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 limitingfield is the same as that obtained by Dawson (1977). In the recurrent case it is aspatially uniform field. The author also give a central limit theorem for the weightedoccupation time of the super Brownian motion with underlying dimension numberd 3, completing the results of Iscoe (1986).     

带移民超布朗运动的占位时中偏差

本文证明了当底空间维数d≥3时,一类带移民超布朗运动占位时过程的中偏差,其移民由Lebesgue 测度控制.可以清楚地看出,中偏差的规范化因子和速度函数恰好介于中心极限定理和大偏差之间,在 这个意义下,中偏差填补了中心极限定理和大偏差之间的空白.     

Functional central limit theorem for super α-stable processes

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α (0 < α ≤ 2), the limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d < 2α, the limiting process is Brownian motion. Zhang Mei, Functional central limit theorem for the super-brownian motion with super-Brownian immigration, J. Theoret. Probab., to appear.     

On Dynamical Systems Perturbed by a Null-Recurrent Fast Motion: The Continuous Coefficient Case with Independent Driving Noises

An ordinary differential equation perturbed by a null-recurrent diffusion will be considered in the case where the averaging type perturbation is strong only when a fast motion is close to the origin. The normal deviations of these solutions from the averaged motion are studied, and a central limit type theorem is proved. The limit process satisfies a linear equation driven by a Brownian motion time changed by the local time of the fast motion.     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).     

Central limit theorems and invariance principles for Lorenz attractors

We prove statistical limit laws for Hölder observationsof the Lorenz attractor, and more generally for geometric Lorenzattractors. In particular, we prove the almost sure invarianceprinciple (approximation by Brownian motion). Standard consequencesof this result include the central limit theorem, the law ofthe iterated logarithm, and the functional versions of theseresults.     

Combinatorial devices for sequential analysis

Summary For Brownian motion, escape probabilities over curved boundaries have been studied in some detail. Since sequential analysis is not concerned with really large samples, the approximation by the Brownian motion is questionable; neither the central limit theorem nor the renewal theorem regulating the overshoot may be appropriate tools. The present paper shows that combinatorial results developed in fluctuation theory have some bearing on the calculation of escape probabilities. The main result is a kind of tangential approximation for random walks crossing a curved boundary.     

Central Limit Theorems for Multiple Skorokhod Integrals

In this paper, we prove a central limit theorem for a sequence of multiple Skorokhod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed.     

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.     

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.     

NA及LNQD随机变量列的几乎处处中心极限定理

本文在二阶矩存在的条件下,证明了NA及LNQD随机变量列的几乎处处中心极限定理,使主要结果成立,其中W为[0,1]上标准Brown运动。     

Large Deviations from the Almost Everywhere Central Limit Theorem

We prove large deviation principles for the almost everywhere central limit theorem, assuming that the i.i.d. summands have finite moments of all orders. The level 3 rate function is a specific entropy relative to Wiener measure and the level 2 rate the Donsker-Varadhan entropy of the Ornstein-Uhlenbeck process. In particular, the rate functions are independent of the particular distribution of the i.i.d. process under study. We deduce these results from a large deviation theory for Brownian motion via Skorokhod's representation of random walk as Brownian motion evaluated at random times. The results for Brownian motion come from the well-known large deviation theory of the Ornstein-Uhlenbeck process, by a mapping between the two processes.     

Approximation of Some Gaussian Processes

We study a class of processes which have a moving average representation with respect to a fixed driving martingale, and can be represented as a mixture of semi-martingale processes. When the driving martingale is Gaussian we obtain a numerically efficient approximation scheme and a central limit theorem (a typical process in this class is fractional Brownian motion).     

Density estimates and central limit theorem for the functional of fractional SDEs

In this paper, we consider a general class of functionals of stochastic differential equations driven by fractional Brownian motion. For this class, we obtain Gaussian estimates for the density and a quantitative central limit theorem. The main tools of the paper are the techniques of Malliavin calculus.     

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In this paper the local functional limit theorem for increments of a Brownian motion is derived with large and small deviations, and the local functional convergence rate for increments of Brownian motion in Holder norm with respect to (r,p)capacity is estimated.     

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??In this paper, we study a class of stochastic Volterra equations, which include the stochastic differential equation driven by fractional Brownian motion. By using a maximal inequality due to It\^o (1979), we establish the central limit theorem for stochastic Volterra equation on the continuous path space, with respect to the uniform norm.     

A Central Limit Theorem for the Generalized Quadratic Variation of the Step Fractional Brownian Motion

This paper gives a central limit theorem for the generalized quadratic variation of the step fractional Brownian motion. We first recall the definition of this process and the statistical results on the estimation of its parameters.     

Hlder范数下关于Brown运动增量的泛函局部收敛速度

本文研究了Brown运动在H?lder范数与容度下的泛函极限问题.利用大偏差小偏差方法,获得了Brown运动增量局部泛函极限的收敛速度,推广了文[4]中的结果.     

Theéeorème limite pour une equation differentielle stochastique anticipative

Ocone and Pardoux have introduced a stochastic differential equation in which the initial condition and the drift depend on the driving Brownian motion in an anticipative way. In this paper we prove a limit theorem for such equations when the Brownian motion is approximated by a sequence of piecewise linear processes