MOMENTS AND LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION IN RANDOM ENVIRONMENTS

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.

CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

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.

A note on asymptotic behavior of Galton-Watson branching processes in random environments

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.

ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

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.

THE ASYMPTOTIC PROPERTIES OF SUPERCRITICAL BISEXUAL GALTON-WATSON BRANCHING PROCESSES WITH IMMIGRATION OF MATING UNITS

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.

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

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.

MOMENTS OF CONTINUOUS-STATE BRANCHING PROCESSES IN LéVY RANDOM ENVIRONMENTS

For continuous-state branching processes in Lévy random environments, the recursion of n-moments and the equivalent condition for the existence of general f-moments are established, where f is a positive continuous function satisfying some standard conditions.

SOME PROPERTIES OF GALTON-WATSON BRANCHING PROCESSES IN VARYING ENVIRONMENTS

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.

A LIMIT THEOREM FOR INTERACTING MEASURE-VALUED BRANCHING PROCESSES

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.

SPECTRAL RADIUSES OF THE GALTON-WATSON BRANCHING PROCESSES

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.

Decomposition of Supercritical Linear-Fractional Branching Processes

It is well known that a supercritical single-type Bienayme-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienayme-Galton-Watson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.

Branching Process based Cascading Failure Probability Analysis for a Regional Power Grid in China with Utility Outage Data

Studying the propagation of cascading failures through the transmission network is key to asses and mitigate the risk faced the energy system. As complex systems the power grid failure is often studied using some probability distributions. We apply 4 well-known probabilistic models, Poisson model, Power Law model, Generalized Poisson Branching process model and Borel-Tanner Branching process model, to a 14-year utility historical outage data from a regional power grid in China, computing probabilities of cascading line outages. For this data, the empirical distribution of the total number of line outages is well approximated by the initial line outages propagating according to a Borel-Tanner branching process. Also for this data, Power law model overestimates, while Generalized Possion branching process and Possion model underestimate, the probability of larger outages. Especially, the probability distribution generated by the Poisson model deviates heavily from the observed data, underestimating the probability of large events (total no. of outages over 5) by roughly a factor of 10-2 to 10-5. The observation is confirmed by a statistical test of model fitness. The results of this work indicate that further testing of Borel-Tanner branching process models of cascading failure is appropriate, and should be further discussed as it outperforms other more traditional models.

顺北油田分支断裂及断控储层的识别与描述研究

顺北油气田是断裂控制的断控油气田,其中不少断层发育有分支断裂。分支断裂在常规地震资料上难以发现,属于断距小或者说是比较隐蔽的断裂。因这些断裂带形态与分布不清,断控储集体特征也不清楚,从而使得储层预测与目标优选困难。如何对这类断裂开展识别和描述,进而开展储层预测和目标评价,本次研究尝试从多种方法的组合使用入手,通过反复试验发现,把时频谱蓝化提高地震分辨率处理与方向性边界保持断层增强处理组合使用,能有效提高地震资料分支断裂的识别能力;把梯度结构张量断裂检测处理与最大似然属性断裂检测处理组合使用能很好地对分支断裂进行描述与刻画;把结构张量属性分析与缝洞增强处理组合使用能很好地对洞穴型储层进行有效刻画;GST(Gradient Structural Tensor)相干属性与不连续性属性联合使用实现了对孔洞型储层的预测;体曲率+最大似然属性组合使用实现了对裂缝型储层预测与描述;在不同类型储层预测基础上,通过不同类型储层的叠合得到的总储层预测。

Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the assertion of Rhyzhov and Skorokhod(Theory Probab.Appl.,1970)on the uniqueness of the solutions to the equation,which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.

Boundary Behaviors for a Continuous-state Nonlinear Neveu’s Branching Process

By generalizing a criterion of Mufa Chen for Markov jump processes,we establish the necessary and sufficient conditions for the extinction,explosion and coming down from infinity of a continuous-state nonlinear Neveu’s branching process.

