Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.

A STRASSEN LAW OF THE ITERATED LOGARITHM FOR PROCESSES WITH INDEPENDENT INCREMENT

Let X={X(t),t 0} be a process with independent increments (PII)such that E=0, D X(t)E 2<∞, lim t→∞D X(t)t=1, and there exists a majoring measure G for the jump △X of X . Under these assumptions, using rather a direct method, a Strassen's law of the iterated logarithm (Strassen LIL) is established. As some special cases,the Strassen LIL for homogeneous PII and for partial sum process of i.i.d.random variables are comprised.

Chung-type Law of the Iterated Logarithm on l^p-valued Gaussian Processes

By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.

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 ,

Chung＇s Law of the Iterated Logarithm for Subfractional Brownian Motion

Let XH = {xH（t）,t ∈ R＋} be a subfractional Brownian motion in Rd. We provide asufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung＇s law of the iterated logarithm for XH.

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 .

THE LOCAL CONTINUITY MODULI FOR TWO CLASSES OF GAUSSIAN PROCESSES

In this article,local continuity moduli for the fractional Wiener process and l ∞\|valued Gaussian processes is discussed.

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.

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.

STRONG APPROXIMATION FOR MOVING AVERAGE PROCESSES UNDER DEPENDENCE ASSUMPTIONS

Let {Xt,t ≥ 1} be a moving average process defined by Xt = ∑^∞ k=0 αkξt-k, where {αk,k ≥ 0} is a sequence of real numbers and {ξt,-∞ 〈 t 〈 ∞} is a doubly infinite sequence of strictly stationary dependent random variables. Under the conditions of {αk, k ≥ 0} which entail that {Xt, t ≥ 1} is either a long memory process or a linear process, the strong approximation of {Xt, t ≥ 1} to a Gaussian process is studied. Finally, the results are applied to obtain the strong approximation of a long memory process to a fractional Brownian motion and the laws of the iterated logarithm for moving average processes.

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.

Exact Rates of Convergence of Functional Limit Theorems for Csorgo-Revesz Increments of a Wiener Process

Let {W(t): t≥0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables sup_(0≤≤T-a_T inf_(f∈S sup_(0≤r≤1 |Y_t, T(x)-f(x)] and inf_(0≤t≤T-a_T sup_(0≤x≤1|Y_(t.T)(x)-f(x)| for any given f∈S, where Y_(t.T)(x)=(W(t+xa_T)-W(t))(2a_T(logTa_T^(-1)+log logT))^(-1/2). We establish a relation between how small the increments are and the functional limit results of Csrg-Revesz increments for a Wiener process. Similar results for partial sums of i.i.d, random variables are also given.

How big are the Csorgo-Revesz increments of two-parameter Wiener processes?

In this paper, we prove a theorem on the set of limit points of the increments of a two-parameter Wiener process via establishing a large deviation principle on the increments of the two-parameter Wiener process.

The almost sure behavior of the oscillation modulus for PL-process and cumulative hazard process under random censorship

The local behavior of oscillation modulus of the product-limit (PL) process and the cumulative hazard process is investigated when the data are subjected to random censoring. Laws of the iterated logarithm of local oscillation modulus for the PL-process and the cumulative hazard process are established. Some of these results are applied to obtain the almost sure best rates of convergence for various types of density estimators as well as the Bahadur-Kiefer type process.

Asymptotic Behavior of the Lipschitz-1／2 Modulus of the PL-process for Truncated and Censored Data

In this paper, we give a detailed description of the local behavior of theLipschitz-1/2 modulus for cumulative hazard process and PL-process when the data are subject to lefttruncation and right censored observations. We establish laws of the iterated logarithm of theLipschitz-1/2 modulus of PL-process and cumulative hazard process. These results for the PL-processare sharper than other results found in the literature, which can be used to establish theasymptotic properties of many statistics.

SOME FUNCTIONAL LIMIT THEOREMS FOR THE INFINITE SERIES OF OU PROCESSES

This paper obtains functional modulus of continuity and Strassen's functional LIL of theinfinite series of independent Ornstein-Uhlenbeck processes, which also imply the Levy's exactmodulus of continuity and LIL of this process respectively.

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.

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.

由强混合序列生成的线性过程重对数律的精确渐近性质

设{εt,t∈Z}为定义在同一概率空间(Ω,F,P)上的严平稳随机变量序列,满足Eε0=0,E|ε_0|~p<∞,对某个p>2,且满足强混合条件.{a_j,j∈Z}为一实数序列,满足sum from -∞ to ∞(|a_j|)<∞,sum from -∞ to ∞(a_j)≠0.令X_t=sum from -∞ to ∞(a_jε_(t-j))(t≥1),S_n=sum from 1 to n(X_t)(n≥1).利用由强混合序列生成的线性过程的弱收敛定理及矩不等式讨论了在bn=O(1/loglogn)的条件下,当∈→0时,P{|S_n|≥(∈+b_n)τ(2nloglogn)^(1/2)}的一类加权级数的收敛性质.

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．