A LAW OF THE ITERATED LOGARITHM FOR NEAREST NEIGHBOR ESTIMATION OF MULTIVARIATE DENSITY FUNCTION

Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).

Strong Approximation Method and the(Functional)Law of Iterated Logarithm for GI/G/1 Queue

In this paper, a unified method based on the strong approximation(SA) of renewal process(RP) is developed for the law of the iterated logarithm(LIL) and the functional LIL(FLIL), which quantify the magnitude of the asymptotic rate of the increasing variability around the mean value of the RP in numerical and functional forms respectively. For the GI/G/1 queue, the method provides a complete analysis for both the LIL and the FLIL limits for four performance functions: The queue length, workload, busy time and idle time processes, covering three regimes divided by the traffic intensity.

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.

Study of the Convergence of the Increments of Gaussian Process

Let be a Gaussian process with stationary increments . Let be a nondecreasing function of t with . This paper aims to study the almost sure behaviour of where with and is an increasing sequence diverging to .

Laws of the Iterated Logarithm for Locally Square Integrable Martingales

Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.

Chung’s functional law of the iterated logarithm for the Brownian sheet

In this paper,we investigate functional limit problem for path of a Brownian sheet,Chung’s functional law of the iterated logarithm for a Brownian sheet is obtained.The main tool in the proof is large deviation and small deviation for a Brownian sheet.

Liminf law of the iterated logarithm for the smooth empirical distribution function

The study of perturbed (smoothed) empirical distribution has received considerable at-tention. The monograph by Reiss gives a particularly lucid exposition of the mathematical attractions of smoothing the empirical distribution function (e.d.f.): it is intuitively appealing; it is easy to compute, it provides increased second-order efficiency (see refs. [2, 3]) and it improves the speed of convergence in Bootstrap (see refs. [4,5]).

A LAW OF THE ITERATED LOGARITHM FOR RANDOM WINDOW-WIDTH KERNEL ESTIMATOR OF A NONPARAMETRIC REGRESSION FUNCTION

In this paper we study the estimation of the regression function.We establish a law ofthe iterated logarithm for the random window-width kernel estimator and,as an application,fora nearest neighbor estimator.These results give sharp pointwise rates of strong consistency ofthese estimators.

A LAW OF THE ITERATED LOGARITHM FOR PROCESSES WITH INDEPENDENT INCREMENTS

By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of the following conditions is satisfied,(2) Suppose the Levy's measure of X may be written as v(dt,ds) = Ft(dx) dV(t) and there is a σ-finite measure G such tnat ,

FUNCTIONAL LAW OF ITERATED LOGARITHM FOR ADDITIVE FUNCTIONALS OF REVERSIBLE MARKOV PROCESSES

Using the forward-backward martingale decomposition and the martingale limit theorems, we establish the functional law of iterated logarithm for an additive functional (At) of a reversible Markov process, under the minimal condition that σ~2(A)= tim BA_t~2/t exists in R. We extend also t →∞ the previous remarkable functional central limit theorem of Kipnis and Varadhan.

LOCAL OSCILLATION MODULUS OF THE UNIFORM EMPIRICAL PROCESS

In this paper,we present a detailed description of the limiting behaviorof local oscillation of the uniform empirical process.As an application,we estab-lish the laws of the iterated logarithm for the“naive”estimator and the nearestneighbor estimator of the density function.When compared to those of Hall andHong,the conditions of the bandwidth imposed here are as weak as possible.

On Global and Local Properties of the Trajectories of Gaussian Random Fields——A Look Through the Set of Limit Points

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近,同时又给出由NA随机变量列产生的线性过程的重对数律.

独立增量过程的一个强逼近定理

该文给出了独立增量过程的一种强逼近定理，由此得到相应的ｓｔｒａｓｓｅｎ重对数律．

扩散过程的拟必然局部Strassen重对数律

应用大偏差 ,得到了扩散过程和重随机积分的拟必然局部 Strassen重对数律 .

k-维Brown运动的泛函重对数定律(英文)

本文研究了k-维Brown运动的泛函样本轨道性质.利用了一致范数在高维连续函数空间生成的拓扑下建立大偏差公式的方法,获得了k-维Brown运动的泛函重对数定律.

Brown运动在容度意义下的局部Strassen重对数律

利用Brown运动在 (r,p) -容度意义下的大偏差 。

二重马氏过程泛函的最佳非线性均方预测

设φ是满足某些条件的 Borel 可测函数,记{Y(t),t∈R^+}为由公式 Y(t)=∫(t-s)dB(s) from t=0 to t所确定的二重 Markov 过程,这里 B(t)是规范化的布朗运动过程。在本文中要研究 Y(t)经函数φ变换而得到的随机泛函 M(t)=φ(Y(t))的最佳非要性均方预测问题.作者在此文中,根据已知观测值 M(s),s≤t 研讨了怎样求 M(t+τ),τ>0的最佳非线性均方预测量■(t,τ)。另一方面,还利用单参数预测算子、布朗运动过程的迭对数定律以及二重马氏过程的统计性质给出了计算预测量■(t,τ)的显式公式和由观测值来确定基础过程值的算法规则。

扩散过程在Hlder范数下的局部Strassen重对数律

在该文中,作者应用扩散过程在Holder范数下的大偏差得到了扩散过程在Holder范数下的局部Strassen重对数律．并且还得到了重It■积分的泛函重对数律．

Wiener过程在Hlder范数下的泛函重对数定律(英文)

本文借助于Hlder范数在函数空间中诱导出的强拓扑下的大偏差公式,得到了Wiener过程在Hlder范数下的泛函重对数定律.