# A functional limit theorem for coin tossing Markov chains

## TL;DR

This paper establishes a functional limit theorem for a class of Markov chains that move up or down by state-dependent amounts, enabling approximation of complex continuous Markov processes, including those with irregular features.

## Contribution

It introduces a new functional limit theorem for Markov chains with state-dependent jumps, extending approximation capabilities to irregular continuous Markov processes.

## Key findings

- Approximate all one-dimensional regular continuous strong Markov processes in natural scale.
- Applicable to processes not characterized by stochastic differential equations.
- Illustrated with sticky Brownian motion and Cantor set slowed Brownian motion.

## 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.