# Persistent Random Walks. II. Functional Scaling Limits

@article{Cnac2018PersistentRW,
title={Persistent Random Walks. II. Functional Scaling Limits},
author={Peggy C{\'e}nac and Arnaud Le Ny and Basile de Loynes and Yoann Offret},
journal={Journal of Theoretical Probability},
year={2018},
volume={32},
pages={633-658}
}
We describe the scaling limits of the persistent random walks (PRWs) for which the recurrence has been characterized in Cénac et al. (J. Theor. Probab. 31(1):232–243, 2018). We highlight a phase transition phenomenon with respect to the memory: depending on the tails of the persistent time distributions, the limiting process is either Markovian or non-Markovian. In the memoryless situation, the limits are classical strictly stable Lévy processes of infinite variations, but the critical Cauchy…
1 Citations

## Figures from this paper

Non-regular g-measures and variable length memory chains
• Mathematics
• 2019
It is well-known that there always exists at least one stationary measure compatible with a continuous g-function g. Here we prove that if the set of discontinuities of the g-function g has null

## References

SHOWING 1-10 OF 51 REFERENCES
Persistent Random Walks. I. Recurrence Versus Transience
• Mathematics
• 2018
We consider a walker on the line that at each step keeps the same direction with a probability which depends on the time already spent in the direction the walker is currently moving. These walks
Topics in self-interacting random walks
In this thesis, we will show results on two different self-interacting random walk models on Z. First, we observe the frog model, an infinite system of interacting random walks, on Z with an
Semi-Markov approach to continuous time random walk limit processes
• Mathematics, Physics
• 2012
Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically
Limit Theorems and Governing Equations for Levy Walks
• Mathematics
• 2014
The Levy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of β-stable
Limit Distributions of Directionally Reinforced Random Walks
• Mathematics
• 1998
Abstract As a mathematical model for time and space correlations in ocean surface wave fields, Mauldin, Monticino, and von Weizsacker (1996,Adv. in Math.117, 239–252), introduced directionally
Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes
• Mathematics
• 2012
A classical random walk $(S_t, t\in\mathbb{N})$ is defined by $S_t:=\displaystyle\sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in\mathbb{N}}$ are a one-order Markov chain,
Limit theorems for continuous-time random walks with infinite mean waiting times
• Mathematics
Journal of Applied Probability
• 2004
A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has
Lévy Processes and Stochastic Calculus
Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random
Diffusion processes and their sample paths
• Mathematics
• 1996
Prerequisites.- 1. The standard BRownian motion.- 1.1. The standard random walk.- 1.2. Passage times for the standard random walk.- 1.3. Hin?in's proof of the de Moivre-laplace limit theorem.- 1.4.