迭代Brown运动的一个Chung型重对数律

X及Y分别为Rd1及Rd2中的相互独立的标准Brown运动,满足X（0）=Y（0）=0．定义,称为一个迭代Brown运动．本文给出了关于Zd1,d2的一个Chung型重对数律．     

Functional Limit Theorems for Multiparameter Fractional Brownian Motion

We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy’s modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed.      

Rates of convergence for increments of Brownian motion

The results of this paper concern rates of convergence for increments of Brownian motion. As a by-product we give some improvements of a result of Bolthausen dealing with Strassen's law of the iterated logarithm.     

A Nonparametric Sequential Test with Power 1 for the Mean of Lévy-stable Laws with Infinite Variance

A nonparametric sequential test with power one for the mean of Lévy-stable laws with infinite variance is given. Our considerations are based on a law of the iterated logarithm for Peng’s estimator [Peng, Stat. Probab. Lett., 52:255–264, 2001] of the mean of heavy-tailed distributions. Our main motivation comes from applications to financial data, and in particular to sequential control of daily asset returns.      

Asymptotic Properties of Ranked Heights in Brownian Excursions

Pitman and Yor^{(20, 21)} recently studied the distributions related to the ranked excursion heights of a Brownian bridge. In this paper, we study the asymptotic properties of the ranked heights of Brownian excursions. The heights of both high and low excursions are characterized by several integral tests and laws of the iterated logarithm. Our analysis relies on the distributions of the ranked excursion heights considered up to some random times.     

Some Limit Results for Transient Subsequence of Brownian Motion on Line

Ｌｅｔ {Ｗ(ｔ) ,0≤ｔ<∞ }ｂｅａｓｔａｎｄａｒｄ ,ｏｎｅ ｄｉｍｅｎｓｉｏｎａｌＢｒｏｗｎｉａｎｍｏｔｉｏｎｏｎ (Ω ,Ｆ ,Ｐ) .Ｉｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔ -∞ =ｌｉｍｉｎｆｔ→∞ Ｗ(ｔ) <ｌｉｍｓｕｐｔ→∞ Ｗ(ｔ) =∞ａｎｄａｃｃｏｒｄｉｎｇｔｏＫａｈａｎｃｅ ([1 ] ,Ｔｈｅｏｒｅｍ1 ,Ｃｈａｐｔｅｒ 1 2 ) ,ｉｆａｓｅｑｕｅｎｃｅ {ｔｎ,ｎ≥ 1 )ｓａｔｉｓｆｉｅｓ∑∞ｎ=11ｔｎ<∞ ,ｔｈｅｎｌｉｍｎ→∞ Ｗ(ｔｎ) =∞ａ .ｓ.Ｗｅｃａｌｌｔｈｅｓｅｑｕｅｎｃｅ {Ｗ (ｔｎ) ,ｎ≥ 1 }ａｔｒａｎｓｉｅｎｔ…     

Brown运动增量在H?lder范数下的局部泛函Chung重对数律

本文利用Brown运动在H?lder范数下的大偏差和小偏差,得到了Brown运动增量在H?lder范数下的局部泛函Chung重对数律.     

A Functional LIL for m-Fold Integrated Brownian Motion

Let {Xm(t),t∈R } be an m-Fold integrated Brownian motion. In this paper, with the help of small ball probability estimate, a functional law of the iterated logarithm (LIL) for Xm(t) is established. This extends the classic Chung type liminf result for this process. Furthermore, a result about the weighted occupation measure for Xm(t) is also obtained.     

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.     

Time Difference on the Brownian Curve

Let X _t and Y _t be respectively the locations of the maximum and minimum, over [0, t], of a real-valued Wiener process. We establish limsup and liminf iterated logarithm laws for , the time difference between the maximum and the minimum, as well as for max(X _t, Y _t) and min(X _t, Y _t).     

A problem of Földes and Puri on the Wiener process

Let be a real-valued Wiener process starting from 0, and be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of as tends to infinity, i.e. they ask: how long does stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for .

     

A note on some laws of the iterated logarithm

Some function space laws of the iterated logarithm for Brownian motion with values in finite and infinite dimensional vector spaces are shown to follow from Hincin's classical law of the iterated logarithm and some martingale techniques. A law of the iterated logarithm for Brownian motion in a differentible manifold is also stated.     

Measuring the Rarely Visited Sites of Brownian Motion

We study the almost sure asymptotic behaviors of the Lebesgue measure of the points which are hardly visited, in the sense of Földes and Révész,⁽⁷⁾ by a linear Wiener process.     

A Functional LIL for m-Fold Integrated Brownian Motion

Abstract Let {X _m (t); t ∈ R ₊} be an m-Fold integrated Brownian motion. In this paper, with the help of small ball probability estimate, a functional law of the iterated logarithm (LIL) for X _m (t) is established. This extends the classic Chung type liminf result for this process. Furthermore, a result about the weighted occupation measure for X _m (t) is also obtained. *Project supported by the National Natural Science Foundation of China (No.10131040) and the Specialized Research Fund for the Doctor Program of Higher Education (No.2002335090).     

Strassen-type Laws of Iterated Logarithm for a Fractional Brownian Sheet

Abstract

We study Strassen-type laws of iterated logarithm for a fractional Brownian sheet including that for small time, which imply most of the former laws of the iterated logarithm and Strassen's laws for one-parameter and two-parameter Wiener processes.     

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Abstract   Let Λ = {λ _k } be an infinite increasing sequence of positive integers with λ _k →∞. Let X = {X(t), t ∈? R ^N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ ^d with d ≥ 4 and let |W _r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Some Limit Results for Moving Sums of Stable Random Variables

We discuss the integral test for a moving sum of an iid symmetric stable sequence with exponent α (0 < α < 2), and the Chover-type LIL is given as a corollary. We also obtain the law of a single logarithm for moving sums of a normal sequence.