  A discrete time quantum walk is considered in which the step lengths are
chosen to be either $1$ or $2$ with the additional feature that the walker is
persistent with a probability $p$. This implies that with probability $p$, the
walker repeats the step length taken in the previous step and is otherwise
antipersistent. We estimate the probability $P(x,t)$ that the walker is at $x$
at time $t$ and the first two moments. Asymptotically, $\langle x^2 \rangle =
t^\nu$ for all $p$. For the extreme limits $p=0$ and $1$, the walk is known to
show ballistic behaviour, i.e., $\nu = 2$. As $p$ is varied from zero to 1, the
system is found in four different phases characterised by the value of $\nu$:
$\nu =2$ at $p=0$, $1 \leq \nu \leq 3/2$ for $0 < p < p_c$, $\nu = 3/2$ for $
p_c < p <1$ and $\nu = 2$ again at $p=1$. $p_c$ is found to be very close to
$1/3$ numerically. Close to $p=0,1$, the scaling behaviour shows a crossover in
time. Associated with this crossover, two diverging timescales varying as $1/p$
and $1/(1-p)$ close to $p=0$ and $p=1$ respectively are detected. Using a
different scheme in which the antipersistence behaviour is suppressed, one gets
$\nu= 3/2$ for the entire region $0 < p< 1$. Further, a measure of the entropy
of entanglement is studied for both the schemes.
