Invariance principle for tempered fractional time series models

2014/07/15 by Farzad Sabzikar, Sabzikar, Farzad
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Fractional Differential Equations Solutions #Probability (math.PR) #Stochastic processes and financial applications #math.PR

paper · pdf · doi:10.48550/arxiv.1407.4109

32 pages, 1 Figure, This new version is the replacement of the previous version "Tempered Hermite Process"; some major revisions implemented throughout

openalex publication_date 2014/07/15 · arxiv created 2014/08/19 · arxiv updated 2014/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, THP¹, which is well-defined for any H>(1)/(2). When (1)/(2)<H<1, we develop the Wiener integral with respect to THP¹ to provide the sufficient condition for the convergence n^-H∑_k=0^+∞f((k)/(n))X^\fracλn_k→ ∫_\rrf(u)Z¹_H,λ(du) in distribution, as n→∞, where X_k is an ARTFIMA time series and Z¹_H,λ is THP¹.

Citations

• Fractional Brownian Motions, Fractional Noises and Applications
• Tempered fractional Brownian motion

Related

• Invariance Principles for Tempered Fractionally Integrated Processes
• Statistical inference for ARTFIMA time series with stable innovations
• On fractional L 'evy processes: tempering, sample path properties and\n stochastic integration
• Testing of tempered fractional Brownian motions
• Functional Limit Theorems of moving averages of Hermite processes and an application to homogenization