  We introduce a one-dimensional random walk, which at each step performs a
reinforced dynamics with probability $\theta$ and with probability $1 -
\theta$, the random walk performs a step independent of the past. We analyse
its asymptotic behaviour, showing a law of large numbers and characterizing the
diffusive and superdiffusive regions. We prove central limit theorems and law
of iterated logarithm based on the martingale approach.
