  Simple random walks are a basic staple of the foundation of probability
theory and form the building block of many useful and complex stochastic
processes. In this paper we study a natural generalization of the random walk
to a process in which the allowed step sizes take values in the set
$\{\pm1,\pm2,\ldots,\pm k\}$, a process we call a random leap. The need to
analyze such models arises naturally in modern-day data science and so-called
"big data" applications. We provide closed-form expressions for quantities
associated with first passage times and absorption events of random leaps.
These expressions are formulated in terms of the roots of the characteristic
polynomial of a certain recurrence relation associated with the transition
probabilities. Our analysis shows that the expressions for absorption
probabilities for the classical simple random walk are a special case of a
universal result that is very elegant. We also consider an important variant of
a random leap: the reflecting random leap. We demonstrate that the reflecting
random leap exhibits more interesting behavior in regard to the existence of a
stationary distribution and properties thereof. Questions relating to
recurrence/transience are also addressed, as well as an application of the
random leap.
