The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in ran...The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in random environment Q（θ） with birth rate λ 〈 death rate μ, the following results are proved, （1） there is an unique q-process in random environment, P^-（θ＊（0）;t） = （p^-（θ^＊（0）;t,i,j）,i,j ≥ 0）, which is ergodic, that is, lim t→∞（θ^＊（0）;t,i,j） = π^-（θ^＊（0）;j） ≥0 does not depend on i ≥ 0 and ∑j≥0π （θ＊（0）;j） = 1, （2） there is a bi-immigration birth and death process in random enjvironment （X^＊ = {X^＊,t ≥ 0},ε^＊ = {εt,t ∈ （-∞, ∞）}） with random transition matrix P^-（θ^＊ （0）;t） such that X^＊ is a strictly stationary process.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
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
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
A general formulation of the stochastic model for random walk in time-random environment and an equivalent definition is established in this paper. Moreover, some basic probability relations similar to the classical c...A general formulation of the stochastic model for random walk in time-random environment and an equivalent definition is established in this paper. Moreover, some basic probability relations similar to the classical case which are very useful in the corresponding research of fractal properties are given. At the end, a typical example is provided to show the recurrence and transience. Key words random environment - random walk in timerandom environment - skew product Markov chain CLC number O 211.6 Foudation item: Supported by the National Natural Science Foundation of China (10371092) and Foundation of Wuhan University.Biography: Zhang Xiao-min (1977-), male, Ph. D candidate, research direction: stochastic processes and random fractal.展开更多
We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the envi...We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.展开更多
This paper introduces some concepts such as q- process in random environment, Laplace transformation, ergodic potential kernel, error function and some basic lemmas.We study the continuity and Laplace transformation o...This paper introduces some concepts such as q- process in random environment, Laplace transformation, ergodic potential kernel, error function and some basic lemmas.We study the continuity and Laplace transformation of random transition function. Finally, we give the sufficient condition for the existence of ergodic potential kernel for homogeneous q- processes in random environments.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
This paper is a continuation of [8]. In Section 1, three kinds of communication are introdnced for two states and the relations among them are investigated. In Section 2, two kinds of period of a state are introdnced ...This paper is a continuation of [8]. In Section 1, three kinds of communication are introdnced for two states and the relations among them are investigated. In Section 2, two kinds of period of a state are introdnced and it is obtained that the period is a 'class property',i.e. two states x and y belong to same class implies the period of x is equal to the period of y.展开更多
This paper is a continuation of [8] and [9]. The author obtains the decomposition of state space X of an Markov chain in random environment by making use of the results in [8] and [9], gives three examples, random wal...This paper is a continuation of [8] and [9]. The author obtains the decomposition of state space X of an Markov chain in random environment by making use of the results in [8] and [9], gives three examples, random walk in random environment, renewal process in random environment and queue process in random environment, and obtains the decompositions of the state spaces of these three special examples.展开更多
This is a study of one dimensional generalized birth-death chains in a random environment (GBDIRE). We give two sufficient conditions of recurrence for GBDIRE.
We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environmen...We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.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
We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in ...We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.展开更多
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
In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) su...In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.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 this article, the authors mainly discuss the law of large number under Kalikow＇s condition for multi-dimensional random walks in random environment with holding times. The authors give an expression to the escape ...In this article, the authors mainly discuss the law of large number under Kalikow＇s condition for multi-dimensional random walks in random environment with holding times. The authors give an expression to the escape speed of random walks in terms of the Lyapounov exponents, which have been precisely used in the context of large deviation.展开更多
基金Supported by the NNSF of China (10371092,10771185) the Foundation of Whuan University
文摘The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in random environment Q（θ） with birth rate λ 〈 death rate μ, the following results are proved, （1） there is an unique q-process in random environment, P^-（θ＊（0）;t） = （p^-（θ^＊（0）;t,i,j）,i,j ≥ 0）, which is ergodic, that is, lim t→∞（θ^＊（0）;t,i,j） = π^-（θ^＊（0）;j） ≥0 does not depend on i ≥ 0 and ∑j≥0π （θ＊（0）;j） = 1, （2） there is a bi-immigration birth and death process in random enjvironment （X^＊ = {X^＊,t ≥ 0},ε^＊ = {εt,t ∈ （-∞, ∞）}） with random transition matrix P^-（θ^＊ （0）;t） such that X^＊ is a strictly stationary process.
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.
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.
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A general formulation of the stochastic model for random walk in time-random environment and an equivalent definition is established in this paper. Moreover, some basic probability relations similar to the classical case which are very useful in the corresponding research of fractal properties are given. At the end, a typical example is provided to show the recurrence and transience. Key words random environment - random walk in timerandom environment - skew product Markov chain CLC number O 211.6 Foudation item: Supported by the National Natural Science Foundation of China (10371092) and Foundation of Wuhan University.Biography: Zhang Xiao-min (1977-), male, Ph. D candidate, research direction: stochastic processes and random fractal.
文摘We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.
基金Supported by the National Natural Science Foundation of China (10371092)
文摘This paper introduces some concepts such as q- process in random environment, Laplace transformation, ergodic potential kernel, error function and some basic lemmas.We study the continuity and Laplace transformation of random transition function. Finally, we give the sufficient condition for the existence of ergodic potential kernel for homogeneous q- processes in random environments.
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This paper is a continuation of [8]. In Section 1, three kinds of communication are introdnced for two states and the relations among them are investigated. In Section 2, two kinds of period of a state are introdnced and it is obtained that the period is a 'class property',i.e. two states x and y belong to same class implies the period of x is equal to the period of y.
基金Supported by the National Natural Science Foundation of China (10371092) and the Foundation of Wuhan University.
文摘This paper is a continuation of [8] and [9]. The author obtains the decomposition of state space X of an Markov chain in random environment by making use of the results in [8] and [9], gives three examples, random walk in random environment, renewal process in random environment and queue process in random environment, and obtains the decompositions of the state spaces of these three special examples.
文摘This is a study of one dimensional generalized birth-death chains in a random environment (GBDIRE). We give two sufficient conditions of recurrence for GBDIRE.
基金partially supported by the National Natural Science Foundation of China(NSFC,11101039,11171044,11271045)a cooperation program between NSFC and CNRS of France(11311130103)+1 种基金the Fundamental Research Funds for the Central UniversitiesHunan Provincial Natural Science Foundation of China(11JJ2001)
文摘We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.
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 Natural Science Foundation of China(11571052,11731012)the Hunan Provincial Natural Science Foundation of China(2018JJ2417)the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering(2018MMAEZD02)。
文摘We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.
基金Supported by the National Natural Science Foundation of China (10871200)
文摘In this article, we obtain the central limit theorem and the law of the iterated logarithm for Galton-Watson processes in i.i.d, random environments.
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.
文摘In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.
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Sponsored by the NSFC (10531070)Research Foundation for Outstanding Young Teachers of China University of Geoscience (Wuhan) (0816)
文摘In this article, the authors mainly discuss the law of large number under Kalikow＇s condition for multi-dimensional random walks in random environment with holding times. The authors give an expression to the escape speed of random walks in terms of the Lyapounov exponents, which have been precisely used in the context of large deviation.
