  In this paper we present some extensions of recent noncentral moderate
deviation results in the literature. In the first part we generalize the
results in \cite{BeghinMacciSPL2022} by considering a general L\'evy process
$\{S(t):t\geq 0\}$ instead of a compound Poisson process. In the second part we
assume that $\{S(t):t\geq 0\}$ has bounded variation and it is not a
subordinator; thus, in some sense, we have the difference of two independent
non-null subordinators. In this way we generalize the results in
\cite{LeeMacci} for Skellam processes.
