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.