Let $Γ$ act on a countable set V with only finitely many orbits. Given a $Γ$-invariant random environment for a Markov chain on V and a random scenery, we exhibit, under certain conditions, an equivalent stationary measure for the environment and scenery from the viewpoint of the random walker. Such theorems have been very useful in investigations of percolation on quasi-transitive graphs.
We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of Révész \cite{r05} concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has "thin" negative tails and is false otherwise.
For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of the iterated logarithm. As a consequence, conditioned on the random scenery, the one-dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.
Suppose that the vertices of the Euclidean lattice Z^d are endowed with a random scenery, obtained by tossing a fair coin at each vertex. A random walker, starting from the origin, replaces the coins along its path by i.i.d. biased coins. For which walks and dimensions can the resulting scenery be distinguished from the original scenery? We find the answer for simple random walk, where it does not depend on dimension, and for walks with a nonzero mean, where a transition occurs between dimensions three and four. We also answer this question for other types of graphs and walks, and raise severa…
We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.
We consider a d-dimensional random walk in random scenery X(n), where the scenery consists of i.i.d. with exponential moments but a tail decay of the form exp(-c t^a) with a<d/2. We study the probability, when averaged over both randomness, that {X(n)>ny}. We show that this probability is of order exp(-(ny)^b) with b=a/(a+1).
For one-dimensional simple random walk in a general i.i.d. scenery and its limiting process we construct a coupling with explicit rate of approximation extending a recent result for Gaussian sceneries due to Khoshnevisan and Lewis. Furthermore we explicity identify the constant in the law of iterated logarithm.
We provide a random walk in random scenery representation of a new class of stable self-similar processes with stationary increments introduced recently by Jung, Owada and Samorodnitsky. In the functional limit theorem they provided, only a single instance of this class arose as a limit. We construct a model in which a significant portion of processes in this new class is obtained as a limit.
Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $ξ=(ξ_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of random variables $(Σ_n)_{n\geq 0}$ where $$Σ_n=\sum_{k=0}^n ξ_{S_k}, n\in\mathbb{N}.$$ Under a weak dependence assumption on the scenery $ξ$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).
We consider Random Walk in Random Scenery, denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $α$ with $1<α$. We present asymptotics for the probability, over both randomness, that $\{X_n>n^β\}$ for $1/2<β<1$. To obtain such asymptotics, we establish large deviations estimates for the the self-intersection local times process.
Random walks in random scenery are processes defined by $$Z_n:=\sum_{k=1}^nω_{S_k}$$ where $S:=(S_k,k\ge 0)$ is a random walk evolving in $\mathbb{Z}^d$ and $ω:=(ω_x, x\in{\mathbb Z}^d)$ is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk $S$ and the random scenery $ω$, almost surely with respect to $ω$, the correctly renormalized sequence $(Z_n)_{n\geq 1}$ is proved to converge in distribution to a centered Gaussian law with explicit variance.
A {\it scenery} is a coloring $ξ$ of the integers. Let $\{S_t\}_{t\geq 0}$ be a recurrent random walk on the integers. Observing the scenery $ξ$ along the path of this random walk, one sees the color $χ_t:=ξ(S_t)$ at time $t$. The {\it scenery reconstruction problem} is concerned with recovering the scenery $ξ$, given only the sequence of observations $χ:=(χ_t)_{t\geq 0}$. The scenery reconstruction methods presented to date require the random walk to have bounded increments. Here, we present a new approach for random walks with unbounded increments which works when the tail of the increment d…
In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of…
This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.
A random walk in random scenery $(Y_n)_{n\in\mathbb{N}}$ is given by $Y_n=ξ_{S_n}$ for a random walk $(S_n)_{n\in\mathbb{N}}$ and iid random variables $(ξ_n)_{n\in\mathbb{Z}}$. In this paper, we will show the weak convergence of the sequential empirical process, i.e. the centered and rescaled empirical distribution function. The limit process shows a new type of behavior, combining properties of the limit in the independent case (roughness of the paths) and in the long range dependent case (self-similarity).
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.
