Convergence of random walks with Markovian cookie stacks to Brownian motion perturbed at extrema
| Autor: | Elena Kosygina, Thomas Mountford, Jonathon Peterson |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Discretization Markov process markovian cookie stacks 01 natural sciences brownian motion perturbed at its extrema 010104 statistics & probability symbols.namesake Mathematics::Probability FOS: Mathematics Statistical physics 0101 mathematics excited random-walks Brownian motion central-limit-theorem times Central limit theorem Mathematics Mesoscopic physics knight theorems Markov chain Probability (math.PR) 010102 general mathematics speed repelling random-walk Coupling (probability) Random walk 60K35 (Primary) 60F17 60J55 (Secondary) bond repulsion excited random walk symbols generalized ray– branching-like processes Statistics Probability and Uncertainty Analysis Mathematics - Probability |
| Popis: | We consider one-dimensional excited random walks (ERWs) with i.i.d. markovian cookie stacks in the non-boundary recurrent regime. We prove that under diffusive scaling such an ERW converges in the standard Skorokhod topology to a multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of the limiting process are given explicitly in terms of those of the cookie markov chain at a single site. While our results extend the results of Dolgopyat and Kosygina (2012, ERWs with boundedly many cookies per site) and Kosygina and Peterson (2016, ERWs with periodic cookie stacks), the approach taken is very different and involves coarse graining of both the ERW and the random environment changed by the walk. Through a careful analysis of the environment left by the walk after each ``mesoscopic'' step, we are able to construct a coupling of the ERW at this ``mesoscopic'' scale with a suitable discretization of the limiting BMPE. The analysis is based on generalized Ray-Knight theorems for the directed edge local times of the ERW stopped at certain stopping times and evolving in both the original random cookie environment and (which is much more challenging) in the environment created by the walk after each ``mesoscopic'' step. 62 pages, 2 figures |
| Databáze: | OpenAIRE |
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