A complete characterization of a correlated Bernoulli process

TL;DR

This paper provides a comprehensive analysis of a correlated Bernoulli process, detailing its asymptotic behavior and limit theorems depending on the correlation parameter, using martingale theory.

Contribution

It offers a complete characterization of the process's asymptotics, including laws of large numbers and central limit theorems, based on the correlation parameter.

Findings

Law of large numbers for heta .5

Central limit theorem for heta .5

Almost sure convergence to a non-degenerate limit for heta > 0.5

Abstract

We present a complete characterization of the asymptotic behaviour of a correlated Bernoulli sequence { which depends on the parameter $\theta \in [0,1]$. A martingale theory based approach will allow} us to prove versions of the law of large numbers, quadratic strong law, law of iterated logarithm, almost sure central limit theorem and functional central limit theorem, in the case $\theta \le 1/2$. For $\theta > 1/2$, we will obtain a strong convergence to a non-degenerated random variable, including a central limit theorem and a law of iterated logarithm for the fluctuations.