Constants in the Weighted Law of the Iterated Logarithm under Long-Range Dependence: Hermite Rank Two
Abstract
At Hermite rank two, the constant in the weighted law of the iterated logarithm under long-range dependence is represented as the largest eigenvalue of a suitably normalized positive integral operator on the unit interval. The same eigenvalue determines the exponential-moment threshold and the logarithmic right-tail rate of the weighted second-chaos limit, which in the unweighted case is the Rosenblatt law. The weighted third spectral moment is evaluated in closed form. Convergent two-sided spectral enclosures are obtained, and in the weight-concentration limit the leading eigenvalue is characterized by a scalar equation with a uniform geometric remainder. At the unweighted memory boundary, the constant decays like the square root of the distance to criticality. Its leading coefficient is determined to $25$ decimal places with certified full-operator residual bounds. After division by the square-root boundary factor, the weight-concentration and memory limits commute. Their common coefficient differs from the unweighted boundary coefficient and lies in the certified interval (1.370323114331, 1.370323114332). The joint asymptotic formula is established with a uniform two-parameter remainder bound.
Related
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BibTeX
@misc{moldavskaya2026constants,
title = {{Constants in the Weighted Law of the Iterated Logarithm under Long-Range Dependence: Hermite Rank Two}},
author = {Elina Moldavskaya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30331},
eprint = {2609.30331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30331},
}
RIS
TY - UNPB
TI - Constants in the Weighted Law of the Iterated Logarithm under Long-Range Dependence: Hermite Rank Two
AU - Elina Moldavskaya
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30331
UR - https://arxiv.org/abs/2609.30331
AB - At Hermite rank two, the constant in the weighted law of the iterated logarithm under long-range dependence is represented as the largest eigenvalue of a suitably normalized positive integral operator on the unit interval. The same eigenvalue determines the exponential-moment threshold and the logarithmic right-tail rate of the weighted second-chaos limit, which in the unweighted case is the Rosenblatt law. The weighted third spectral moment is evaluated in closed form. Convergent two-sided spectral enclosures are obtained, and in the weight-concentration limit the leading eigenvalue is characterized by a scalar equation with a uniform geometric remainder. At the unweighted memory boundary, the constant decays like the square root of the distance to criticality. Its leading coefficient is determined to $25$ decimal places with certified full-operator residual bounds. After division by the square-root boundary factor, the weight-concentration and memory limits commute. Their common coefficient differs from the unweighted boundary coefficient and lies in the certified interval (1.370323114331, 1.370323114332). The joint asymptotic formula is established with a uniform two-parameter remainder bound.
ER -