This paper addresses the estimation of the second-order structure of a manifold cross-time random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert-Schmidt operator norm are derived beyond structural assumptions. Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive-moving average (SPHARMA(p,q)) processes.

翻译：本文采用Ruiz-Medina（2022）提出的函数型时间序列框架，研究具空间变化长程依赖（LRD）的流形交叉时间随机场（RF）二阶结构的估计问题。在无需结构性假设的条件下，推导了Hilbert-Schmidt算子范数下积分周期图算子的渐近无偏性条件。在高斯背景下，基于半参数函数谱框架实现了长记忆算子的弱相合估计。同时分析了投影流形过程在流形不同尺度上同时呈现短程依赖（SRD）与LRD的情形。通过模拟研究，以多重分形积分球面函数自回归滑动平均（SPHARMA(p,q)）过程为例，展示了两种估计方法的性能。