L\'evy processes as weak limits of rough Heston models

We show weak convergence of the time-$t$ marginals for the integrated variance in a re-scaled rough Heston model to an Inverse Gaussian L\'{e}vy process. This shows we can obtain such a limit without having to impose that the true Hurst exponent $H$ for the model is $\frac{1}{2}$ as in [Abi Jaber, & De Carvalho, 2024], or that $H\searrow -\frac{1}{2}$ as in [Abi Jaber, Attal, & Rosenbaum, 2025], so the result potentially has increased financial relevance. We later extend the analysis to the case where $V$ has jumps, showing weak convergence of the finite-dimensional distributions of the integrated variance to a deterministic time-change of the first-passage time process to lower barriers for a more general class of spectrally positive L\'evy processes. This convergence result is then strengthened to a functional setting, namely on the space of c\`adl\`ag functions on the non-negative half-line endowed with the $M_1$ topology.