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In this paper we present some extensions of recent noncentral moderatedeviation results in the literature. In the first part we generalize theresults 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 weassume that $\{S(t):t\geq 0\}$ has bounded variation and it is not asubordinator; thus, in some sense, we have the difference of two independentnon-null subordinators. In this way we generalize the results in\cite{LeeMacci} for Skellam processes.

Noncentral moderate deviations for time-changed L\'evy processes with inverse of stable subordinators