A complete characterization of a correlated Bernoulli process

Abstract

We present a complete characterization of the asymptotic behaviour of a correlated Bernoulli sequence that depends on the parameter $\mathit{\theta }\in [0,1)$. A martingale theory based approach allows 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 $\mathit{\theta }\le 1∕2$. For $\mathit{\theta }>1∕2$, we obtain a strong convergence to a non-degenerated random variable, including a central limit theorem and a law of iterated logarithm for the fluctuations.