Weak Convergence Theorem of a Nonnegative Random Walk to Sticky Reflected Brownian Motion

We obtain a convergence theorem of a 1-dimensional sticky reflected random walk with state space R ₊. It behaves like a random walk if it is away from the origin. Once it reaches 0, it stays at 0 for a while and is then repelled to the positive region. We consider its tightness and a martingale problem for a discontinuous function in order to construct a weak convergence theorem.     

Inequalities for the Moments and Distribution of the Ladder Height of a Random Walk

We obtain upper bounds for the tail distribution of the first nonnegative sum of a random walk and for the moments of the overshoot over an arbitrary nonnegative level if the expectation of jumps is positive and close to zero. In addition, we find an estimate for the expectation of the first ladder epoch.     

The Weak Approximation Theorem for Valuations

(1) We prove a Weak Approximation Theorem for valuations that are not necessarily independent.(2) We study the existence of intersections of finite families of valuation rings having a prescribed divisibility group and prescribed residue fields.     

Moments of Absorption Time for a Conditioned Random Walk

A random walk on the set of integers {0,1,2,...,a} with absorbing barriers at 0 and a is considered. The transition times from the integers z (0<z<a) are random variables with finite moments. The nth moment of the time to absorption at a, D_z,n, conditioned on the walk starting at z and being absorbed at a, is discussed, and a difference equation with boundary values and initial values for D_z,n is given. It is solved in several special cases. The problem is motivated by questions from biology about tumor growth and multigene evolution which are discussed.     

Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^＊ -mixing random fields,     

Computing the Distribution of the Maximum of Gaussian Random Processes

The aim of this paper is to propose an Splus program to calculate bounds for the distribution of the maximum of a smooth Gaussian process on a fixed interval. We generalize the results given in Azaïs et al. (1999) to the case of the absolute value of the Gaussian process and to the non-homogeneous case. Our method relies on calculations of the first three terms of the Rice's series. Some applications are given to illustrate the method and the performances of the program. The corresponding Splus functions are available at the URL: http://www.lsp.ups-tlse.fr/Cdelmas/software.html.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Rates of Convergence for Random Walk Summation

We prove a Marcinkiewicz-Zygmund type strong law of large numbersfor random walk summation methods. We show that the rate ofconvergence of this type of sums is equivalent to the existenceof moments of the summands.     

Valleys and the Maximum Local Time for Random Walk in Random Environment

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ^*(n) = max _x ξ(n, x). It is known that lim sup is a positive constant a.s. We prove that lim inf is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.     

关于随机函数的Weierstrass逼近定理

定义了随机系数多项式的概念之后,将函数的Weierstrass逼近定理推广到了随机函数上.     

The Critical Case of the Cramer-Lundberg Theorem on the Asymptotic Tail Behavior of the Maximum of a Negative Drift Random Walk

We study the asymptotic tail behavior of the maximum M = max{0,S _n ,n ≥ = 1} of partial sums S _n = ξ₁ + ? + ξ _n of independent identically distributed random variables ξ₁,ξ₂,... with negative mean. We consider the so-called Cramer case when there exists a β > 0 such that E e ^βξ1 = 1. The celebrated Cramer-Lundberg approximation states the exponential decay of the large deviation probabilities of M provided that Eξ₁ e ^βξ1 is finite. In the present article we basically study the critical case Eξ₁ e ^βξ1 = ∞.     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

We consider a branching random walk on \({\mathbb {R}}\) with a stationary and ergodic environment \(\xi =(\xi _n)\) indexed by time \(n\in {\mathbb {N}}\). Let \(Z_n\) be the counting measure of particles of generation n and \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]\), and establish a moderate deviation principle for the measures \(Z_n\).     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

随机加权U-过程的条件弱收敛

假定Ｆ是一个由函数组成的集合．在这篇文章中,我们研究了指标集Ｆ上２阶的随机加权Ｕ－过程的条件弱收敛性质,导出了Ｕ－过程的随机加权逼近．     

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??The local limit theorems for the minimum of a random walk with Markovian increments is given, with using Presman's factorization theory. This result implies the asymptotic behaviour of the survival probability for a critical branching process in Markovian depended random environment.     

多目标minimax问题的极大熵逼近收敛性

本文利用极大熵逼近函数，展开了多目标ｍｉｎｉｍａｘ问题的逼近方法的研究，并讨论了该逼近方法的收敛性，所得结果是目前已有的结果进一步拓广．     

广义高斯分布的参数估计及其收敛性质

广义高斯分布是一类以Gaussian分布、Laplacian分布为特例的对称分布 ,它在信号处理和图像处理等领域都有广泛的应用 .本文采用矩估计方法讨论广义高斯分布的形状参数和尺度参数的估计问题 ,首先导出了矩和参数的关系表达式 ,然后由此提出参数估计方法 ,并对参数估计的收敛性质进行了分析 ,最后利用模拟实验对本文所提方法进行了验证 .