Limit Theorems For Random Walks Research Articles

Asymptotic Properties of Extremal Markov Processes Driven by Kendall Convolution

This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected with the Kendall convolution. In particular, we provide general formulas for the finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and associated continuous-time stochastic processes.

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Let ξ1, ξ2,… be i.i.d. random variables of zero mean and finite variance and η1, η2,… positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, α∈(0,1). The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump ξk occurs; if the present position of the Markov chain is nonpositive, then the jump ηk occurs. We prove functional limit theorems for this and two closely related Markov chains under Donsker’s scaling. The weak limit is a nonnegative process (X(t))t≥0 satisfying a stochastic equation dX(t)=dW(t)+dUα(LX(0)(t)), where W is a Brownian motion, Uα is an α-stable subordinator which is independent of W, and LX(0) is a local time of X at 0. Also, we explain that X is a Feller Brownian motion with a ‘jump-type’ exit from 0.

Quenched local central limit theorem for random walks in a time-dependent balanced random environment

We prove a quenched local central limit theorem for continuous-time random walks in {mathbb {Z}}^d, dge 2, in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.

Limit Theorems for the One-Dimensional Random Walk with Random Resetting to the Maximum

The first part of this paper is devoted to study a model of one-dimensional random walk with memory to the maximum position described as follows. At each step the walker resets to the rightmost visited site with probability $$r \in (0,1)$$ and moves as the simple random walk with remaining probability. Using the approach of renewal theory, we prove the laws of large numbers and the central limit theorems for the random walk. These results reprove and significantly enhance the analysis of the mean value and variance of the process established in Majumdar et al. (Phys Rev E 92:052126, 2015). In the second part, we expand the analysis to the situation where the memory of the walker decreases over time by assuming that at the step n the resetting probability is $$r_n = \min \{rn^{-a}, \tfrac{1}{2}\}$$ with r, a positive parameters. For this model, we first establish the asymptotic behavior of the mean values of $$X_n$$ -the current position and $$M_n$$ -the maximum position of the random walk. As a consequence, we observe an interesting phase transition of the ratio $${{\mathbb {E}}}[X_n]/{{\mathbb {E}}}[M_n]$$ when a varies. Precisely, it converges to 1 in the subcritical phase $$a\in (0,1)$$ , to a constant $$c\in (0,1)$$ in the critical phase $$a=1$$ , and to 0 in the supercritical phase $$a>1$$ . Finally, when $$a>1$$ , we show that the model behaves closely to the simple random walk in the sense that $$\frac{X_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } {\mathcal {N}}(0,1)$$ and $$\frac{M_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } \max _{0 \le t \le 1} B_t$$ , where $${\mathcal {N}}(0,1)$$ is the standard normal distribution and $$(B_t)_{t\ge 0}$$ is the standard Brownian motion.

Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

We establish a quenched local central limit theorem for the dynamic random conductance model on {mathbb {Z}}^d only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi’s iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights.

Critical branching processes in random environment with immigration: survival of a single family

We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event \(\mathcal {A}_{i}(n)\) that all individuals alive at time n are offsprings of the immigrant which joined the population at time i. We study the asymptotic probability of this extreme event when n is large and i follows different asymptotics which may be related to n (i fixed, close to n, or going to infinity but far from n). In order to do so, we establish some limit theorems for random walks conditioned to stay positive or nonnegative, which are of independent interest.

Central limit theorems for random walks associated with hypergeometric functions of type BC

Consider the non-compact Grassmann manifolds $G/K$ over the fields $\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge 1$ and dimension parameter $p \gt q$. The associated spherical functions are Heckman–Opdam hypergeometric functions of type $BC$, whe

On Weak Limiting Distributions for Random Walks on a Spider

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider.

Functional central limit theorem for random walks in random environment defined on regular trees

We study Random Walks in an i.i.d. Random Environment (RWRE) defined on b-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the process is defined on b-regular trees, with b≥4, substantially improving the results of the first author (see Theorem 3 of Collevecchio (2006)).

Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero

We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are change of measure and some conditional limit theorems for random walks.

Three problems solved by Sébastien Gouëzel

We present three results of Sebastien Gouezel's: the local limit theorem for random walks on hyperbolic groups, a multiplicative ergodic theorem for non-expansive mappings (joint work with Anders Karlsson), and the description of the essential spectrum of the Laplacian on \begin{document}$ SL(2,{\mathbb R}) $\end{document} orbits in the moduli space (joint work with Artur Avila).

