Invariance principle for tempered fractional time series models

TL;DR

This paper establishes an invariance principle for ARTFIMA time series, showing that their scaled partial sums converge to a tempered Hermite process, extending the understanding of limit behaviors in tempered fractional models.

Contribution

It introduces a new invariance principle for ARTFIMA processes, defining the limiting tempered Hermite process of order one and developing the Wiener integral for it.

Findings

Convergence of scaled partial sums to the tempered Hermite process.

Development of Wiener integral with respect to THP^{1}.

Provides conditions for distributional convergence of ARTFIMA sums.

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 $THP^{1}$.