Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and $L^1$ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire and Bacry \emph{et al.} in the simpler framework of Mandelbrot cascades.
We revisit the multifractal analysis of $\R^d$-valued branching random walks averages by considering subsets of full Hausdorff dimension of the standard level sets, over each infinite branch of which a quantified version of the Erdös-Rényi law of large numbers holds. Assuming that the exponential moments of the increments of the walks are finite, we can indeed control simultaneously such sets when the levels belong to the interior of the compact convex domain $I$ of possible levels, i.e. when they are associated to so-called Gibbs measures, as well as when they belong to the subset $(\partial{…
In the present work, some density estimations associated with vector-valued quasi-Ahlfors measures are developed within the mixed multifractal analysis framework. The principle idea based on the fact that being quasi-Ahlfors is sufficient to conduct a mixed multifractal analysis for vector-valued measures. In the present work, we introduced a multifractal density for finitely many measures and showed that such density may be estimated well by means of the mixed multifractal measures. Such estimation induces an exact computation of the multifractal spectrum of the vector-valued quasi-Ahlfors me…
The multifractal formalism for measures in its original formulation is checked for special classes of measures such as doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In the present work, a large class of measures satisfying a weak condition known as quasi Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausd…
We consider a $\mathbb{R}^d$-valued branching random walk with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. With the help of the uniform convergence of martingale and the multifractal analysis, we establish a large deviation result for the measures $Z_n$ as well as a moderate deviation principle.
This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classica…
Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence," and is kept fixed in between those times. This model interpolates between that of a homogenous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the correspond…
Metric stability for random walks (with applications in renormalization theory)
考虑确定性随机游动F：I x Z -> I x Z，定义为F（x，n）=（f（x），K（x）+n），其中f是区间I和K：I->Z上的扩展马尔可夫映射。本文研究了当|n|趋于无穷大时，与F拓扑共轭的G（x，n）=（f_n（x），L（x，n）+n）型扰动类的遍历性（如常返性和瞬时性）、几何性和多重分形性的普适性（稳定性）.我们给出了这些结果在研究区间上的（广义）无限可重整映射之间的共轭的正则性和一维映射的野吸引子的存在性中的应用。
Consider an $\R^d$-valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets $E(K)$ of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points $K$. This goes beyond multifractal analysis, which only considers those level sets when $K$ ranges in the set of singletons $\{α\}$, $α\in\R^d$. We also give a $0$-$\infty$ l…
In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein-Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter $H>1/2$.
For Sinaï's walk (X_k) we show that the empirical measure of the environment seen from the particle (\bar\w_k) converges in law to some random measure S. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of (\bar\w_k) . When the limit in this "in law" ergodic theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show …
We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show the almost sure convergence of our estimate in the diffusive, critical and superdiffusive regimes. The local asymptotic normality of our statistical procedure is established in the diffusive regime, while the local asymptotic mixed normality is proven…
We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(μ,μ)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $η$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary …
