A fixed-time stable limit theorem for non-normalized functionals on irregular time grids

Let an Itô semimartingale be observed at times satisfying ${\tau }_{i+1}^n-{\tau }_i^n=(n{\theta }_{{\tau }_i^n}{)}^{-1}$, where $\theta$ is adapted and càdlàg, and $\theta$ and its reciprocal are bounded on compact intervals. Under the usual hypothesis (H) on the semimartingale, we prove fixed-time stable convergence for test functions whose Hessian is $o(\text{norm}x)$ at zero. The limit is a sum of jump contributions with sampling scale ${\theta }_{T-}^{-1}$ and separate pre-jump and post-jump volatilities. The proof establishes joint stable convergence of the local Poisson positions and Brownian increments on the endogenous grid, and controls infinitely many small jumps by a uniform Itô estimate. Three examples identify the left-limit sampling scale and show why positivity and càdlàg regularity alone cannot replace local control of the reciprocal sampling intensity.