Limit Theorems for Tempered Linear Processes with Innovations in the Domain of Attraction of a Stable Law

Abstract

We study the partial-sum behavior of tempered linear processes \[ X_{N,n}=\sum_{j=1}^{\infty}e^{-\lambda_Nj}\frac{\ell(j)}{j}\varepsilon_{n-j}, \qquad \lambda_N\downarrow0, \] where $\ell$ is slowly varying and the innovations belong to the domain of attraction of an $\alpha$-stable law with $1<\alpha\leq2$. The filter $j^{-1}\ell(j)$ represents the logarithmic boundary between summable and power-law long-memory coefficients. Let \[ Q_N=\sum_{j=1}^{N}e^{-\lambda_Nj}\frac{\ell(j)}{j}, \qquad L(N)=\sum_{j=1}^{N}\frac{\ell(j)}{j}. \] We prove that the partial-sum process, normalized by $B_NQ_N$, converges to the stable L\'{e}vy motion associated with the innovations. Moreover, \[ Q_N\sim L(N)\quad\text{if }N\lambda_N=O(1), \qquad Q_N\sim L(1/\lambda_N)\quad\text{if }N\lambda_N\to\infty. \] Then weak, moderate, and strong tempering have the same first-order L\'{e}vy 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.