Let $(Z_n)_{n\in\N}$ be a $d$-dimensional {\it random walk in random scenery}, i.e., $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$ and $(Y(z))_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and finite variance. We identify the speed and the rate of the logarithmic decay of $¶(\frac 1n Z_n>b_n)$ for various choices of sequences $(b_n)_n$ in $[1,\infty)$. Depending on $(b_n)_n$ and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions.…
It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ ξ_j $ for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ ξ_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion …
We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramer's condition. We prove moderate deviation principles in dimensions two and larger, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case of dimension four and larger we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In dimension three and larger, an important ingredient in the proofs…
Let {ξ(n)}_{n\in Z} be a two-color random scenery, that is, a random coloring of Z in two colors, such that the ξ(i)'s are i.i.d. Bernoulli variables with parameter \tfrac12. Let {S(n)}_{n\in N} be a symmetric random walk starting at 0. Our main result shows that a.s., ξ\circ S (the composition of ξand S) determines ξup to translation and reflection. In other words, by observing the scenery ξalong the random walk path S, we can a.s. reconstruct ξup to translation and reflection. This result gives a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in a…
Let $G$ be an infinite connected graph with vertex set $V$. Let $\{S_n: n \in \mathbb N_0 \}$ be the simple random walk on $G$ and let $\{ ξ(v) : v \in V \}$ be a collection of i.i.d. random variables which are independent of the random walk. Define the random walk in random scenery as $T_n = \sum_{k=0}^n ξ(S_k)$, and the normalization variables $V_n = (\sum_{k=0}^n ξ^2(S_k))^{1/2}$ and $L_{n,2} = (\sum_{v \in V} \ell^2_n(v))^{1/2}$. For $G= \mathbb Z^d$ and $G = \mathbb T_d$, the $d$-ary tree, we provide large deviations results for the self-normalized process $T_n \sqrt{n}/(L_{n,2}V_n)$ unde…
A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new o…
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(ξ_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We suppose that the distributions of $X_1$ and $ξ_0$ belong to the normal domain of attraction of strictly stable distributions with index $α\in[1,2]$ and $β\in(0,2)$ respectively. We are interested in the asymptotic behaviour as $|a|$ goes to infinity of quantities of the form $\sum_{n\ge 1}{\mathbb E}[h(Z_n-a)]$ (when $(Z_n)_n$ is transient) or $\sum_{n\ge 1}{\mathbb E}[h(Z_n)-h(Z_n-a)]$ (when $(Z_…
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(ξ_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $α\in (0,2]$ and $β\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $α\neq 1$ and as $n\to \infty$, of $n^{-δ}Z_n$, for some suitable $δ>0$ depending on $α$ and $β$. Here we are interested in the convergence…
This paper considers 1-dimensional generalized random walks in random scenery. That is, the steps of the walk are generated by an arbitrary stationary process, and also the scenery is a priori arbitrary stationary. Under an ergodicity condition--which is satisfied in the classical case--a simple proof of the distinguishability of periodic sceneries is given.
In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case where the random walk has increments 0, +1 or -1 with probability epsilon, (1-epsilon)p and (1-epsilon)(1-…
We answer the question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the Fölner property almost surely.
We prove large deviations principles in large time, for the Brownian occupation time in random scenery. The random scenery is constant on unit cubes, and consist of i.i.d. bounded variables, independent of the Brownian motion. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched'' and ``annealed'' settings.
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(ξ_y,y\in{\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb Z}^d$ and $\mathbb R$ respectively. We suppose that the distributions of $X_1$ and $ξ_0$ belong to the normal basin of attraction of stable distribution of index $α\in(0,2]$ and $β\in(0,2]$. When $d=1$ and $α\ne 1$, a functional limit theorem has been established in \cite{KestenSpitzer} and a local limit theorem in \cite{BFFN}. In this paper, we establish the convergence o…
. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On $\mathbb {Z}^d$, $d\geq 1$, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by $\mathbb {Z}$, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random wal…
We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process.These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n^(1/4)).When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n^(1/8)).
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{ξ_{S_k}\leq s\}}-s\big) \end{equation*} where $(ξ_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $ξ$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1α}S_n)_{n\geq 1}$ converges in distr…
A scenery $f$ on a finite group $G$ is a function from $G$ to $\{0,1\}$. A random walk $v(t)$ on $G$ is said to be reconstructive if the distributions of 2 sceneries evaluated on the random walk with uniform initial distribution are identical only if one scenery is a shift of the other scenery. Previous results gave a sufficient condition for reconstructivity on finite abelian groups. This paper gives a ready generalization of this sufficient condition to one for reconstructivity on finite non-abelian groups but shows that no random walks on finite non-abelian groups satisfy this sufficient co…
Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk. We assume that the random variables Y_z have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of Pr(Z_n>t_n n) for all sequences (t_n)_{n\in\N} satisfying a certain lower bound. This complements previous results, where it was assumed that Y_z has exponential moments of all orders. In contr…
Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $ξ= (ξ_x)_{x \in \Z^d}$ of i.i.d.\ random variables, which is called the \emph{random scenery}, and a random walk $S = (S_n)_{n \in \N}$ evolving in $\Z^d$, independent from the scenery. The RWRS $Z = (Z_n)_{n \in \N}$ is then defined as the accumulated scenery al…
We present a Darboux-Wiener type lemma and apply it to obtain an exact asymptotic for the variance of the self-intersection of one and two-dimensional random walks. As a corollary, we obtain a central limit theorem for random walk in random scenery conjectured by Kesten and Spitzer in 1979.
Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $α$-stable motions called local time fractional stable motions. When $α=2$, these processes are precisely fractional Brownian motions with $1/2<H<1$. Motivated by random walks in alternating scenery, we find a "complementary" family of symmetric $α$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $α=2$, one gets fractional Brownian motions with $0<H<1/2$.
For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form of measure-theoretic duality" between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in "alternating" scenery, thus hinting at a mechanism…
We are interested in the behaviour of the range of the Campanino and Pétritis random walk, namely a simple random walk on the lattice $\mathbb Z^2$ with random orientations of the horizontal layers. We also study the range of random walks in random scenery, from which the asymptotic behaviour of the range of the first coordinate of the Campanino and Pétritis random walk can be deduced.
Probabilistic models of random walks in random sceneries give rise to examples of probability-preserving dynamical systems. A point in the state spaces consists of a walk-trajectory and a scenery, and its `motion' corresponds to shifting the time-origin. These models were proposed as natural examples of non-Bernoulli K-automorphisms by Adler, Ornstein and Weiss. This was proved in a famous analysis by Kalikow using Ornstein's Very Weak Bernoulli characterization of Bernoulli processes. Since then, various authors have generalized this construction to give other examples, including some smooth …
We establish limit theorems for U-statistics indexed by a random walk on Z^d and we express the limit in terms of some Levy sheet Z(s,t). Under some hypotheses, we prove that the limit process is Z(t,t) if the random walk is transient or null-recurrent ant that it is some stochastic integral with respect to Z when the walk is positive recurrent. We compare our results with results for random walks in random scenery.
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where basically $(X_k,k\ge 1)$ and $(ξ_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that $X_1$ is $\ZZ$-valued, centered and with finite moments of all orders. We also assume that $ξ_0$ is $\ZZ$-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that $(n^{-3/4}Z_{[nt]},t\ge 0)$ converges in distribution as $n\to \infty$ toward some self-similar process $(Δ_t,t\ge 0)$ called Brownian motion in random scenery. In a previous pa…
We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a 2d-random walk in different situations: when the r.f. is iid with a second order moment (random sceneries), or when it is generated by the action of commuting automorphisms of a torus. We consider also a quenched version of the FCLT when the random walk is replaced by a Lorentz process in the random scenery.
We consider the sums $T_N=\sum_{n=1}^N F(S_n)$ where $S_n$ is a random walk on $\mathbb Z^d$ and $F:\mathbb Z^d\to \mathbb R$ is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that $T_N$ always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on $F$, we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for…
We extend the notion of the associated random walk and the Wald martingale in random walks where the increments are independent and identically distributed to the more general case of stationary ergodic increments. Examples are given where the increments are Markovian or Gaussian, and an application in queueing is considered.
In this paper, we consider the state-dependent reflecting random walk on a half-strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.
In this paper, we consider the $(1,R)$ state-dependent reflecting random walk (RW) on the half line, allowing the size of jumps to the right at maximal $R$ and to the left only 1. We provide an explicit criterion for positive recurrence and the explicit expression of the stationary distribution based on the intrinsic branching structure within the walk. As an application, we obtain the tail asymptotic of the stationary distribution in the "near critical" situation.
We introduce a variation of strong stationary times for random walks on the symmetric group. Rather than proceed in the usual fashion of accumulating larger and larger blocks of cards which may be in any order, we wait for pairs of cards to `interact', and bound the mixing time by the time taken for enough interactions to occur.
We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non-trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d.\ environments.
In this paper, we consider the $(L,1)$ state-dependent reflecting random walk (RW) on the half line, which is a RW allowing jumps to the left at a maxial size $L$. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution.As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the $(L,1)$-random walk.
Our model is a constrained homogeneous random walk in a nonnegative orthant Z_+^d. The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie. In this paper we show that, for this type of random walks, computing the stationary probability exactly is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this mo…
In [4], we examined the use of coupling to obtain bounds on the mixing time of statistics on Markov chains. In the present paper, we consider the same general problem, but using strong stationary times rather than coupling. We discuss various types of behaviour that may occur when this is attempted, and analyse a variety of examples.
We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming the upward jump size distributions have light tails, we completely find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the …
In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series appr…
We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes.
