We study the path behaviour of a simple random walk on the 2-dimensional comb lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all horizontal edges off the x-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour.
A local limit theorem for nonlattice multidimensional random walks in cones
CERADE ESAIP；LAREMA, UMR CNRS 6093, Université d’Angers, France（昂热大学）；Institut Denis Poisson UMR 7013, Université de Tours, Université d’Orléans, CNRS France（图尔大学）；Institute of Mathematics, Vietnam Academy of Science and Technology, Vietnam（越南科学技术院数学研究所）
We look at random walks in Dirichlet environment. It was known that in dimension $d\geq 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order $κ$ for some explicit $κ$. We show that the walk, after renormalization, actually converges to a $κ$-stable completely asymmetric Levy Process.
Strong ratio limit theorems associated with a broad class of spread out random walks on unimodular groups were proved in the preceding paper, where these random walks were assumed to have the convergence parameter $R=1$. In the present paper, we study the case of an arbitrary $R\ge1$ and clarify the role of the condition that the group is unimodular.
We prove a conjecture of Toth and Veto about the weak convergence of the self repelling random walk with directed edges under diffusive scaling to a uniform distribution.
We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen from the position of the walker, i.e the environment process. We also obtain ergodicity of the uniquely ergodic measure of the environment process as well as continuity as a function of the jump rates of the walker. As a consequence we obtain several limit theorems, such as law of large numbers, Einstein relation, central limit theorem and concentration propert…
In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a k-dimensional random walk conditioned to stay in a strict order at all times. Moreover, they have shown that the rescaled random walk converges to the Dyson Brownian motion. In the present paper we find the optimal moment assumptions for the construction of the conditional random walk and generalise the limit theorem for this conditional process.
We consider random walks perturbed at zero which behave like (possibly different) random walks with i.i.d. increments on each half lines and restarts at $0$ whenever they cross that point. We show that the perturbed random walk, after being rescaled in a proper way, converges to a skew Brownian motion whose parameter is defined by renewal functions of the simple random walks and the transition probabilities from $0$.
The continuous time random walks (CTRWs) are typically defned in the way that their trajectories are discontinuous step fuctions. This may be a unwellcome feature from the point of view of application of theese processes to model certain physical phenomena. In this article we propose alternative definition of continuous time random walks with continuous trajectories. We also give the functional limit theorem for sequence of such random walks. This result requires the use of strong Skorohod M1 topology instead of Skorohod J1 topology, which is usually used in limit theorems for ordinary CTRW pr…
We consider simple random walk on a realization of an Erdős-Rényi graph that is asymptotically almost surely (a.a.s.) connected. We show a Central Limit Theorem (CLT) for the average starting hitting time, i.e. the expected time it takes the random walker on average to first hit a vertex $j$ when starting in a fixed vertex $i$. The average is taken with respect to $π_j$, the invariant measure of the random walk.
We consider the random walk among random conductances on Z^d. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t^{-1/10} for d < 3, and speed t^{-1/5} otherwise, up to logarithmic corrections.
We consider a symmetric random walk on the $ν$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction to diffusive behaviour appears in any dimension along with a long-range correction in the one-dimensional case.
In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.
In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results …
Recently, Ishiwata, Kawabi and Kotani [2] proved two kinds of central limit theorems for non-symmetric random walks on crystal lattices from the view point of discrete geometric analysis. In the present paper, we obtain yet another kind of the central limit theorem for them. Our argument is based on a measure-change technique due to Alexopoulos [1].
Unlike classical simple random walks, one-dimensional random walks in random environments (RWRE) are known to have a wide array of potential limiting distributions. Under certain assumptions, however, it is known that CLT-like limiting distributions hold for the walk under both the quenched and averaged measures. We give upper bounds on the rates of convergence for the quenched central limit theorems for both the hitting time and position of the RWRE with polynomial rates of convergence that depend on the distribution on environments.
