In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for th...In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.展开更多
This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properti...This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.展开更多
It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued br...It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.展开更多
Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual ...Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.展开更多
Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) ...Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.展开更多
A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equati...A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.展开更多
The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the sp...The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the spectral radius of Jacobi matrix of its generating function.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
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
In this paper, we study the large deviation for a supercritical branching process with immigration controlled by a sequence of non-negative integer-valued independently identical distributed random variables, improvin...In this paper, we study the large deviation for a supercritical branching process with immigration controlled by a sequence of non-negative integer-valued independently identical distributed random variables, improving the previous results for non immigration processes. We rely heavily on the detail description and limit property of the generating function of immigration processes.展开更多
We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every var...We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.展开更多
For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive ...For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P(Zn=kn) with kn=o(mn) as n→∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.展开更多
Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(C...Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.展开更多
In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an ...In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process.展开更多
The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asympto...The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asymptotic estimate of the survival probability at generation n.展开更多
We consider a supercritical branching process (Zn) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the nat...We consider a supercritical branching process (Zn) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn|ξ], the convergence rates of W - Wn (by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Zn).展开更多
The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodi...The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.展开更多
A two-dimensional stochastic integral equation system with jumps is studied. We first prove its unique weak solution is a two-type continuous-state branching process with immigration. Then the comparison property of t...A two-dimensional stochastic integral equation system with jumps is studied. We first prove its unique weak solution is a two-type continuous-state branching process with immigration. Then the comparison property of the solution is established. These results imply the existence and uniqueness of the strong solution of the stochastic equation system.展开更多
A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching ...A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.展开更多
We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positi...We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.展开更多
文摘In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.
基金supported by NNSF of China(6053408070571079)Open Fundation of SKLSE of Wuhan University (2008-07-03)
文摘This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.
文摘It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.
文摘Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.
基金partially supported by the National Nature Science Foundation of China(11601286,11501146)。
文摘Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.
基金supported by the National Key R&D Program of China(2020YFA0712900)the National Natural Science Foundation of China(11531001).
文摘A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.
文摘The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the spectral radius of Jacobi matrix of its generating function.
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.
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In this paper, we study the large deviation for a supercritical branching process with immigration controlled by a sequence of non-negative integer-valued independently identical distributed random variables, improving the previous results for non immigration processes. We rely heavily on the detail description and limit property of the generating function of immigration processes.
基金Supported by National Natural Science Foundtation of China (Grant No. 10771185)
文摘We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11871103)the National Key Research and Development Program of China(No.2020YFA0712900)Research Foundation for Youth Scholars of Beijing Technology and Business University(Grant No.PXM2019_014213_000007).
文摘For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P(Zn=kn) with kn=o(mn) as n→∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.
基金supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciencessupported by the Royal Society Newton International Fellowship and the EU-funded Hungarian(Grant No.EFOP-3.6.1-16-2016-00008)。
文摘Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.
基金Supported by MPG and CAS joint doctoral promotion program and the Klaus Tschira Stiftung through the International Max-Planck Research School
文摘In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process.
基金Supported by the National Natural Science Foundation of China(No.11301133 and 11471218)the Natural Science Foundation of Hebei province(No.A2014202236 and A2014202052)
文摘The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asymptotic estimate of the survival probability at generation n.
基金Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11171044, 11101039) and the Natural Science Foundation of Hunan Province (Grant No. 11JJ2001).
文摘We consider a supercritical branching process (Zn) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn|ξ], the convergence rates of W - Wn (by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Zn).
基金Supported by NSFC(Grant Nos.1131003,11626245 and 11571043)
文摘The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.
文摘A two-dimensional stochastic integral equation system with jumps is studied. We first prove its unique weak solution is a two-type continuous-state branching process with immigration. Then the comparison property of the solution is established. These results imply the existence and uniqueness of the strong solution of the stochastic equation system.
文摘A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.
基金Acknowledgements The authors would like to give their sincere thanks to Professor Zenghu Li for encouragement and helpful discussion. They also would like to acknowledge the Laboratory of Mathematics and Complex Systems (Ministry of Education, China) for providing them the research facilities. This work was supported in part by the National Natural Science Foundation of China (Grants Nos. 11201030, 11071021, 11126037), the Specialized Research Fund for the Doctoral Program of Higher Education (20110003120003), and Ministry of Education (985 Project).
文摘We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.