Random Walk Conditioned to Stay Positive

A random walk that is certain to visit (0, ) has associatedwith it, via a suitable h-transform, a Markov chain called ‘randomwalk conditioned to stay positive’, which is defined properlybelow. In continuous time, if the random walk is replaced byBrownian motion then the analogous associated process is Bessel-3.Let (x) = log log x. The main result obtained in this paper,which is stated formally in Theorem 1, is that, when the randomwalk has zero mean and finite variance, the total time for whichthe random walk conditioned to stay positive is below x ultimatelylies between Lx²/(x) and Ux²(x), for suitable (non-random) positiveL and finite U, as x goes to infinity. For Bessel-3, the bestL and U are identified.     

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.     

A Functional Limit Theorem for Random Walk Conditioned to Stay Non-Negative

In this paper we consider an aperiodic integer-valued randomwalk S and a process S* that is a harmonic transform of S killedwhen it first enters the negative half; informally, S* is ‘Sconditioned to stay non-negative’. If S is in the domainof attraction of the standard normal law, without centring,a suitably normed and linearly interpolated version of S convergesweakly to standard Brownian motion, and our main result is thatunder the same assumptions a corresponding statement holds forS*, the limit of course being the three-dimensional Bessel process.As this process can be thought of as Brownian motion conditionedto stay non-negative, in essence our result shows that the interchangeof the two limit operations is valid. We also establish somerelated results, including a local limit theorem for S*, anda bivariate renewal theorem for the ladder time and height process,which may be of independent interest.     

First Passage Time Distribution for Linear Functions of a Random Walk

In this paper, theorems about asymptotic behavior of the local probabilities of crossing the linear boundaries by a perturbed random walk are proved.     

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.     

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 \(Z_n\).     

Second-Order Term of Cover Time for Planar Simple Random Walk

Journal of Theoretical Probability - We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length n. Dembo et al. (Ann Math 160:433–464, 2004)...     

On the Inverse Local Time Process of a Plane Random Walk

Periodica Mathematica Hungarica -     

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.     

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.     

Long Excursions of a Random Walk

In Csáki et al. ⁽¹⁾ and Révész and Willekens⁽⁹⁾ it was proved that the length of the longest excursion among the first n excursions of a plane random walk is nearly equal to the total sum of the lenghts of these excursions. In this paper several results are proved in the same spirit, for plane random walks and for random walks in higher dimensions.     

带停留对称随机游动局部时的渐进行为

该文研究带停留对称随机游动,模型的局部时可写成一个非退化两物种分支过程的泛函.由这个表示建立模型局部时的收敛性质.     

时间随机环境下随机游动的渐近行为

本文给出了可数状态空间中时间随机环境下随机游动的一个统一的模型 .对于最常见的情况 ,即d维最近邻域随机环境下随机游动 ,如果环境是严平稳的 ,则在一定条件下 ,该随机游动满足强大数定律和中心极限定理 .特别地 ,当环境独立同分布时 ,我们可以得到更为具体的结果 ,该结果类似于经典的随机游动的相应结论 .     

Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S _n} _n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n _i for which respectively We thereby obtain conditions for to be a strong limit point of {S _n} or {S _n /n}. The first of these properties is shown to be equivalent to for some sequence a _i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between and for an increasing function fand suitable sequences n _i and a _i. These sorts of properties are of interest in sequential analysis. Known conditions for and (divergence through the whole sequence n) are also simplified.     

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

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Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is ??Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is $0s. This result characterizes the slowdown property of the walk.     

Asymptotic Behavior of the Mean Sojourn Time for a Random Walk to be in a Domain of Large Deviations

Siberian Advances in Mathematics - We study the asymptotic behavior of the mean of sojourn time for a homogeneous random walk defined on $$ [0,n]$$ to be above a receding curvilinear boundary in a...     

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.