Small drift limit theorems for random walks

arXiv: Probability, 2016

We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are "small", (ii) a linear motion with a random slope if modifications are "large", and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are "moderate".

Stochastic Processes and their Applications, 2009

The long time asymptotics of the time spent on the positive side are discussed for one-dimensional diffusion processes in random environments. The limiting distributions under the log-log scale are obtained for the diffusion processes in the stable medium as well as for the Brox model. Similar problems are discussed for random walks in random environments and it is proved that the limiting laws are the same as in the case of diffusions.

1994

We summarize many limit theorems for systems of independent simple random walks in Z. These theorems are classi ed into four classes: weak convergence, moderate deviations, large deviations and enormous deviations. A hierarchy of relations is pointed out and some open problems are posed. Extensions to function spaces are also mentioned.

Lithuanian Mathematical Journal, 1987

Electronic Communications in Probability, 2017

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

Journal of Theoretical Probability

A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish limit theorems for n step local and occupation times.

A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed "locally perturbating set") are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. Assuming that the particle cannot jump over the membrane we prove that the weak scaling limit of the particle position is a skew Brownian motion with parameter $\gamma\in [-1,1]$. The path of a skew Brownian motion is obtained by taking each excursion of a reflected Brownian motion, independently of the others, positive with probability $2^{-1}(1+\gamma)$ and negative with probability $2^{-1}(1-\gamma)$. To prove the weak convergence result we offer a new approach which is based on the martingale characterization of a skew Brownian ...

Stochastic Processes and their Applications, 2013

We study discrete-time stochastic processes (X t ) on [0, ∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form s≤t X α s , α > 0. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of R and to a class of multidimensional 'centrally biased' random walks on R d ; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al. We benefitted in the early stages of this project from enjoyable discussions with Iain MacPhee, who sadly passed away on 13th January 2012; we dedicate this paper to Iain, in memory of our valued colleague and in gratitude for his generosity.

Advances in Applied Probability, 2011

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn, n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn, n ≥ 0} is a sequence of independent and identically distributed random variables.

Theory of Stochastic Processes

We consider a random walk Ŝ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that {Ŝ(nt)/√n} has no weak limit in D([0,1]); alternatively, the weak limit is a reflected Brownian motion.