Glivenko--Cantelli Theorems for Integrated Volatility Functionals in Pure-Jump Semimartingales with an Application to Cryptocurrency Markets
Abstract
We develop a two-step procedure for estimating integrated volatility functionals, defined through the occupation measure of the latent spot volatility process, when the asset price is a pure-jump semimartingale. In the first step, block-based estimators formed from absolute powers of high-frequency increments uniformly approximate local averages of powers of volatility. In the second step, these estimates are aggregated into an empirical occupation measure. Since price increments have infinite variance in this setting, arguments based on local Gaussianity are unavailable, and the uniform theory instead rests on maximal inequalities tailored to the stable regime. We establish Glivenko--Cantelli-type uniform consistency over classes of bounded monotone, Lipschitz-in-parameter, and locally Hölder test functions. These results deliver consistent estimation of volatility occupation times and quantiles, together with an argmax-consistency theory for M-estimators built on nonparametrically recovered latent processes. We further propose a stability-based rule for selecting the power index of the volatility estimator, which tracks an infeasible ex ante optimal choice closely in Monte Carlo experiments. An application to high-frequency cryptocurrency markets illustrates the framework in a jump-dominated, heavy-tailed environment.
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BibTeX
@misc{chen2026glivenkocantelli,
title = {{Glivenko--Cantelli Theorems for Integrated Volatility Functionals in Pure-Jump Semimartingales with an Application to Cryptocurrency Markets}},
author = {Dachuan Chen and Jia Li and Tian Xie and Chengxin Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07301},
eprint = {2609.07301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07301},
}
RIS
TY - UNPB
TI - Glivenko--Cantelli Theorems for Integrated Volatility Functionals in Pure-Jump Semimartingales with an Application to Cryptocurrency Markets
AU - Dachuan Chen
AU - Jia Li
AU - Tian Xie
AU - Chengxin Yan
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07301
UR - https://arxiv.org/abs/2609.07301
AB - We develop a two-step procedure for estimating integrated volatility functionals, defined through the occupation measure of the latent spot volatility process, when the asset price is a pure-jump semimartingale. In the first step, block-based estimators formed from absolute powers of high-frequency increments uniformly approximate local averages of powers of volatility. In the second step, these estimates are aggregated into an empirical occupation measure. Since price increments have infinite variance in this setting, arguments based on local Gaussianity are unavailable, and the uniform theory instead rests on maximal inequalities tailored to the stable regime. We establish Glivenko--Cantelli-type uniform consistency over classes of bounded monotone, Lipschitz-in-parameter, and locally Hölder test functions. These results deliver consistent estimation of volatility occupation times and quantiles, together with an argmax-consistency theory for M-estimators built on nonparametrically recovered latent processes. We further propose a stability-based rule for selecting the power index of the volatility estimator, which tracks an infeasible ex ante optimal choice closely in Monte Carlo experiments. An application to high-frequency cryptocurrency markets illustrates the framework in a jump-dominated, heavy-tailed environment.
ER -