Long-memory Hermite functional processes in functional time series spectral asymptotics
Abstract
Functional Non-Central Limit Theorems (NCLTs) are derived, in the functional spectral domain (see RuizMedina, 2022), for square integrable transformations, mapping an H_{1}-valued Gaussian stationary functional sequence into a nonlinear H_{2}-valued sequence, where H_{1} and H_{2} are separable Hilbert spaces. Linear and nonlinear operator-based scalings are considered, leading to different limit LRD Hermite functional processes. Under bounded linear operator scalings, an infinite-dimensional version of Theorem 5.3.1 in Pipiras and Taqqu (2017) is obtained, involving a functional reduction theorem, extending Theorem 5.3.3 in Pipiras and Taqqu (2017). Alternatively, when an operator-valued exponent based nonlinear scaling is applied, we pay attention to the LRD dominant set at each active output direction, in the leading chaos. Both scaling scenarios allow the derivation of NCLTs under similar settings of conditions.
Related
- Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field
- Novel Unified Variants, Properties, and Applications of Ostrowski, Jensen, and Hermite–Hadamard Inequalities for Generalized (η, p, h)-Convex Stochastic Processes
- Malliavin Smoothness of Hermite Processes of Arbitrary Order and Their Wiener Integrals
- A stable and efficient Galerkin spectral method with asymmetrically-weighted Hermite functions for the Vlasov-Poisson system
- Uniform Local Asymptotics for Lévy Processes with Subexponential Jumps
BibTeX
@misc{duker2026longmemory,
title = {{Long-memory Hermite functional processes in functional time series spectral asymptotics}},
author = {M. C. Düker and P. Zoubouloglou and M. D. Ruiz-Medina},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06011},
eprint = {2611.06011},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06011},
}
RIS
TY - UNPB
TI - Long-memory Hermite functional processes in functional time series spectral asymptotics
AU - M. C. Düker
AU - P. Zoubouloglou
AU - M. D. Ruiz-Medina
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06011
UR - https://arxiv.org/abs/2611.06011
AB - Functional Non-Central Limit Theorems (NCLTs) are derived, in the functional spectral domain (see RuizMedina, 2022), for square integrable transformations, mapping an H_{1}-valued Gaussian stationary functional sequence into a nonlinear H_{2}-valued sequence, where H_{1} and H_{2} are separable Hilbert spaces. Linear and nonlinear operator-based scalings are considered, leading to different limit LRD Hermite functional processes. Under bounded linear operator scalings, an infinite-dimensional version of Theorem 5.3.1 in Pipiras and Taqqu (2017) is obtained, involving a functional reduction theorem, extending Theorem 5.3.3 in Pipiras and Taqqu (2017). Alternatively, when an operator-valued exponent based nonlinear scaling is applied, we pay attention to the LRD dominant set at each active output direction, in the leading chaos. Both scaling scenarios allow the derivation of NCLTs under similar settings of conditions.
ER -