Youth Scholars of Ministry of Education of China(12YJCZH217)the Natural Science Foundation of Anhui Province(1308085MA03)
