A strong invariance principle for two-dimensional random walk in random scenery

We present a strong approximation of two-dimensional Kesten–Spitzer random walk in random scenery by Brownian motion.     

Quenched invariance principle for simple random walk on percolation clusters

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤ ^d with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.     

Quenched invariance principle for simple random walk on discrete point processes

We consider the simple random walk on random graphs generated by discrete point processes. This random walk moves on graphs whose vertex set is a random subset of a cubic lattice and whose edges are lines between any consecutive vertices on lines parallel to each coordinate axis. Under the assumption that the discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, i.e., for almost every configuration of the point process, the path distribution of the walk converges weakly to that of a Brownian motion.     

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

An almost sure invariance principle for random walks in a space-time random environment

We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L² averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain. T. Sepp?l?inen was partially supported by National Science Foundation grant DMS-0402231.     

An invariance principle for lattices of dependent random variables

Summary It is shown that for -mixing arrays of Banach space valued random vectors the central limit theorem implies the invariance principle. Applying this result to lattices of random variables a higher dimensional invariance principle under dependence assumptions is obtained.Dedicated to Professor Leopold Schmetterer     

An invariance principle for isotropic diffusions in random environment

We investigate in this work the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more, we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media.     

Moments and distribution of the local time of a two-dimensional random walk

Let ?(n,x)$?(n,x)$ be the local time of a random walk on Z²${\mathbb{Z}}^2$. We prove a strong law of large numbers for the quantity _Ln(α)=_∑x∈Z2?(n,x)^α$L_n\left(\alpha \right)={\sum }_{x\in {\mathbb{Z}}^2}?{(n,x)}^{\alpha }$ for all α≥0$\alpha \ge 0$. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.     

Strassen's invariance principle for random subsequences

Summary Necessary and sufficient conditions are given for Strassen's invariance principle ([6]) in case a random subsequence is considered. A few iterated logarithm laws for random subsequences are derived as corollaries. These results generalize, in particular, those obtained by Chow et al. ([1]).     

(ρ)-混合序列的不变原理

给出一类较广泛的(ρ)-混合序列,并证明了在一定的矩条件下,(ρ)-混合序列的不变原理成立.