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

Abstract

Let an It\^o semimartingale be observed at times satisfying $\tau_{i+1}^n-\tau_i^n=(n\theta_{\tau_i^n})^{-1}$, where $\theta$ is adapted and c\`adl\`ag, 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 $o(\norm{x})$ 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 It\^o estimate. Three examples identify the left-limit sampling scale and show why positivity and c\`adl\`ag regularity alone cannot replace local control of the reciprocal sampling intensity.