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Non-Gaussian fluctuations of randomly trapped random walks

Part of: Markov processes Limit theorems Special processes Stochastic processes

Published online by Cambridge University Press:  08 October 2021

Adam Bowditch
Adam Bowditch*
Affiliation:
University College Dublin
*
*Postal address: University College Dublin, School of Mathematics and Statistics, Belfield, Dublin 4, Ireland. Email: [email protected]
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Abstract

In this paper we consider the one-dimensional, biased, randomly trapped random walk with infinite-variance trapping times. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton–Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

Keywords

Trap modelrandom environmentanomalous fluctuationsGalton–Watson treeLévy processesfunctional convergence

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; L'evy processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 3 , September 2021 , pp. 801 - 838
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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