THE LAW OF THE ITERATED LOGARITHM FOR SPATIAL AVERAGES OF THE STOCHASTIC HEAT EQUATION

Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)and the functional LIL for a linear additive functional of the form∫[0,R]u(t,x)dx and the nonlinear additive functionals of the form∫[0,R]g(u(t,x))dx,where g:R→R is nonrandom and Lipschitz continuous,as R→∞for fixed t>0,using the localization argument.

PRECISE RATES IN THE LAW OF THE ITERATED LOGARITHM FOR R/S STATISTICS

Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q（n）be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q（n）≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B（t）-inf0≤t≤sB（t）,and B（t） is a Brownian bridge.

The Law of the Iterated Logarithm for the Sums of φ-Mixing Sequences with Duple Suffixes

Hu Shuhe gets a sufficient condition on the law of the iterated logarithm for the sums of φ-mixing sequences with duple suffixes. This paper greatly improves his condition.

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.

The Law of the Iterated Logarithm for Independent Random Variables with Multidimensional Parameters and Its Application

For a set of i.i.d.r.v. indexed by positive integer d-dimensional lattice points, and for some general normalizing sequence, we determine necessary and sufficient conditions for the law of iterated logarithm. As its application, we give conditions for the existence of moments of the supremum of normed partial sums.

PRECISE RATE IN THE LAW OF ITERATED LOGARITHM FOR ρ-MIXING SEQUENCE

Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞n=1ρ 2/d(2n)<∞, where d=2,if -1<b<0 and d>2(b+1),if b≥0. It is proved that,for any b>-1, limε0ε 2(b+1)∞n=1(loglogn)bnlognP{M n≥εσ2nloglogn}= 2(b+1)πГ(b+3/2)∞k=0(-1)k(2k+1) 2b+2,where Г(·) is a Gamma function.

THE LAW OF ITERATED LOGARITHM FOR R/S STATISTICS

A law of iterated logarithm for R/S statistics with the help of the strong approximations of R/S statistics by functions of a Wiener process is shown.

A LAW OF ITERATED LOGARITHM FOR THE MLE IN A RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION

In this article, a law of iterated logarithm for the maximum likelihood estimator in a random censoring model with incomplete information under certain regular conditions is obtained.

Precise Rates in the Law of the Iterated Logarithm under Dependence Assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.

Invariance principles for the law of the iterated logarithm under G-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm(LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen's invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications.

THE LAW OF THE ITERATED LOGARITHM OF THE KAPLAN-MEIER INTEGRAL AND ITS APPLICATION

For right censored data, the law of the iterated logarithm of the Kaplan-Meier integral is established. As an application, the authors prove the law of the iterated logarithm for weighted least square estimates of randomly censored linear regression model.

On the law of the iterated logarithm for Brownian motion on compact manifolds

By taking a functional analytic point of view,we consider a family of distributions(continuous linear functionals on smooth functions),denoted by{μt,t>0},associated to the law of the iterated logarithm for Brownian motion on a compact manifold.We give a complete characterization of the collection of limiting distributions of{μt,t>0}.

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent;（ii） For all constants λ ∈ [0, ∞）,lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.

A note on the cluster set of the law of the iterated logarithm under sub-linear expectations

In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-normalized law of the iterated logarithm is established.

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).

Precise asymptotics in the law of the logarithm for random fields in Hilbert space

Consider the positive d-dimensional lattice Z^d（d≥2） with partial ordering ≤, let {XK; K∈Z＋^d} be i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with mean zero and covariance operator ∑ and set partial sums SN =∑K≤nXK,K,N∈Z＋^d. Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε↓ by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of （Gut and Spataru, 2003）.

A NONCLASSICAL LAW OF ITERATED LOGARITHM FOR NEGATIVELY ASSOCIATED RANDOM VARIABLES

A nonclassical law of iterated logarithm that holds for a stationary negatively associated sequence of random variables with finite variance is proved in this paper. The proof is based on a Rosenthal type maximal inequality and the subsequence method.This result extends the work of Klesov,Rosalsky (2001) and Shao,Su (1999).

A Method of Disc Inclination Correction Based on the Inversion Model of Rotation Law

Under the traditional dynamic model,the conventional method for solving the rotation angle of a rigid body is to use the fixed-axis rotation law of the rigid body,but the known rotation shaft position must be used as a prerequisite.In practical work,for the rotation of a rigid body under multiple forces,solving the shaft is often a difficult problem.In this paper,we consider the rigid body of the disc is subjected to the force of uneven magnitude from multiple angles,the position of the rotating shaft is obtained by iterative inversion through the rigid body rotation law and the dichotomy method.After the position of the shaft is determined,we establish a differential equation model based on the law of rigid body rotation,the rotation angle of the rigid body thus being solved based on this model.Furthermore,an optimization algorithm such as genetic algorithm is used to search for a correction scheme to return the rigid body to equilibrium at any given deflection angle.The model and method are based on computer to explore the law of rotation,the practical application of them play an important role in studying the concentric drum movement and the balance of handling furniture.

Study on the Log-Linear Velocity Profile of Near-Bed Tidal Currentin Estuarine and Coastal Waters

Many observed data show that the near-bed tidal velocity profile deviates from the usual logarithmic law. The amount of deviation may not be large, but it results in large errors when the logarithmic velocity profile is used to calculate the bed roughness height and friction velocity （or shear stress）. Based on their investigation, Kuo et al. （1996） indicate that the deviation amplitude may exceed 100%. On the basis of fluid dynamic principle, the profile of the near-bed tidal velocity in estuarine and coastal waters is established by introducing Prandtl＇ s mixing length theory and Von Kannan selfsimilarity theory. By the fitting and calculation of the near-bed velocity profde data observed in the west Solent, England, the results are compared with those of the usual logarithmic model, and it is shown that the present near-bed tidal velocity profile model has such advantages as higher fitting precision, and better inner consistency between the roughness height and friction velocity. The calculated roughness height and friction velocity are closer to reality. The conclusions are validated that the logarithmic model underestimates the roughness height and friction velocity during tidal acceleration and overestimates them during tidal deceleration.

PRECISE ASYMPTOTICS IN SELF-NORMALIZED SUMS OF ITERATED LOGARITHM FOR MULTIDIMENSIONALLY INDEXED RANDOM VARIABLES

In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑n1/｜n｜（log｜n｜dP（｜Sn/Vn｜≥ε√log log｜n｜） and ∑n（logn｜）b/｜n｜（log｜n｜）^d-1P（｜Sn/Vn｜≥ε√log n）,as ε↓0,is established.