Limit theorems for additive functionals of continuous time random walks

Abstract

For a continuous-time random walk $X=\{X_t,t\ge 0\}$ (in general non-Markov), we study the asymptotic behavior, as $t\to \infty$, 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 Lévy motion. We further study a situation where $X$ is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )=e^{-{\sum }_{y\in \gamma }\varphi (x-y)},$ where $\varphi :\mathbb{R}\to [0,\infty )$ 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".

References in corpus (1)

• Convergence of functionals of sums of r.v.s to local times of fractional stable motions

Cited by in corpus (1)

• Perpetual Integral Functionals of Multidimensional Stochastic Processes