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

Let an Itô semimartingale be observed at times satisfying $τ_{i+1}^n-τ_i^n=(nθ_{τ_i^n})^{-1}$, where $θ$ is adapted and càdlàg, and $θ$ 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