Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field
Abstract
We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order q≥2, every admissible anisotropic monomial kernel gγ, and every fixed 0<h≤1/2, we prove that N2h+θ−1VN converges to a Rosenblatt process with self-similarity exponent 1−θ, without subtracting any chaos projection. Thus the visible roughness h determines the normalization, whereas the latent singularity exponent θ determines the memory of the limit. The proof uses a sharp paired-filter threshold and charged diagram power counting. At h=0, the bare kernel has logarithmically divergent energy. Variance-normalized h-regularization and a hard endpoint cutoff have the same residue and converge in finite-dimensional distributions to the universal Gaussian field Gt=(ζt−ζ0)/√2. Although G has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to √3B. In the isotropic case, simultaneous limits hN↓0 are governed by λN=hNN1/2−θ: λN→0 yields a Brownian limit, λN→λ∈(0,∞) yields an independent Brownian-Rosenblatt sum, and λN→∞ yields a Rosenblatt limit after division by λN. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.
Related
- Generalized Splines and Gaussian Processes
- Variational Splines and Generalized Gaussian Processes
- Power variations of critical Gaussian multiplicative chaos and their spectral applications
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
- Quadratic-size Random Permutations are Universal
BibTeX
@misc{assaad2026quadratic,
title = {{Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field}},
author = {Obayda Julien Assaad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27789},
eprint = {2609.27789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27789},
}
RIS
TY - UNPB
TI - Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field
AU - Obayda Julien Assaad
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27789
UR - https://arxiv.org/abs/2609.27789
AB - We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order q≥2, every admissible anisotropic monomial kernel g_γ, and every fixed 0<h≤1/2, we prove that N^2h+θ−1V_N converges to a Rosenblatt process with self-similarity exponent 1−θ, without subtracting any chaos projection. Thus the visible roughness h determines the normalization, whereas the latent singularity exponent θ determines the memory of the limit. The proof uses a sharp paired-filter threshold and charged diagram power counting. At h=0, the bare kernel has logarithmically divergent energy. Variance-normalized h-regularization and a hard endpoint cutoff have the same residue and converge in finite-dimensional distributions to the universal Gaussian field G_t=(ζ_t−ζ_0)/√2. Although G has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to √3B. In the isotropic case, simultaneous limits h_N↓0 are governed by λ_N=h_NN^1/2−θ: λ_N→0 yields a Brownian limit, λ_N→λ∈(0,∞) yields an independent Brownian-Rosenblatt sum, and λ_N→∞ yields a Rosenblatt limit after division by λ_N. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.
ER -