We consider a branching system of random walk in random environment (in location) in $\mathbb{N}$. We will give the exact limit value of $\frac{M_{n}}{n}$, where $M_{n}$ denotes the minimal position of branching random walk at time $n$. A key step in the proof is to transfer our branching random walks in random environment (in location) to branching random walks in random environment (in time), by use of Bramson's "branching processes within a branching process" .
In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.
We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.
We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.
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 of investigations based on a study of the Radon-Nikodym density of the ERW distribution wit…
We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at …
It is well known that random walks in one dimensional random environment can exhibit subdiffusive behavior due to presence of traps. In this paper we show that the passage times of different traps are asymptotically independent exponential random variables with parameters forming, asymptotically, a Poisson process. This allows us to prove weak quenched limit theorems in the subdiffusive regime where the contribution of traps plays the dominating role.
We study the asymptotic behaviour of occupation times of a transient random walk in quenched random environment on a strip in a sub-diffusive regime. The asymptotic behaviour of hitting times, which is a more traditional object of study, is the exactly same. As a particular case, we solve a long standing problem of describing the asymptotic behaviour of a random walk with bounded jumps on a one-dimensional lattice
We survey some geometrical properties of trajectories of $d$-dimensional random walks via the application of functional limit theorems. We focus on the functional law of large numbers and functional central limit theorem (Donsker's theorem). For the latter, we survey the underlying weak convergence theory, drawing heavily on the exposition of Billingsley, but explicitly treat the multidimensional case. Our two main applications are to the convex hull of a random walk and the centre of mass process associated to a random walk. In particular, we establish the limit sets of the convex hull in the…
We consider a random walk $S_τ$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $τ$ appropriately scaled random walk $S_τ$ converges in the Skorohod space to the symmetric $α$-stable process $B^α$. We also prove asymptotic formula for the transition function of $S_τ$ similar to the Pólya's asymptotic formula for $B^α$.
We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous…
We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.
We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have finite second moment. In a quenched environment, an environment dependent centring is determined which is necessary to achieve a central limit theorem. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive and prove a tight bound on the bias required to obtain such limiting behaviour.
We introduce a new random walk with unbounded memory obtained as a mixture of the Elephant Random Walk and the Dynamic Random Walk which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture the distribution of the increments of the resulting random process is time dependent. We prove a strong law of large numbers for the DERW and, in a particular case, we provide an explicit expression for its speed. Finally, we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold.
We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $L^{2+\varepsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.
We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest. We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.
The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Schütz and Trimper (2004). It has a parameter $α\in (-1,1)$: The case $α=0$ corresponds to the simple symmetric random walk, and when $α>0$ (resp. $α<0$), the mean displacement of the walker at time $n$ grows (resp. vanishes) like $n^α$. The walk admits a phase transition at $α=1/2$ from the diffusive behavior to the superdiffusive behavior. In this paper, we study the rate of the moment convergence in the central limit theorem for the position of the walker wh…
First, we prove a \emph{local almost sure central limit theorem} for lattice random walks in the plane. The corresponding version for random walks in the line was considered by the author in \cite{5}. This gives us a quantitative version of Pólya's Recurrence Theorem \cite{6}. Second, we prove a \emph{local almost sure central limit theorem} for (not necessarly lattice) random walks in the line or in the plane, which will also give us quantitative recurrence results. Finally, we prove an \emph{almost sure central limit theorem} for multidimensional (not necessarly lattice) random walks. This i…
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.
We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We construct a renewal structure from which a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure follow. We further prove an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.
We consider a branching random walk with immigration in a random environment, where the environment is a stationary and ergodic sequence indexed by time. We focus on the asymptotic properties of the sequence of measures $(Z_n)$ that count the number of particles of generation $n$ located in a Borel set of real line. In the present work, a series of limit theorems related to the above counting measures are established, including a central limit theorem, a moderate deviation principle and a large deviation result as well as a convergence theorem of the free energy.
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 in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish limit theorems on local and occupation times in n steps.
We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines …
