AI 中文总结
该研究针对新息属于稳定律吸引域的回火线性过程，推导了其部分和过程的极限定理，明确了不同回火强度下的归一化关系与极限性质，推广了未回火的对数边界定理。
AI 中文摘要
我们研究回火线性过程$X_{N,n}=\sum_{j=1}^{\infty}e^{-\lambda_Nj}\frac{\ell(j)}{j}\varepsilon_{n-j}$（其中$\lambda_N\downarrow0$，$\ell$是慢变函数，新息属于$\alpha$-稳定律的吸引域且$1<\alpha\leq2$）的部分和行为。滤波器$j^{-1}\ell(j)$代表可求和与幂律长记忆系数之间的对数边界。令$Q_N=\sum_{j=1}^{N}e^{-\lambda_Nj}\frac{\ell(j)}{j}$，$L(N)=\sum_{j=1}^{N}\frac{\ell(j)}{j}$，我们证明由$B_NQ_N$归一化的部分和过程收敛到与新息相关的稳定莱维运动。此外，当$N\lambda_N=O(1)$时$Q_N\sim L(N)$，当$N\lambda_N\to\infty$时$Q_N\sim L(1/\lambda_N)$。弱、中、强回火具有相同的一阶莱维极限，但归一化方式不同；在弱和中回火状态下，由$B_N\ell(N)$归一化的二阶余项，在有限维分布中收敛到对数回火稳定过程，在高斯有限矩情形下该收敛是泛函的。这些结果推广了未回火的对数边界定理，补充了现有回火线性过程的不变性原理。
英文摘要
We study the partial-sum behavior of tempered linear processes \[ X_{N,n}=\sum_{j=1}^{\infty}e^{-λ_Nj}\frac{\ell(j)}{j}\varepsilon_{n-j}, \qquad λ_N\downarrow0, \] where $\ell$ is slowly varying and the innovations belong to the domain of attraction of an $α$-stable law with $1<α\leq2$. The filter $j^{-1}\ell(j)$ represents the logarithmic boundary between summable and power-law long-memory coefficients. Let \[ Q_N=\sum_{j=1}^{N}e^{-λ_Nj}\frac{\ell(j)}{j}, \qquad L(N)=\sum_{j=1}^{N}\frac{\ell(j)}{j}. \] We prove that the partial-sum process, normalized by $B_NQ_N$, converges to the stable Lévy motion associated with the innovations. Moreover, \[ Q_N\sim L(N)\quad\text{if }Nλ_N=O(1), \qquad Q_N\sim L(1/λ_N)\quad\text{if }Nλ_N\to\infty. \] Then weak, moderate, and strong tempering have the same first-order Lévy limit but different normalizations. In the weakly and moderately tempered regimes, the second-order remainder, normalized by $B_N\ell(N)$, converges in finite-dimensional distributions to a logarithmically tempered stable process; in the Gaussian finite-moment case the convergence is functional. These results extend the untempered logarithmic-boundary theorem and complement existing invariance principles for tempered linear processes.
Comments37 pages
