  For the standard elephant random walk, Laulin (2022) studied the case when
the increment of the random walk is not uniformly distributed over the past
history instead has a power law distribution. We study such a problem for the
unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg
(2014). Depending on the memory parameter $p$ and the power law exponent
$\beta$, we obtain three distinct phases in one such phase the elephant travels
only a finite distance almost surely, and the other two phases are
distinguished by the speed at which the elephant travels.
