共查询到20条相似文献,搜索用时 15 毫秒
1.
Remco van der Hofstad Frank den Hollander Wolfgang König 《Probability Theory and Related Fields》2003,125(4):483-521
In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge
to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based
on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces,
and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large
deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation
rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach
followed in our earlier work, which was based on functional analytic arguments applied to variational representations and
only worked in a very limited number of situations.
We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example
(1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be
extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling
results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture
by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on
a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and
find the scaling as the width tends to infinity.
Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003
Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60
Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality 相似文献
2.
D. A. Yarotskii 《Mathematical Notes》1999,66(3):372-383
A nonhomogeneous random walk on the grid ℤ1 with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points
is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process
on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables
and Brownian motion.
Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 459–472, September, 1999. 相似文献
3.
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. 相似文献
4.
Robert M. Anderson 《Israel Journal of Mathematics》1976,25(1-2):15-46
In a recent paper [10], Peter A. Loeb showed how to convert non-standard measure spaces into standard ones and gave applications to probability theory. We apply these results to Brownian Motion and Itô integration. We first develop a number of new tools about Loeb spaces. We then show that Brownian Motion can be obtained as the Loeb process corresponding to a non-standard random walk obtained from a*-finite number of coin tosses. This permits a very constructive proof of a special case of Donsker's Theorem. The Itô integral with respect to this Brownian Motion is a non-standard Stieltjes integral with respect to the random walk. As a consequence, an easy proof of Itô’s Lemma is possible. The results in this paper were announced in [1]. 相似文献
5.
Lixin Zhang 《中国科学A辑(英文版)》2001,44(5):619-630
This paper is to prove that, if a one-dimensional random walk can be approximated by a Brownian motion, then the related random
walk in a general independent scenery can be approximated by a Brownian motion in Brownian scenery. 相似文献
6.
Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion
Antoine Lejay 《Statistical Inference for Stochastic Processes》2018,21(3):539-551
We study the asymptotic property of simple estimator of the parameter of a skew Brownian motion when one observes its positions on a fixed grid—or equivalently of a simple random walk with a bias at 0. This estimator, nothing more than the maximum likelihood estimator, is based only on the number of passages of the random walk at 0. It is very simple to set up, is consistent and is asymptotically mixed normal. We believe that this simplified framework is helpful to understand the asymptotic behavior of the maximum likelihood of the skew Brownian motion observed at discrete times which is studied in a companion paper. 相似文献
7.
A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics. 相似文献
8.
The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space–time, resampling the random walk increments, leads to some natural dynamics. In this paper we consider the corresponding dynamics for the Brownian web. In particular, pairs of coupled Brownian webs are studied, where the second web is obtained from the first by perturbing according to these dynamics. A stochastic flow of kernels, which we call the erosion flow, is obtained via a filtering construction from such coupled Brownian webs, and the N-point motions of this flow of kernels are identified. 相似文献
9.
A. N. Borodin 《Probability Theory and Related Fields》1986,72(2):231-250
Summary Many results are known about the convergence of some processes to Brownian local time. Among such processes are the process of occupation times of Brownian motion, the number of downcrossings of Brownian motion over smaller and smaller intervals before timet, the number of visits of the recurrent integer-valued random walk to some point duringn steps and others. In this paper we consider the asymptotic behaviour of the differences between Brownian local time and some of the processes which converge to it. 相似文献
10.
Lamberton 《Applied Mathematics and Optimization》2002,45(3):283-324
One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian
motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution
of the approximating random walk. 相似文献
11.
Lamberton 《Applied Mathematics and Optimization》2008,45(3):283-324
One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian
motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution
of the approximating random walk. 相似文献
12.
Uniform random mappings of an n-element set to itself have been much studied in the combinatorial literature. We introduce a new technique, which starts by specifying a coding of mappings as walks with ± 1 steps. The uniform random mapping is thereby coded as a nonuniform random walk, and our main result is that as n→∞ the random walk rescales to reflecting Brownian bridge. This result encompasses a large number of limit theorems for “global” characteristics of uniform random mappings. © 1994 John Wiley & Sons, Inc. 相似文献
13.
Wensheng Wang 《Probability Theory and Related Fields》2003,126(2):203-220
Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries.
In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence
towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery.
Received: 17 April 2002 / Revised version: 11 October 2002 /
Published online: 15 April 2003
Research supported by NSFC (10131040).
Mathematics Subject Classification (2002): 60J55, 60J15, 60J65
Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery 相似文献
14.
We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index H > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed
on the basis of the fractional Brownian motion.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1040–1046, August, 2007. 相似文献
15.
ZHANG Lixin 《中国科学A辑(英文版)》2001,44(5)
This paper is to prove that, if a one-dimensional random wa lkcan be approximated by a Brownian motion, then the related random walk in a g eneral independent scenery can be approximated by a Brownian motion in Brownian scenery. 相似文献
16.
In this paper we obtain a closed form expression of the expected exit time of a Brownian motion from equilateral triangles.
We consider first the analogous problem for a symmetric random walk on the triangular lattice
and show that it is equivalent to the ruin problem of an appropriate three player game.
A suitable scaling of this random walk allows us to exhibit explicitly the relation between
the respective exit times. This gives us the solution of the related Poisson equation. 相似文献
17.
In this paper we obtain a closed form expression of the expected exit time of a Brownian motion from equilateral triangles.
We consider first the analogous problem for a symmetric random walk on the triangular lattice
and show that it is equivalent to the ruin problem of an appropriate three player game.
A suitable scaling of this random walk allows us to exhibit explicitly the relation between
the respective exit times. This gives us the solution of the related Poisson equation. 相似文献
18.
We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself. 相似文献
19.
The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula. 相似文献
20.
In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion. 相似文献