# Limit theorems for additive functionals of continuous time random walks

## TL;DR

This paper investigates the long-term behavior of additive functionals of continuous-time random walks, establishing convergence to stable local times and analyzing effects of random environments on the asymptotic distribution.

## Contribution

It extends limit theorems for additive functionals to non-Markovian continuous-time random walks and incorporates random environments with Poisson shot-noise potentials.

## Key findings

- Convergence to local time of an $ ext{alpha}$-stable Lévy motion for jump distributions in the domain of attraction.
- Identification of weak limits with both quenched and averaged components in random environments.
- Generalization of asymptotic behavior results to non-Markovian processes with environmental randomness.

## Abstract

For a continuous-time random walk $X=\{X_t,t\ge 0\}$ (in general non-Markov), we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X_s)ds$, $t\ge 0$. Similarly to the Markov situation, assuming that the distribution of jumps of $X$ belongs to the domain of attraction to $\alpha$-stable law with $\alpha>1$, we establish the convergence to the local time at zero of an $\alpha$-stable L\'evy motion. We further study a situation where $X$ is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda(x,\gamma)= e^{-\sum_{y\in \gamma} \phi(x-y)},$ where $\phi\colon\mathbb R\to [0,\infty)$ is a bounded function decaying sufficiently fast, and $\gamma$ is a homogeneous Poisson point process, independent of $X$. We find that in this case the weak limit has both "quenched" component depending on $\Lambda$, and a component, where $\Lambda$ is "averaged".