CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

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.

SOME CENTRAL LIMIT THEOREMS FOR SUPER BROWNIAN MOTION

The author proves a central limit theorem for the critical super Brownian motion, which leads to a Gaussian random field. In the transient case the limiting field is the same aa that obtained by Dawson (1977). In the recurrent case it is a spatially uniform field. The author also give a central limit theorem for the weighted occupation time of the super Brownian motion with underlying dimension number d less than or equal to 3, completing the results of Iscoe (1986).

LIMIT THEOREMS FOR A GALTON-WATSON PROCESS IN THE I.I.D. RANDOM ENVIRONMENT

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.

Some generalized limit theorems concerning delayed sums of random sequence

For a sequence of arbitrarily dependent m-valued random variables （Xn） n∈N , the generalized strong limit theorem of the delayed average is investigated. In our proof, we improved the method proposed by Liu [6] . As an application, we also studied some limit properties of delayed average for inhomogeneous Markov chains.

LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES

We use tridiagonal models to study the limiting behavior of β-Laguerre and β-Jacobi ensembles,focusing on the limiting behavior of the extremal eigenvalues and the central limit theorem for the two ensembles.For the central limit theorem of β-Laguerre ensembles,we follow the idea in[1]while giving a modified version for the generalized case.Then we use the total variation distance between the two sorts of ensembles to obtain the limiting behavior of β-Jacobi ensembles.

Some Strong Limit Theorems for Even-odd Markov Chain Fields Indexed by Trees

First,a class of strong limit theorems are proved by constructing two nonnegative martingales.Then they are applied to the study of all kinds of even-odd Markov chain fields and Markov chain fields defined in the paper.Finally,some strong limit theorems for the even-odd Markov chain fields and Markov chain fields are obtained.

Limit Theorems for Critical Galton-Watson Processes with Immigration Stopped at Zero

In this paper,we consider a critical Galton-Watson branching process with immigration stopped at zero W.Some precise estimation on the probability generating function of the n-th population are obtained,and the tail probability of the life period of W is studied.Based on above results,two conditional limit theorems for W are established.

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.

Functional Limit Theorems for C-R Increments of k-Dimensional Brownian Motion in Holder Norm

In this paper. based on large deviation formulas established in stronger topology generated by Hlder norm, we obtain the functional limit theorems for C-R increments of k-dimensional Brownian motion in Hlder norm

Limit Theorems of a Class of Super-Uniformly Elliptic Diffusions

In this paper, we investigate super-uniformly elliptic diffusions {Xt,t ≥ 0} with its branching mechanism given by ψ(z) = γz1+β(0 < ≤ 1), and, when the initial value X0(dx) is one kind of invariant measures of the underlying processes, we show that if dimension d satisfies βd ≤ 2, then the random measures Xt will converge to the null in distribution and if βd > 2, then Xt will converge to a nondegenerative random measure in the same sense.

Limit Theorems for Inverse Process Tn of Hawkes Process

We consider linear Hawkes process Nt and its inverse process Tn. The limit theorems for Nt are well known and studied by many authors. In this paper, we study the limit theorems for Tn. In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for Tn. The main tool of the proof is based on immigration-birth representation and the observations on the relation between N＋ and Tn

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.

Multi-dimensional Central Limit Theorems and Laws of Large Numbers under Sublinear Expectations

In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.

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.

Lindeberg's central limit theorems for martingale-like sequences under sub-linear expectations

The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes,especially stochastic integrals and differential equations.In this paper,the central limit theorem and the functional central limit theorem are obtained for martingale-like random variables under the sub-linear expectation.As applications,the Lindeberg's central limit theorem is obtained for independent but not necessarily identically distributed random variables,and a new proof of the Lévy characterization of a GBrownian motion without using stochastic calculus is given.For proving the results,Rosenthal's inequality and the exponential inequality for the martingale-like random variables are established.

Functional Limit Theorems for d-dimensional FBM in H lder Norm

In this paper, we obtain functional limit theorems for d-dimensional FBM in HSlder norm via estimating large deviation probabilities for d-dimensional FBM in HSlder norm.

On Almost Sure Max-limit Theorems of Complete and Incomplete Samples from Stationary Sequences

Let Mn denote the partial maximum of a strictly stationary sequence （Xn）. Suppose some of the random variables of （Xn） can be observed and let Mn stand for the maximum of observed random variables from the set {X1,..., Xn}. In this paper, the almost sure limit theorems related to random vector （Mn, Mn） are considered in terms of i.i.d, case. The related results are also extended to weakly dependent stationary Gaussian sequence as its covariance function satisfies some regular conditions.

The Limit Theorems for Maxima of Stationary Gaussian Processes with Random Index

Let {X（t）, t ≥ 0} be a standard（zero-mean, unit-variance） stationary Gaussian process with correlation function r（·） and continuous sample paths. In this paper, we consider the maxima M（T） = max{X（t）, t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M（TT） exists under some additional conditions related to the correlation function r（·）.

Almost Sure Central Limit Theorems for Heavily Trimmed Sums

We obtain an ahnost sure central limit theorem(ASCLT)for heavily trimmed sums.We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums of i.i.d,random variables with EX_1=0,EX_1~2=1.

Central limit theorems for power variation of Gaussian integral processes with jumps

This paper presents limit theorems for realized power variation of processes of the form Xt=t0φsdGs+ξt as the sampling frequency within a fixed interval increases to infinity.Here G is a Gaussian process with stationary increments,ξis a purely non-Gaussian L′evy process independent from G,andφis a stochastic process ensuring that the integral is well defined as a pathwise Riemann-Stieltjes integral.We obtain the central limit theorems for the case that both the continuous term and the jump term are presented simultaneously in the law of large numbers.