  In a recent work we introduced a semi-Markovian discrete-time generalization
of the telegraph process. We referred this random walk to as squirrel random
walk (SRW). The SRW is a discrete-time random walk on the one-dimensional
infinite lattice where the step direction is reversed at arrival times of a
discrete-time renewal process and remains unchanged at uneventful time
instants. We first recall general notions of the SRW. The main subject of the
paper is the study of the SRW where the step direction switches at the arrival
times of a generalization of the Sibuya discrete-time renewal process (GSP)
which only recently appeared in the literature. The waiting time density of the
GSP, the `generalized Sibuya distribution' (GSD) is such that the moments are
finite up to a certain order $r\leq m-1$ ($m \geq 1$) and diverging for orders
$r \geq m$ capturing all behaviors from broad to narrow and containing the
standard Sibuya distribution as a special case ($m=1$). We also derive some new
representations for the generating functions related to the GSD. We show that
the generalized Sibuya SRW exhibits several regimes of anomalous diffusion
depending on the lowest order $m$ of diverging GSD moment. The generalized
Sibuya SRW opens various new directions in anomalous physics.
