长程相依下加权重对数律中的常数：Hermite 秩二

中文摘要 AI 辅助

在 Hermite 秩为二的情况下，长程相依下加权重对数律中的常数被表示为单位区间上适当归一化的正积分算子的最大特征值。该特征值还决定了加权二阶混沌极限的指数矩阈值和对数右尾速率，在无加权情形下该极限即为 Rosenblatt 定律。加权第三谱矩以闭式形式给出。获得了收敛的双侧谱包络，并且在权重集中极限下，主特征值由一个带有均匀几何余项的标量方程刻画。在无加权记忆边界处，该常数以到临界距离的平方根形式衰减。其前导系数被确定到小数点后 25 位，并带有经过认证的全算子残差界。在除以平方根边界因子后，权重集中极限与记忆极限可交换。它们的公共系数不同于无加权边界系数，并位于经过认证的区间 (1.370323114331, 1.370323114332) 内。联合渐近公式以带有均匀双参数余项界的形式建立。

英文摘要

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.

发表机构

• Technion–Israel Institute of Technology（以色列理工学院）

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