Almost sure central limit theorems for random fields

A general almost sure limit theorem is presented for random fields. It is applied to obtain almost sure versions of some (functional) central limit theorems. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Central limit theorems for supercritical branching Markov processes

In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching mechanisms. Compared with the results of [1], our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non-degenerate.     

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.     

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.     

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.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

Limit theorems for random transformations and processes in random environments

I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

     

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.     

Classical limit theorems for measure-valued Markov processes

Laws of large numbers, central limit theorems, and laws of the iterated logarithm are obtained for discrete and continuous time Markov processes whose state space is a set of measures. These results apply to each measure-valued stochastic process itself and not simply to its real-valued functionals.     

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.     

Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes

We obtain sufficient criteria for central limit theorems (CLTs) for ergodic continuous-time Markov chains (CTMCs). We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.     

Infinitely dimensional control Markov branching chains in random environments

First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification of β-MBCRE according to the different standards.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

相伴随机变量序列的泛函型几乎处处中心极限定理

对于均值为零的平稳相伴随机变量序列,首先证明了在L(n)=EX_1~2 2 sum from n to j=2 Cov(X_1,X_j)是一个缓变函数的条件下的泛函型几乎处处中心极限定理.另外还给出了正则化部分和函数的对数平均几乎处处收敛性.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

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）.     

Small time central limit theorems for semimartingales with applications

We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews.     

A local limit theorem for independent random variables

A local limit theorem is given for independent noninteger random variables under a condition which is more general than one previously given, and which reduces, in the case of identically distributed random variables, to a well-known result.     

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in d‐dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 262–286, 2017