Precise Asymptotics in Limit Theorems for a Supercritical Branching Process with Immigration in a Random Environment

Let(Zn)be a supercritical branching process with immigration in an independent and identically distributed random environment.Under necessary moment conditions,we show the exact convergence rate in the central limit theorem on log Zn and establish the corresponding local limit theorem by using the moments of the natural submartingale and the convergence rates of its logarithm.By similar approach and with the help of a change of measure,we also present the so-called integrolocal theorem and integral large deviation theorem to characterize the precise asymptotics of the upper large deviations.

我国南方西南和中东部区域两次持续性低温雨雪过程与关键环流系统的关系研究

利用NCEP再分析资料和国家气象信息中心提供的753站逐日气温和降水资料,对比分析了我国南方西南和中东部区域两次持续10 d以上的低温雨雪过程,结果表明:(1)两次过程中欧亚大陆中高纬东亚大槽均加深,但环流形势有差异。西南过程呈现"北高南低"形势,关键脊区在贝加尔湖,而中东部区域过程"北高南低"和"西高东低"形势共存,关键脊区从乌拉尔山延伸至贝加尔湖。两次过程异常的环流与北大西洋向东传播的波列有关。(2)西南过程关键脊区提前过程3 d发展并东移至贝加尔湖,形成稳定形势;而中东部区域过程关键脊区提前过程一周发展,在开始日达最强。两次过程均伴随蒙古高压东移南压使地面降温,500 hPa关键脊区超前蒙古高压2 d变化。西南过程降温主要受到冷平流和绝热冷却影响,而中东部区域过程主要受到冷平流的影响。(3)西南过程水汽来自孟加拉湾,只受南支槽支配。中东部区域过程水汽来自孟加拉湾、南海和西太平洋,由南支槽和西太平副热带高压的共同影响。两次过程水汽正收支主要来自南边界。

Stochastic Model of Dengue: Analysing the Probability of Extinction and LLN

In this article, we develop and analyze a continuous-time Markov chain (CTMC) model to study the resurgence of dengue. We also explore the large population asymptotic behavior of probabilistic model of dengue using the law of large numbers (LLN). Initially, we calculate and estimate the probabilities of dengue extinction and major outbreak occurrence using multi-type Galton-Watson branching processes. Subsequently, we apply the LLN to examine the convergence of the stochastic model towards the deterministic model. Finally, theoretical numerical simulations are conducted exploration to validate our findings. Under identical conditions, our numerical results demonstrate that dengue could vanish in the stochastic model while persisting in the deterministic model. The highlighting of the law of large numbers through numerical simulations indicates from what population size a deterministic model should be considered preferable.

插穗预处理和激素处理对金叶银杏硬枝扦插生根的影响

以观赏良种金叶银杏‘万年金’为材料,研究了插穗预处理(冷藏、蔗糖、硼酸)和激素(NAA、IBA、NAA+IBA)对金叶银杏硬枝扦插生根的影响,分析不同处理下的不定根长、不定根数、生根率的差异,筛选出较优的预处理和激素处理,为金叶银杏硬枝扦插育苗技术体系建立提供数据支撑。结果表明:(1)基于隶属函数综合评价,排名前三的预处理依次是冷藏10 d、5.0%蔗糖、0.1%硼酸,其中冷藏10 d处理同步提高了金叶银杏插穗不定根长、不定根数和生根率,5.0%蔗糖处理仅提高了不定根长,0.1%硼酸处理仅提高了不定根数。(2)基于隶属函数综合评价,排名前四的激素处理依次是IBA800、NAA400+IBA400、IBA500、NAA200+IBA200,四个处理均同步显著提高了不定根长、不定根数和生根率;IBA对金叶银杏扦插生根有明显的促进作用,优于NAA和NAA+IBA。(3)基于隶属函数综合评价,IBA500处理1 h对金叶银杏硬枝扦插生根效果较优。

随机环境中带移民分枝过程的Hoeffding型不等式

考虑到自然界人口迁徙和疾病传播等现象,引入移民因素(Yn)的影响,令 {Zn,n≥0}为独立同分布环境ξ=(ξn)n≥0的一个上临界带移民分枝过程,对统计量log Zn0+n/Zn0进行研究,利用logZ_(n)的分解以及Hoeffding不等式,建立随机环境中带移民分枝过程的一个偏差不等式。