Invariance principle for tempered fractional time series models

Abstract

Autoregressive tempered fractionally integrated moving average (ARTFIMA) time series is a useful model for velocity data in turbulence flows. In this paper, we obtain an invariance principle for the partial sum of an ARTFIMA process. The limiting process is called tempered Hermite process of order one, $THP^{1}$, which is well-defined for any $H>\frac{1}{2}$. When $\frac{1}{2}<H<1$, we develop the Wiener integral with respect to $THP^{1}$ to provide the sufficient condition for the convergence \begin{equation*} n^{-H}\sum_{k=0}^{+\infty}f\Big(\frac{k}{n}\Big)X^{\frac{\lambda}{n}}_{k}\rightarrow \int_{\rr}f(u)Z^{1}_{H,\lambda}(du) \end{equation*} in distribution, as $n\to\infty$, where $X_{k}$ is an ARTFIMA time series and $Z^{1}_{H,\lambda}$ is $THP^{1}$.