A fixed-time stable limit theorem for non-normalized functionals on irregular time grids
Abstract
Let an Itô semimartingale be observed at times satisfying τi+1n−τin=(nθτin)−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 whose Hessian is o(normx) at zero. The limit is a sum of jump contributions with sampling scale θT−−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.
Related
- Irregular Hodge Bundles for Deligne--Mumford Landau--Ginzburg Families with Fixed Pole Orders
- On the convolution theorem for directionally pathwise differentiable functionals
- A New Minimax Theorem for a Class of C1-Functionals
- Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays
- Stieltjes-generated moment functionals and orthogonality: a transfer principle and an image theorem
BibTeX
@misc{guo2026fixedtime,
title = {{A fixed-time stable limit theorem for non-normalized functionals on irregular time grids}},
author = {Yi Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09570},
eprint = {2610.09570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A fixed-time stable limit theorem for non-normalized functionals on irregular time grids
AU - Yi Guo
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09570
UR - https://arxiv.org/abs/2610.09570
AB - 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 whose Hessian is o(normx) at zero. The limit is a sum of jump contributions with sampling scale θ_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.
ER -