Topological reconstruction of compact supports of dependent stationary random variables

TL;DR

This paper extends topological support reconstruction methods from i.i.d. to dependent stationary random variables, demonstrating convergence and providing novel theoretical results with illustrative examples and simulations.

Contribution

It introduces new convergence results for the Hausdorff distance between dependent stationary random vectors and their supports, generalizing previous i.i.d. support reconstruction theories.

Findings

Convergence of random vector clouds to their supports in Hausdorff distance.

A new topological reconstruction theorem for dependent stationary variables.

Illustrative examples including the M"{o}bius Markov chain with simulations.

Abstract

In this paper we extend results on reconstruction of probabilistic supports of random i.i.d variables to supports of dependent stationary $\mathbb R^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the M\"{o}bius Markov chain on the circle is treated at the end with simulations.