随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

The branching chain with drift in space-time random environment (I): Model,Markov property,moments

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment (BCDSTRE), i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see "Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145-168 (1975)").     

A class of two-sex branching processes with reproduction phase in a random environment

We model the demographic dynamics of populations with sexual reproduction where the reproduction phase occurs in a non-predictable environment and we assume the immigration/out-migration of mating units in the population. We introduce a general class of two-sex branching processes where, in each generation, the number of mating units which take part in the reproduction phase is randomly determined and the offspring probability distribution changes over time in a random environment. We provide several probabilistic results about the limit behaviour of populations whose dynamics is modelled by such a class of stochastic processes. In particular, we provide sufficient conditions for the almost sure extinction of the population or for its survival with a positive probability. As illustration, we include some simulated examples.     

Asymptotic properties of supercritical branching processes in random environments

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）.     

Harmonic moments of branching processes in random environments

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.     

Weighted moments of the limit of a branching process in a random environment

Let (Z _n ) be a supercritical branching process in an independent and identically distributed random environment ζ = (ζ ₀, ζ ₁,…), and let W be the limit of the normalized population size Z _n / $\mathbb{E}$ (Z _n |ζ). We show a necessary and sufficient condition for the existence of weighted moments of W of the form $\mathbb{E}$ , where α ≥ 1 and ? is a positive function slowly varying at ∞.     

Second order limit theorems for the Markov branching process in random environments

In a Markov branching process with random environments, limiting fluctuations of the population size arise from the changing environment, which causes random variation of the ‘deterministic’ population prediction, and from the stochastic wobble around this ‘deterministic’ mean, which is apparent in the ordinary Markov branching process. If the random environment is generated by a suitable stationary process, the first variation typically swamps the second kind. In this paper, environmental processes are considered which, in contrast, lead to sampling and environmental fluctuation of comparable magnitude. The method makes little use either of stationarity or of the branching property, and is amenable to some generalization away from the Markov branching process.     

Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

Let \{Z_n,\mathrm{}n\ge \mathrm{0}\}be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(Z_{n+n_{\mathrm{0}}}/Z_{n_{\mathrm{0}}}) uniformly in n_{\mathrm{0}}\in \mathbb{Z},which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n_{\mathrm{0}}=\mathrm{}\mathrm{0}. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(Z_{n+n_{\mathrm{0}}}/Z_{n_{\mathrm{0}}}) and n.     

Extremum of a time-inhomogeneous branching random walk

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean \mathrm{1}+{B(\mathrm{1}+n)}^{-\beta } for \beta \in (\mathrm{0},\mathrm{1}) and ‘displacement’ {{\displaystyle \xi }}_n with a drift {A(\mathrm{1}+n)}^{-\mathrm{2}\alpha } for \alpha \in (\mathrm{0},\mathrm{1}/\mathrm{2}), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ {{\displaystyle \xi }}_n is strictly positive or negative for \left|A\right|\rangle \mathrm{0} but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter \beta and \alpha.     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in 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 environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

Central limit theorems for a supercritical branching process in a random environment

For a supercritical branching process (Z_n) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population W_n=Z_n/E[Z_n|ξ] to its limit W_∞: we show a central limit theorem for W_∞−W_n with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for W_n+k−W_n for each fixed k∈N^∗.     

Limit theorems for a supercritical branching process with immigration in a random environment

Let (Z_n) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size W_n converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Z_n.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

On sums of independent random variables without power moments

In 1952 Darling proved the limit theorem for the sums of independent identically distributed random variables without power moments under the functional normalization. This paper contains an alternative proof of Darling’s theorem, using the Laplace transform. Moreover, the asymptotic behavior of probabilities of large deviations is studied in the pattern under consideration.     

Phase transition of the minimum degree random multigraph process

We study the phase transition of the minimum degree multigraph process. We prove that for a constant h_g ≈︁ 0.8607, with probability tending to 1 as n → ∞, the graph consists of small components on O(log n) vertices when the number of edges of a graph generated so far is smaller than h_gn, the largest component has order roughly n^2/3 when the number of edges added is exactly h_gn, and the graph consists of one giant component on Θ(n) vertices and small components on O(log n) vertices when the number of edges added is larger than h_gn. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1∕2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.     

Fixed points of smoothing transformation in random environment

At each time n\in \mathbb{N},\mathrm{l}\mathrm{e}\mathrm{t}{\overline{Y}}^{(n)}(\xi )=(y_1^{(n)}(\xi ),y_2^{(n)}(\xi ),\cdots ) be a random sequence of non-negative numbers that are ultimately zero in a random environment\xi \text{=}（{\xi }_n{）}_{n\in \mathbb{N}}. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.\xi ,Z(\xi )\stackrel{\text{d}}{=}{\sum }_{i\in \mathbb{N}+}{y}_i^{(\mathrm{0})}(\xi )Z_i^{(\mathrm{1})}(\xi ),where ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝ are random variables in random environment which satisfy that for any environment\xi; under P_{\xi }; ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝are independent of each other and {\bar{Y}}^{(\mathrm{0})}(\xi ), and have the same conditional distribution P_{\xi }(Z_i^{(1)}(\xi )\in \cdot )=P_{T\xi }(Z(T\xi )\in \cdot ) where T is the shift operator. This extends the classical results of J. D. Biggins [J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.