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Limit theorems for additive functionals of continuous time random walks

Published online by Cambridge University Press:  28 May 2020

Yuri Kondratiev
Affiliation:
Bielefeld University, Germany (kondrat@math.uni-bielefeld.de)
Yuliya Mishura
Affiliation:
Taras Shevchenko National University of Kyiv, Ukraine (myus@univ.kiev.ua; zhora@univ.kiev.ua)
Georgiy Shevchenko
Affiliation:
Taras Shevchenko National University of Kyiv, Ukraine (myus@univ.kiev.ua; zhora@univ.kiev.ua)

Abstract

For a continuous-time random walk X = {Xt, t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$, t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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