C-tightness of scaled Hawkes-based auxiliary processes

Establish C-tightness, in the sense of Jacod–Shiryaev, for the sequences of auxiliary processes Λ_T(t), X_T(t), and Z_T(t) defined in equations (L_T(t)), (BX_T(t)), and (Z_T(t)), respectively, which arise from scaling limits of the multivariate Hawkes process used to model high-frequency trading.

Background

In deriving scaling limits, the authors prove Meyer–Zheng tightness for the processes built from the Hawkes intensities but note that prior works claimed C-tightness. They explicitly state that they could not obtain C-tightness themselves.

Showing C-tightness would strengthen the tightness mode to ensure limits with continuous paths in the Skorokhod topology, aligning the result with claims in the cited literature.