AI 中文总结
本文提出灰色球随机场（GSRFs），将高斯随机场作为特例，通过多元从属子驱动的高斯混合构造，推导了其统计性质，证明其为重尾等待时间的连续时间随机游走的缩放极限，拓宽了球面上可处理的各向同性随机场类别。
AI 中文摘要
我们引入一类新的、易于处理的球面上非高斯各向同性随机场，为简洁起见将其命名为灰色球随机场（Grey Spherical Random Fields, GSRFs）。该框架是经典高斯理论的灵活扩展，同时保持强各向同性，并将高斯随机场作为极限特例嵌入其中。其构造基于随机球谐系数，这些系数是由多元从属子（服从Mittag-Leffler分布或更一般分布）在独立随机时刻取值驱动的高斯混合。该表示使我们能够推导任意阶的显式特征函数、混合矩和累积量。我们还研究了高频渐近行为，确定了适当归一化的球谐系数收敛到高斯尺度混合的一般条件，由此得到包括稳定分布、Gamma分布和逆高斯分布在内的多个重要模型作为特例。通过证明GSRFs作为具有重尾等待时间的多元连续时间随机游走的缩放极限出现，为所提出的模型提供了微观基础。我们进一步获得了衡量球谐系数偏离高斯性的定量界。所得框架大幅拓宽了球面上解析可处理的各向同性随机场的类别，同时保留了显式概率表示和灵活的依赖结构。
英文摘要
We introduce a new and tractable class of non-Gaussian isotropic random fields on the sphere, which we call, for brevity, Grey Spherical Random Fields (GSRFs). The proposed framework provides a flexible extension of the classical Gaussian theory, while preserving strong isotropy and embedding Gaussian random fields as a limiting special case. The construction is based on random spherical harmonic coefficients obtained as Gaussian mixtures driven by multivariate subordinators evaluated at independent random times (with Mittag-Leffler or more general distribution). This representation allows us to derive explicit characteristic functions, mixed moments and cumulants of arbitrary order. We also investigate asymptotic high-frequency behaviour and identify general conditions under which suitably normalized spherical harmonic coefficients converge to Gaussian scale mixtures, recovering several important models, including stable, Gamma and inverse Gaussian cases, as particular examples. Proving that GSRFs emerge as scaling limits of multivariate continuous-time random walks, with heavy-tailed waiting times, provides a microscopic foundation for the proposed models. We further obtain quantitative bounds measuring the deviation from Gaussianity of the spherical harmonic coefficients. The resulting framework substantially broadens the class of analytically tractable isotropic random fields on the sphere, while preserving explicit probabilistic representations and flexible dependence structures.
Comments34 pages
