Endre Csáki, Wolfgang König, Zhan Shi
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.
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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.
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 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…
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)).
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…
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…
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…
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.
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…
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 consider transient random walks in random environment on $\z$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment assuming that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible and either with a finite state space or with the transition kernel dominated above and below by a probability measure.
We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramér condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new …
We consider transient random walks in random environment on Z in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following a previous article by three of the authors, where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. The case o…
The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known on the context in which such processes can arise. To our knowledge, discretisation and convergence theorems are available only in the case of stable Lévy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and that …
By decomposing the random walk path, we construct a multitype branching process with immigration in random environment for corresponding random walk with bounded jumps in random environment. Then we give two applications of the branching structure. Firstly, we specify the explicit invariant density by a method different with the one used in Brémont [3] and reprove the law of large numbers of the random walk by a method known as the environment viewed from particles". Secondly, the branching structure enables us to prove a stable limit law, generalizing the result of Kesten-Kozlov-Spitzer [11] …
Scaling Limits of Line Models in Degenerate Environment
本文研究了退化环境中线模型的缩放极限，证明了超扩散行为并探讨了非爆炸条件及缩放极限猜想。
We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal domain of attraction of an $α$-stable distribution with $0 < α< 1$. This is therefore an example of a random walk in a Lévy random medium. Specifically, it is a generalization of a process known in the physical literature as Lévy-Lorentz gas. We prove that the annealed version of the process is superdiffusive with scaling exponent $1/(α+ 1)$ and identify the li…
We consider transient nearest-neighbor random walks in random environment on Z. For a set of environments whose probability is converging to 1 as time goes to infinity, we describe the fluctuations of the hitting time of a level n, around its mean, in terms of an explicit function of the environment. Moreover, their limiting law is described using a Poisson point process whose intensity is computed. This result can be considered as the quenched analog of the classical result of Kesten, Kozlov and Spitzer [Compositio Math. 30 (1975) 145-168].
Ladder costs for random walks in Lévy random media
本文研究 Lévy 随机介质中随机游走的首达阶梯高度与长度分布律，通过构建键上随机环境随机游走（RWRSB）模型，推导广义 Spitzer-Baxter 恒等式并利用局部时间表示，刻画了首出时间前环境值之和的尾部分布，且结果可推广至整个阶梯变量序列。
In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance \cite{beig} and \cite{acc}. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of …
We consider a diffusion process $X$ in a random Lévy potential $\mathbb{V}$ which is a solution of the informal stochastic differential equation \begin{eqnarray*}\cases{dX_t=dβ_t-{1/2}\mathbb{V}'(X_t) dt,\cr X_0=0,}\end{eqnarray*} ($β$ B. M. independent of $\mathbb{V}$). We study the rate of convergence when the diffusion is transient under the assumption that the Lévy process $\mathbb{V}$ does not possess positive jumps. We generalize the previous results of Hu--Shi--Yor for drifted Brownian potentials. In particular, we prove a conjecture of Carmona: provided that there exists $0<κ<1$ such t…
Branching random walks and Minkowski sum of random walks
本研究证明d≥5维下存活临界分支随机游走值域与两独立简单随机游走值域的闵可夫斯基和交等价，推广了Wiener香肠体积大数律的离散版本，并建立了容量与分支容量的新关系。
Two fundamental theorems by Spitzer/Erickson and Kesten/Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk $(S_{n})_{n\ge 0}$ with iid increments $X_{1},X_{2},\ldots$ and the existence of moments of various related quantities like the first passage into $[x,\infty)$ and the last exit time from $(-\infty,x]$ for arbitrary $x\in\mathbb{R}_{\geqslant}$ are studied in the Markov-modulated situation when the $X_{n}$ are governed by a positive recurrent Markov chain $M=(M_{n})_{n\ge 0}$ on a countable state space $\mathcal{S}$, thus …
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