学习近不稳定重尾霍克斯过程中的外生速率但无法学习临界距离

中文摘要 AI 辅助

在Jaisson和Rosenbaum的近不稳定重尾机制中，线性霍克斯过程的强度收敛于一个粗糙的平方根Volterra过程。我们研究长度为$T$的记录揭示了关于临界相对距离$1-\rho$和外生速率$\mu$的哪些信息。对于前者，信息由$\mathcal I_T=T\mu(1-\rho)$决定，而非事件数量；尽管事件数量以$T^{2\gamma}$发散，$\mathcal I_T$保持有界，因此不存在对临界相对距离的估计量是均匀局部一致的。对于后者，Fisher信息发散，因为粗糙极限在零处有原子：渐近信息几何在外生方向上是奇异的，且与轻尾情形不同，没有Feller型阈值。我们证明联合最大似然估计器$(\mu,\rho)$（其似然函数是联合凹的）以随机速率$\{Q_T/\log\log Q_T\}^{1/2}$一致地恢复$\mu$，该速率由观测信息$Q_T$给出，而其相对边际的估计仅是紧的：外生速率被学习到，尽管与之耦合的内生参数未被学习。对于空起始记录，$Q_T$发散，因为从零开始的粗糙极限在零处花费正时间；在起始后很长时间观测的窗口上，尽管分支比仍然未知，可学习性以趋于1的概率成立（随着窗口增长），这通过粗糙极限的遍历性及其平稳律的原子实现。我们还证明了平稳强度向平稳粗糙Volterra过程的有限维收敛。

英文摘要

In the nearly unstable heavy-tailed regime of Jaisson and Rosenbaum, where the intensity of a linear Hawkes process converges to a rough square-root Volterra process, we ask what a record of length $T$ reveals about the relative distance to criticality $1-ρ$ and the exogenous rate $μ$. For the first, the information is governed by $\mathcal I_T=Tμ(1-ρ)$, not by the number of events; $\mathcal I_T$ stays bounded although the event count diverges like $T^{2γ}$, so no estimator of the relative distance to criticality is uniformly locally consistent. For the second, the Fisher information diverges because the rough limit has an atom at zero: the asymptotic information geometry is singular in the exogenous direction and, unlike the light-tailed case, has no Feller-type threshold. We prove that the joint maximum likelihood estimator of $(μ,ρ)$, whose likelihood is jointly concave, recovers $μ$ consistently at the random rate $\{Q_T/\log\log Q_T\}^{1/2}$ given by the observed information $Q_T$, while its estimate of the relative margin is only tight: the exogenous rate is learned although the endogenous parameter it is coupled to is not. For empty-start records $Q_T$ diverges because the rough limit started at zero spends positive time at zero; on windows observed long after the start, still with the branching ratio unknown, learnability holds with probability tending to one as the window grows, through the ergodicity of the rough limit and the atom of its stationary law. We also prove finite-dimensional convergence of the stationary intensity to the stationary rough Volterra process.

发表机构

• Universidad del Desarrollo（智利发展大学）

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