Second Order Expansions for the Moments of Minimum Point of an Unbalanced Two-sided Normal Random Walk

In this paper, the second order expansions for the first two moments of the minimum point of an unbalanced two-sided normal random walk are obtained when the drift parameters approach zero. The basic technique is the uniform strong renewal theorem in the exponential family. The comparison with numerical values shows that the approximations are very accurate. It is shown, particularly, that the first moment is significantly different from its continuous Brownian motion analog while the second moments are the same in the first order. The results can be used to study properties of the maximum likelihood estimator for the change point.     

一类自回避型随机游动的漂移速度

考虑一种直线上的随机游动模型:边的权重根据随机游动者的经过次数而相应递减.我们考察这个过程中随机游动者的漂移速度,讨论了随机游动者的位置与步数的关系.利用计算机模拟得到数据进行拟合,定量地确定出变量之间的关系.在某种条件下,随机游动者的位置的对数与步数的对数之比是一定的,不会随着边的权值递减的快慢不同而发生改变.这一点是值得进一步考虑的.     

On the Escape Probability for a Left or Right Continuous Random Walk

A formula for the escape probability in a left or right continuous random walk is derived in a simple manner, under natural conditions.AMS Subject Classification: 60G50.     

随机游动的阶梯高度和最大值及其在风险理论中的应用

该文研究了均值为负的实值随机游动的阶梯高度及最大值, 在指数估计的条件不满足的情况下,得到了它们分布的局部渐近估计和尾渐近估计, 并将这些结果应用到风险理论中的Sparre Andersen 风险模型上, 得到了一些关于破产概率的新结果.     

修正的Newman-Watts小世界及其上随机游走的混合时

在一个常规构建的图中加"长边(shortcuts)"会得到一个小世界模型,这是经典的构造小世界模型的方法.最近,吴宪远在文[Internet Mathematics,DOI:10.1080/15427951,2015.101208]中指出,在加"长边"过程中加的所有边,只有与图的直径成正比才会对小世界模型的构造起决定性作用.我们依据此文的加边机制,对体积为n~d的d(d≥1)维格点图,只添加起决定性作用的长边,得到的小世界模型修正了原始的Newman-Watts小世界模型,并证明该模型的直径和混合时是log n阶的.     

Asymptotics of the Stationary Distribution of an Oscillating Random Walk

We obtain some theorems on the asymptotics of the stationary distribution of an oscillating random walk with two levels of switching provided that the distance between the levels tends to infinity.     

Asymptotic Expansions for the Distribution of the Crossing Number of a Strip by Sample Paths of a Random Walk

The complete asymptotic expansions are obtained for the distribution of the crossing number of a strip in n steps by sample paths of an integer-valued random walk with zero mean. We suppose that the Cramer condition holds for the distribution of jumps and the width of strip increases together with n; the results are proven under various conditions on the width growth rate. The method is based on the Wiener–Hopf factorization; it consists in finding representations of the moment generating functions of the distributions under study, the distinguishing of the main terms of the asymptotics of these representations, and the subsequent inversion of the main terms by the modified saddle-point method.     

带形上近临界随机游动的常返暂留性

本文研究带形上的近临界随机游动,借助游动常返暂留性判别准则的显式表达,通过带扰动的线性差分系统的解的渐近性理论,以及矩阵的范数性质,在扰动矩阵不同的阶的条件下,给出了游动常返暂留性的判别.     

Exit and Return of a Simple Random Walk

Let V be a finite subset of Z^m. Estimates are obtained for the average probability that a simple random walk, starting at a point x in V, exits V before it returns to x. The average is taken over all points x in V. Mathematics Subject Classifications (2000) 60G50, 60J45     

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 |.     

组合分布在求解直线上n步随机游动首次通过问题上的应用

基于组合过程模型给出其轨道对目标集的首次通过概率及首中点的分布函数 ,并由此给出直线上n步随机游动的首次通过概率及首中点分布函数的一类显式 .     

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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.     

带形上随机环境中随机游动的内蕴分枝结构

揭示了带形上随机环境中随机游动的内蕴分枝结构一带移民的多物种分枝过程.利用内蕴分枝结构,可精确表达游动的首次击中时.给出了内蕴分枝结构的如下两个应用:(1)计算出首次击中时的均值,给出游动大数定律速度的显示表达,(2)得到从粒子角度看环境的马氏链不变测度的密度函数的显示表达,进而可用另一种"站在粒子看环境"的方法直接证明游动的大数定律.     

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

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 = ∞.     

A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ ∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.     

The Weak Convergence Theorem for the Distribution of the Maximum of a Gaussian Random Walk and Approximation Formulas for its Moments

In this study, asymptotic expansions of the moments of the maximum (M(β)) of Gaussian random walk with negative drift (???β), β?>?0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(β)?≡?2?β?M(β) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter β?∈?(0.5, 3.2] using meta-modeling.     

On Factorization Components of the Sojourn Times for Semicontinuous Random Walks in a Strip

We find an explicit form of the factorization components of the sojourn time for an upper semicontinuous random walk in a strip.     

Estimates for the Tail Probability of the Supremum of a Random Walk with Independent Increments

The authors investigate the tail probability of the supremum of a random walk with independent increments and obtain some equivalent assertions in the case that the increments are independent and identically distributed random variables with O-subexponential integrated distributions. A uniform upper bound is derived for the distribution of the supremum of a random walk with independent but non-identically distributed increments, whose tail distributions are dominated by a common tail distribution with an O-subexponential integrated distribution.     

Error Estimation and Optimization of the Functional Algorithms of a Random Walk on a Grid Which Are Applied to Solving the Dirichlet Problem for the Helmholtz Equation

We consider the algorithms of a random walk on a grid which are applied to global solution of the Dirichlet problem for the Helmholtz equation (the direct and conjugate methods). In the metric space C we construct some upper error bounds and obtain optimal values (in the sense of the error bound) of the parameters of the algorithms (the number of nodes and the sample size).