A general framework of stochastic model for a Markov chain in a space-time random environment is introduced, here the environment ξ^*:={ξ1,x∈N,x∈ X}is a random field. We study the dependence relations between th...A general framework of stochastic model for a Markov chain in a space-time random environment is introduced, here the environment ξ^*:={ξ1,x∈N,x∈ X}is a random field. We study the dependence relations between the environment and the original chain, especially the "feedback". Some equivalence theorems and law of large numbers are obtained.展开更多
We consider a population-size-dependent branching chain in a general random environment.We give suffcident conditions for certain extinction and for non-certain extinction.The chain exhibits different asymptotic accor...We consider a population-size-dependent branching chain in a general random environment.We give suffcident conditions for certain extinction and for non-certain extinction.The chain exhibits different asymptotic according to supk,θmk,θ1, mk,θn→1 as k →∞, n→∞, infk,θmk,θ1.展开更多
Suppose {Xn} is a random walk in time-random environment with state space Z^d, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the r...Suppose {Xn} is a random walk in time-random environment with state space Z^d, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.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
The concepts of Markov process in random environment, q-matrix in random environment, and q-process in random environment are introduced. The minimal q-process in random environment is constructed and the necessary an...The concepts of Markov process in random environment, q-matrix in random environment, and q-process in random environment are introduced. The minimal q-process in random environment is constructed and the necessary and sufficient conditions for the uniqueness of q-process in random environment are given.展开更多
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
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
This article is a continuation of[9].Based on the discussion of random Kolmogorov forward(backward)equations,for any given q-matrix in random environment, Q(θ)=(q(θ;x,y),x,y∈X),an infinite class of q-proces...This article is a continuation of[9].Based on the discussion of random Kolmogorov forward(backward)equations,for any given q-matrix in random environment, Q(θ)=(q(θ;x,y),x,y∈X),an infinite class of q-processes in random environments satisfying the random Kolmogorov forward(backward)equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward(backward)equation are constructed.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
First of all, we introduces the concept of m-irreducible of Markov chain in random environment. Then under the condition of m-irreducible, the relationship of recurrent and positive recurrent between two states is stu...First of all, we introduces the concept of m-irreducible of Markov chain in random environment. Then under the condition of m-irreducible, the relationship of recurrent and positive recurrent between two states is studied. We also give several conditions that can imply a state is recurrent and positive recurrent. And then the period of a state is discussed and we obtained that under the condition of m-irreducible, for any two states in x, they have the same period.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
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
We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form dev...We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form deviations of the 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 branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the converge...We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the convergence of the free energy n-llog Zn(t), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn(t)/E[Zn(t)ξ].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
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 consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environm...We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.展开更多
The concepts of branching chain in random environmnet and canonical branching chain in random environment are introduced. Moreover the existence of these chains is proved. Finally the exact formulas of mathematical ex...The concepts of branching chain in random environmnet and canonical branching chain in random environment are introduced. Moreover the existence of these chains is proved. Finally the exact formulas of mathematical expectation and variance of branching chain in random environment are also given.展开更多
A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved.In order to prove this theorem,the entropy function h is i...A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved.In order to prove this theorem,the entropy function h is introduced and the Shannon-McMillan-Breiman theorem for the Markov chain in a Markovian environment is shown. It's well-known that a Markov process in a Markovian environment is generally not a standard Markov chain,so an example of Poisson approximation for a process which is not a Markov process is given.On the other hand,when the environmental process degenerates to a constant sequence,a Poisson limit theorem for countable Markov chains,which is the generalization of Pitskel's result for finite Markov chains is obtained.展开更多
基金Supported by the National Natural Science Foundation of China (10371092)
文摘A general framework of stochastic model for a Markov chain in a space-time random environment is introduced, here the environment ξ^*:={ξ1,x∈N,x∈ X}is a random field. We study the dependence relations between the environment and the original chain, especially the "feedback". Some equivalence theorems and law of large numbers are obtained.
基金supported by the National Natural Science Foundation of China (10771185,10926036)Zhejiang Provinicial Natural Science Foundation of China (Y6090172)
文摘We consider a population-size-dependent branching chain in a general random environment.We give suffcident conditions for certain extinction and for non-certain extinction.The chain exhibits different asymptotic according to supk,θmk,θ1, mk,θn→1 as k →∞, n→∞, infk,θmk,θ1.
基金Project supported by NNSF of China (10371092)Foundation of Wuhan University
文摘Suppose {Xn} is a random walk in time-random environment with state space Z^d, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.
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...
文摘The concepts of Markov process in random environment, q-matrix in random environment, and q-process in random environment are introduced. The minimal q-process in random environment is constructed and the necessary and sufficient conditions for the uniqueness of q-process in random environment are given.
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.
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the NNSF of China(10371092,10771185,10471148)the Foundation of Wuhan University
文摘This article is a continuation of[9].Based on the discussion of random Kolmogorov forward(backward)equations,for any given q-matrix in random environment, Q(θ)=(q(θ;x,y),x,y∈X),an infinite class of q-processes in random environments satisfying the random Kolmogorov forward(backward)equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward(backward)equation are constructed.
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 (10371092)the Foundation of Wuhan University
文摘First of all, we introduces the concept of m-irreducible of Markov chain in random environment. Then under the condition of m-irreducible, the relationship of recurrent and positive recurrent between two states is studied. We also give several conditions that can imply a state is recurrent and positive recurrent. And then the period of a state is discussed and we obtained that under the condition of m-irreducible, for any two states in x, they have the same period.
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.
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Supported by National Natural Science Foundation of China (Grant No. 11026088), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y6100663), ZJPEDF (Grant No. Y200906909)
文摘We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form deviations of the 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 The authors would like to thank the anonymous referees for valuable comments and remarks. This work was partially supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT. NSRIF. 2015102), the National Natural Science Foundation of China (Grant Nos. 11171044, 11101039), and by the Natural Science Foundation of Hunan Province (Grant No. 11JJ2001).
文摘We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the convergence of the free energy n-llog Zn(t), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn(t)/E[Zn(t)ξ].
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).
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The author would like to thank the referees for comments on conditions (C1) and (C2). This work was supported in part by the National Natural Science Foundation of China (Grant No. 11171262) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130141110076).
文摘We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.
文摘The concepts of branching chain in random environmnet and canonical branching chain in random environment are introduced. Moreover the existence of these chains is proved. Finally the exact formulas of mathematical expectation and variance of branching chain in random environment are also given.
文摘A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved.In order to prove this theorem,the entropy function h is introduced and the Shannon-McMillan-Breiman theorem for the Markov chain in a Markovian environment is shown. It's well-known that a Markov process in a Markovian environment is generally not a standard Markov chain,so an example of Poisson approximation for a process which is not a Markov process is given.On the other hand,when the environmental process degenerates to a constant sequence,a Poisson limit theorem for countable Markov chains,which is the generalization of Pitskel's result for finite Markov chains is obtained.