  A discrete time quantum walker is considered in one dimension, where at each
step, the translation can be more than one unit length chosen randomly. In the
simplest case, the probability that the distance travelled is $\ell$ is taken
as $P(\ell) = \alpha \delta(\ell-1) + (1-\alpha) \delta (\ell-2^n)$ with $n
\geq 1$. Even the $n=1$ case shows a drastic change in the scaling behaviour
for any $\alpha \neq 0,1$. Specifically, $\langle x^2\rangle
  \propto t^{3/2}$ for $0 < \alpha < 1$, implying the walk is slower compared
to the usual quantum walk. This scaling behaviour, which is neither
conventional quantum nor classical, can be justified using a simple form for
the probability density. The decoherence effect is characterized by two
parameters which vanish in a power law manner close to $\alpha =0$ and $1$ with
an exponent $\approx 0.5$. It is also shown that randomness is the essential
ingredient for the decoherence effect.
