Area anomaly in the rough path Brownian scaling limit of hidden Markov walks

TL;DR

This paper investigates how hidden Markov walks converge to Brownian motion with an area anomaly in rough path topology, revealing new structures related to time correlations and embeddings of discrete processes.

Contribution

It introduces the concept of an area anomaly in the rough path limit of hidden Markov walks and uncovers a combinatorial structure generalizing occupation times.

Findings

Hidden Markov walks converge to Brownian motion with an area anomaly.

The area anomaly encodes time-correlation information of the discrete models.

A new combinatorial structure related to occupation times is identified.

Abstract

We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.