具有幂律强度的长记忆马尔可夫链
Long-memory Markov chains with power-law intensities
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中文总结 AI 辅助
研究具有幂律强度动态的自激发点过程，构建允许有限维马尔可夫状态表示的模型，在温和条件下建立相关性质，证明潜在状态过程正 Harris 常返，模拟结果显示该模型在稳定区域边界附近有长记忆行为。
中文摘要 AI 辅助
我们引入了一个具有幂律强度动态的自激发点过程，它允许有限维马尔可夫状态表示。该模型在保留幂律霍克斯过程的局部跳跃和斜率更新结构的同时，用控制强度动态的非线性马尔可夫链取代了全局历史依赖性。在一般状态空间框架下，我们在到达间隔时间分布的温和正则条件下建立了不可约性、非周期性和T链性质。在明确的稳定性条件下，我们进一步证明了潜在状态过程是正 Harris 常返的，确保了唯一不变分布的存在。基于局部 Whittle 估计器的模拟结果表明，所提出的马尔可夫强度模型在稳定区域边界附近表现出长记忆行为。
英文摘要
We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.
