发表机构
Faculty of Engineering, Universidad del Desarrollo（发展大学工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对长记忆霍克斯过程的分支比率推断，提出极小极大下界与块分散估计器，并给出自适应规则及诚实置信区间。
AI 中文摘要
自激点过程的分支比率 $\rho$ 决定了到临界边界的分支裕度 $1-\rho$。我们研究从有限记录中对平稳霍克斯过程（其核完全单调且慢弛豫质量在零速率附近服从包络）的 $\rho$ 进行推断。慢激发可被交换为外生移民；对 Kullback--Leibler 率的谱界以及对未观测过去的耦合论证给出了阶为 $T^{-\gamma/(2\gamma+1)}$ 的极小极大下界，其中包络为幂律且尾指数为 $\gamma$。一种块分散估计器（块长随 $T^{1/(2\gamma+1)}$ 增长）达到该速率。通过簇表示和泊松空间上的二阶 Poincaré 不等式获得的定量中心极限定理，产生了具有接近极小极大期望长度的诚实置信区间。对于未知的 $\gamma$，一种 Lepski 型规则以对数代价在概率上自适应；记忆类别之间的自由自适应是不可能的，且诚实区间无法适应更轻的记忆。模拟与速率一致。
英文摘要
The branching ratio $ρ$ of a self-exciting point process determines the branching margin $1-ρ$ to the critical boundary. We study inference on $ρ$ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback--Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order $T^{-γ/(2γ+1)}$ for a power-law envelope with tail exponent $γ$. A block-dispersion estimator whose block length grows as $T^{1/(2γ+1)}$ attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown $γ$, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.
