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Related papers
Some Limit Theorems for Heights of Random Walks on a Spider
Journal of Theoretical Probability, 2015
A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker can go on the legs in n steps. The heights on the legs are also investigated when the number of legs goes to infinity.
Some limit theorems for heights of random walks on spider
2014
A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker can go on the legs in n steps. The heights on the legs are also investigated when the number of legs goes to infinity.
ESAIM: Probability and Statistics, 2011
A spider consists of several, say N , particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on Z as underlying random walk. We suppose the environment ω = (ω x) x∈Z to be elliptic, with positive drift and nestling, so that there exists a unique positive constant κ such that E[((1 − ω 0)/ω 0) κ ] = 1. The restriction rules are kept very general; we only assume transitivity and irreducibility of the spider. The main result is that the speed of a spider is positive if κ/N > 1 and null if κ/N < 1. In particular, if κ/N < 1 a spider has null speed but the speed of a (single) RWRE is positive.
The maximal height of simple random walks revisited
Journal of Statistical Planning and Inference, 2002
In a recent paper, Katzenbeisser and Panny (J. Appl. Probab. 33 (1996) 311) derived distributional results for a number of the so-called simple random walk statistics deÿned on a simple random walk (in the sense of Cox and Miller, The Theory of Stochastic Processes, Methuen, London, 1968), starting at zero and leading to state l after n steps, where l is arbitrary, but ÿxed. In the present paper, the random walk statistics D + n is the one-sided maximum deviation and Qn the number of times where the maximum is achieved, are considered and distributional results are presented, when it is irrespective, where the random walk terminates after n steps. Thus, the results can be seen as generalizations of some well-known results about (purely) binomial random walks, given e.
Curious properties of simple random walks
Journal of statistical physics, 1993
Consider a particle performing an unrestricted random walk on the line of integers Z starting at the origin. Strictly speaking, one deals with an ensemble of random walkers; talking about a single one is just a convenient abbreviation. Let the probabilities of a step to the right ...
On the number of times a simple random walk reaches a onnegative height
Communications in Statistics. Stochastic Models, 2000
The purpose of this note is to derive distributional properties of the random variable associated with the numb e rof visits to state r, r 0 during the interval 0,n] of a simple random walk. The random walk is de ned in the sense of Cox and Miller, allowing for three step-types, arbitrary probabilities for these steps, and arbitrary terminating state after n steps. It is also shown that some well known results can b eobtained as specializations of two general Theorems.
Lengths and heights of random walk excursions
Consider a simple symmetric random walk on the line. The parts of the random walk between consecutive returns to the origin are called excursions. The heights and lengths of these excursions can be arranged in decreasing order. In this paper we give the exact and limiting distributions of these ranked quantities. These results are analogues of the corresponding results of Pitman and Yor for Brownian motion.
Random walks with different directions
Probability Theory and Related Fields, 2015
As an extension of Polya's classical result on random walks on the square grids (Z d), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after n steps is at most n −d/2−d/(d−2)+o(1) , which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is n −ω(1) , which is much worse than in higher dimensions. In dimension 3, we prove an upper bound of order n −4+o(1). We find a new conjecture concerning incidences between spheres and points in R 3 , which, if holds, would improve the bound to n −9/2+o(1) , which is consistent to the d ≥ 4 case. This conjecture resembles Szemerédi-Trotter type results and is of independent interest.
How uniformly a random walker covers a finite lattice
Physica A: Statistical Mechanics and its Applications, 1993
We study the distribution of the number of visits a random walker makes at a given site on a finite lattice with N sites, during a very long walk which visits each site a large number of times. For regular hypercubic lattices in all dimensions we find normal central limit behavior, but with anomalously large variance in ~<2 dimensions. In 2 dimensions, the ratio of the variance over the average number of visits increases logarithmically with N, while it increases ~N in one dimension. We confront this with the case of self-repelling (also called "true self-avoiding") walks. There, the variance remains bounded for all times, and increases logarithmically with N in 2 dimensions.
On the recurrence of some random walks in random environment
arXiv (Cornell University), 2014
This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely often, for any k. We also consider direct products of one-dimensional recurrent RWRE with another RWRE or with a RW. We point out the that models involving one-dimensional recurrent RWRE are more recurrent than the corresponding models involving simple symmetric walk.