  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.
