A functional limit theorem for coin tossing Markov chains

2019/02/17 by Stefan Ankirchner, Thomas Kruse, Ankirchner, Stefan +3 · 2 citations
Mathematics · #60F17 #60J25 #60J60 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F17 #msc:60J25 #msc:60J60

paper · pdf · doi:10.48550/arxiv.1902.06249

To appear in Ann. Inst. Henri Poincaré Probab. Stat

arxiv created 2020/05/11 · arxiv updated 2020/05/13

Abstract

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability 1/2, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process in natural scale can be approximated with such Markov chains arbitrarily well. The functional limit theorem applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. Our results allow to practically approximate such processes with irregular behavior; we illustrate this with Markov processes exhibiting sticky features, e.g., sticky Brownian motion and a Brownian motion slowed down on the Cantor set.

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