Limit Theorems for Random Walks with Absorption

We introduce a class of absorption mechanisms and study the behavior of real-valued centered random walks with finite variance that do not get absorbed. Our main results serve as a toolkit which allows obtaining persistence and scaling limit results for many different examples in this class. Further, our results reveal new connections between results in Kemperman (The passage problem for a stationary Markov chain. Statistical research monographs, The University of Chicago Press, Chicago, 1961) and Vysotsky (Stoch Processes Appl 125(5):1886–1910, 2015).

Conditioned local limit theorems for random walks defined on finite Markov chains

Let $$(X_n)_{n\geqslant 0}$$ be a Markov chain with values in a finite state space $${\mathbb {X}}$$ starting at $$X_0=x \in {\mathbb {X}}$$ and let f be a real function defined on $${\mathbb {X}}$$. Set $$S_n=\sum _{k=1}^{n} f(X_k)$$, $$n\geqslant 1$$. For any $$y \in {\mathbb {R}}$$ denote by $$\tau _y$$ the first time when $$y+S_n$$ becomes non-positive. We study the asymptotic behaviour of the probability $${\mathbb {P}}_x \left( y+S_{n} \in [z,z+a],\, \tau _y > n \right) $$ as $$n\rightarrow +\infty .$$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order $$n^{3/2}$$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability $${\mathbb {P}}_x \left( \tau _y = n \right) $$ as $$n\rightarrow +\infty $$.

Quenched central limit theorem for random walks in doubly stochastic random environment

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, $\mathcal{L}^{2+\varepsilon}$ (rather than $\mathcal{L}^{2}$) integrability condition on the stream tensor. On the way we extend Nash’s moment bound to the nonreversible, divergence-free drift case, with unbounded ($\mathcal{L}^{2+\varepsilon}$) stream tensor. This paper is a sequel of [Ann. Probab. 45 (2017) 4307–4347] and relies on technical results quoted from there.

Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups

In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac–Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in \[23]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building.

Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices

We prove a central limit theorem under diffusive scaling for the displacement of a random walk on ZdZd in stationary and ergodic doubly stochastic random environment, under the H−1H−1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in Lmax{2+δ,d}Lmax{2+δ,d}, with δ>0δ>0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschlager [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC

The spherical functions of the non-compact Grassmann manifolds $$G_{p,q}({\mathbb {F}})=G/K$$ over the (skew-)fields $${\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}$$ with rank $$q\ge 1$$ and dimension parameter $$p>q$$ can be described as Heckman–Opdam hypergeometric functions of type BC, where the double coset space G /  / K is identified with the Weyl chamber $$ C_q^B\subset {\mathbb {R}}^q$$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rosler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $$p\in [2q-1,\infty [$$ , and that associated commutative convolution structures $$*_p$$ on $$C_q^B$$ exist. In this paper, we study the associated moment functions and the dispersion of probability measures on $$C_q^B$$ with the aid of this generalized integral representation. This leads to strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $$(C_q^B, *_p)$$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitly. For integers p, all results have interpretations for G-invariant random walks on the Grassmannians G / K. Besides the BC-cases, we also study the spaces $$GL(q,{\mathbb {F}})/U(q,{\mathbb {F}})$$ , which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $$q=1$$ , the results of this paper are well known in the context of Jacobi-type hypergroups on $$[0,\infty [$$ .

Limit theorems for random walks

We consider a random walk Sτ 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 Sτ converges in the Skorohod space to the symmetric α-stable process Bα. We also prove asymptotic formula for the transition function of Sτ similar to the Pólya’s asymptotic formula for Bα.

Small drift limit theorems for random walks

AbstractWe show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

Long time asymptotics of non-symmetric random walks on crystal lattices

In the present paper, we study long time asymptotics of non-symmetric random walks on crystal lattices from a view point of discrete geometric analysis due to Kotani and Sunada [11,25]. We observe that the Euclidean metric associated with the standard realization of the crystal lattice, called the Albanese metric, naturally appears in the asymptotics. In the former half of the present paper, we establish two kinds of (functional) central limit theorems for random walks. We first show that the Brownian motion on the Euclidean space with the Albanese metric appears as the scaling limit of the usual central limit theorem for the random walk. Next we introduce a family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We then capture the Brownian motion with a constant drift of the asymptotic direction on the Euclidean space with the Albanese metric associated with the symmetrized random walk through another kind of central limit theorem for the family of random walks. In the latter half of the present paper, we give a spectral geometric proof of the asymptotic expansion of the n-step transition probability for the non-symmetric random walk. This asymptotic expansion is a refinement of the local central limit theorem obtained by Sunada [22,23] and is a generalization of the result in [11] for symmetric random walks on crystal lattices to non-symmetric cases.