State space reconstruction of Markov chains via autocorrelation structure

Understanding the state space of observed Markov processes is essential for advancing causal inference in a wide range of scientific fields. This paper demonstrates how the previously unknown state space can be reconstructed by exploring the spectrum of the time-delay embedding matrix derived from the autocorrelation sequence of the observed series. It also highlights that the eigenvector associated with the smallest eigenvalue can provide valuable insights into the hidden data generation process itself. The presented results provide a deeper understanding of the complex dynamics of Markov chains and hold promise for enhancing various scientific applications.