A Law of the Iterated Logarithm for Extreme Queue Length in Multiphase Queues

The object of this research in queueing theory is the Law of the Iterated Logarithm (LIL) under the conditions of heavy traffic in Multiphase Queueing Systems (MQS). In this paper, the LIL is proved for extreme values of important probabilistic characteristics of the MQS investigated as well as maxima and minima of the summary queue length of customers and maxima and minima of the queue length of customers. Also, the paper presents a survey on the works for extreme values in queues and the queues in heavy traffic.      

加权U—统计量的一个重对数律

证明了关于独立同分布随机变量序列的加权Ｕ－统计量的一个重对数律,类似于献「３」证明了一个加权Ｕ－统计量的解耦不等式。     

两参数OU过程的钟重对数律（英）

在本文中，我们证明了两参数OU过程的钟重对数律。     

The Law of the Iterated Logarithm for Extreme Waiting Time in Multiphase Queueing Systems

The objective of this research in the queueing theory is the law of the iterated logarithm (LIL) under the conditions of heavy traffic in multiphase queueing systems (MQS). In this paper, the LIL is proved for the extreme values of some important probabilistic characteristics of the MQS, namely, maxima and minima of the summary waiting time of a customer, and maxima and minima of the waiting time of a customer.     

A Law of the Iterated Logarithm for Directed Last Passage Percolation

Let \({\widetilde{H}}_N\), \(N \ge 1\), be the point-to-point last passage times of directed percolation on rectangles \([(1,1), ([\gamma N], N)]\) in \({\mathbb {N}}\times {\mathbb {N}}\) over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some \(\alpha _{\sup } >0\),

$$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$

with probability one, and that \(\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}\) provided a commonly believed tail bound holds. The result is in contrast with the normalization \((\log N)^{2/3}\) for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result on the liminf with speed \((\log \log N)^{1/3}\) is also discussed.

     

A Functional Law of the Iterated Logarithm for U-Statistic Type Processes

We prove a functional law of the iterated logarithm for U-statistics type processes. The result is used to determine the almost sure set of limit points for change-point estimators.     

Strassen-Type Law of the Iterated Logarithm for Self-normalized Sums

Let {X,X _n ,n≥1} be a sequence of independent identically distributed random variables with EX=0 and assume that EX ² I(|X|≤x) is slowly varying as x→∞. In this paper it is shown that a Strassen-type law of the iterated logarithm holds for self-normalized sums of such random variables, i.e., when X is in the domain of attraction of the normal law.     

Iterated Logarithm Law for Anticipating Stochastic Differential Equations

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations

The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array of i.i.d. random variables with mean and variance and set . Let denote the empirical distribution function of Z_{1, n} ,..., Z _{N, n} and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for , where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.     

Law of the Iterated Logarithm for Unstable Gaussian Autoregressive Models

We investigate the asymptotic properties of one-dimensional Gaussian autoregressive processes of the second order. We prove the law of the iterated logarithm in the case of an unstable autoregressive model.     

Laws of the Iterated Logarithm for the Local U-Statistic Process

Laws of the iterated logarithm are established for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that should be of separate interest. The local U-statistic process is based upon an estimator of the density of a function of several i.i.d. variables proposed by Frees (J. Am. Stat. Assoc. 89, 517–525, 1994). As a consequence, our results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the L _p norms for these estimators. Research of E. Giné partially supported by NSA Grant H98230-04-1-0075. Research of D.M. Mason partially supported by NSA Grant MDA904-02-1-0034 and NSF Grant DMS-0503908.     

非同分布NA列的对数律和重对数律

给出了非同分布NA列满足对数律和重对数律的一些矩条件，而文[50-[7]中的部分结果可以成为其特殊情形并得到加强．     

A generalization of the Iterated Logarithm Law for weighted sums with infinite variance

Summary A generalization of the classical Law of the Iterated Logarithm (LIL) is obtained for the weighted i.i.d. case consisting of sequences { _n Y _n } where the weights { _n } are nonzero constants and {Y _n} are i.i.d. random variables. If Y is symmetric but not necessarily square integrable and if the weights satisfy a certain growth rate, conditions are given which guarantee that { _n Y _n} obey a Generalized Law of the Iterated Logarithm (GLIL) in the sense that almost certainly for some positive conslants a _n . Teicher has shown that such weights entail the classical LIL when EY ²< and Feller has treated the GLIL when _n =1 and EY ²=. The main finding here asserts that if {q_n} satisfies q _n ² =nG(q_n)loglogq _n where G is a specified slowly varying function, asymptotically equivalent to the truncated second moment of Y, and if a certain series converges, then the GLIL obtains with where .     

Rates of Convergence and Law of the Iterated Logarithm for U-Processes

We study in this paper some limit theorems for U-processes. We calculate rates of convergence in the central limit theorem of nondegenerate U-processes under metric entropy with bracketing condition. In application, we improve upon the law of the iterated logarithm of Arcones. All calculations use the Ossiander chaining procedure.     

关于可交换随机变量序列的重对数律

本文讨论了可交换随机变量序列{Xn:n≥1}的重对数律。     

Strassen's Law of the Iterated Logarithm under Sub-linear Expectations

We establish the Strassen's law of the iterated logarithm（LIL for short） for independent and identically distributed random variables with ■[X₁] =■[X₁] = 0 and C_V[X₁～2] < ∞ under a sublinear expectation space with a countably sub-additive capacity V. We also show the LIL for upper capacity with σ = ■ under some certain conditions.     

The Law of the Iterated Logarithm for Finitely Inhomogeneous Random Walks

The Hartman–Wintner–Strassen law of the iterated logarithm states that if X ₁, X ₂,… are independent identically distributed random variables and S _n =X ₁+???+X _n , then

$\limsup_{n}S_{n}/\sqrt{2n\log \log n}=1\quad \text{a.s.},\qquad \liminf_{n}S_{n}/\sqrt{2n\log \log n}=-1\quad \text{a.s.}$

if and only if EX ₁ ² =1 and EX ₁=0. We extend this to the case where the X _n are no longer identically distributed, but rather their distributions come from a finite set of distributions.