Renewal model for dependent binary sequences

TL;DR

This paper introduces a renewal-based model for generating dependent binary sequences with controllable correlation decay, demonstrating its effectiveness and theoretical properties like entropy and limit theorems.

Contribution

It develops a renewal process framework for modeling dependent binary sequences, analyzing correlation structures, and establishing theoretical properties such as entropy and limit theorems.

Findings

Can model subexponential decay of correlations

Demonstrates maximum entropy principle for the model

Establishes law of large numbers and CLT for observables

Abstract

We suggest to construct infinite stochastic binary sequences by associating one of the two symbols of the sequence with the renewal times of an underlying renewal process. Focusing on stationary binary sequences corresponding to delayed renewal processes, we investigate correlations and the ability of the model to implement a prescribed autocovariance structure, showing that a large variety of subexponential decay of correlations can be accounted for. In particular, robustness and efficiency of the method are tested by generating binary sequences with polynomial and stretched-exponential decay of correlations. Moreover, to justify the maximum entropy principle for model selection, an asymptotic equipartition property for typical sequences that naturally leads to the Shannon entropy of the waiting time distribution is demonstrated. To support the comparison of the theory with data, a law of large numbers and a central limit theorem are established for the time average of general observables.