arXiv subjects

Farzad Sabzikar

Publications and source records attributed to Farzad Sabzikar.

10 recordsLinked to original sources

arXiv2025

Testing of tempered fractional Brownian motions

We propose here a testing methodology based on the autocovariance, detrended moving average, and time-averaged mean-squared displacement statistics for tempered fractional Brownian motions (TFBMs) which are related to the notions of semi-long range dependence and transient anomalous diffusion. In this framework, we consider three types of TFBMs: two with a tempering factor incorporated into their moving-average representation, and one with a tempering parameter added to the autocorrelation formula. We illustrate their dynamics with the use of quantile lines. Using the proposed methodology, we provide a comprehensive power analysis of the tests. It appears that the tests allow distinguishing between the tempered processes with different Hurst parameters.

stat.ME↗

arXiv2021

Statistical inference for ARTFIMA time series with stable innovations

Autoregressive tempered fractionally integrated moving average with stable innovations modifies the power-law kernel of the fractionally integrated time series model by adding an exponential tempering factor. The tempered time series is a stationary model that can exhibits semi-long-range dependence. This paper develops the basic theory of the tempered time series model, including dependence structure and parameter estimation.

stat.AP↗

arXiv2020

How does tempering affect the local and global properties of fractional Brownian motion?

The present paper investigates the effects of tempering the power law kernel of moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer-Major theorem, asymptotic behavior of the 3rd and 4th cumulants of fBm and the optimal fourth moment theorem.

math.PR↗

arXiv2020

Asymptotic theory for regression models with fractional local to unity root errors

This paper develops the asymptotic theory for parametric and nonparametric regression models when the errors have a fractional local to unity root (FLUR) model structure. FLUR models are stationary time series with semi-long range dependence property in the sense that their covariance function resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a noncentrality parameter. When this parameter is sample size dependent, the asymptotic normality for these regression models admit a wide range of stochastic processes with behavior that includes long, semi-long, and short memory processes.

math.ST↗

arXiv2019

On fractional Lévy processes: tempering, sample path properties and stochastic integration

We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP $I\!I$, respectively). TFLP and TFLP $I\!I$ make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP $I\!I$ display semi-long range dependence. We establish the sample path properties of TFLP and TFLP $I\!I$. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP $I\!I$, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.

math.PR↗

arXiv2018

Tempered fractional Brownian motion: wavelet estimation, modeling and testing

The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-à-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a canonical model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays semi-long range dependence (hyperbolic and quasi-exponential decays over moderate and large scales, respectively), a phenomenon that has been observed in wide a range of applications from wind speeds to geophysics to finance. In this paper, we use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology to the analysis of geophysical flow data shows that tfBm provides a much closer fit than fBm.

math.ST↗

arXiv2017

Invariance Principles for Tempered Fractionally Integrated Processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in $α$-stable $(1< α\le 2)$ i.i.d. innovations and related tempered linear processes with vanishing tempering parameter $λ\sim λ_*/N$. We show that the limit of the partial sums process takes a different form in the weakly tempered ($λ_* = 0$), strongly tempered ($λ_* = \infty$), and moderately tempered ($0<λ_* < \infty$) cases. These results are used to derive the limit distribution of the OLS estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.

math.PR↗

arXiv2017

Tempered fractional Brownian and stable motions of second kind

Meerschaert and Sabzikar [12], [13] introduced tempered fractional Brownian/stable motion (TFBM/TFSM) by including an exponential tempering factor in the moving average representation of FBM/FSM. The present paper discusses another tempered version of FBM/FSM, termed tempered fractional Brownian/stable motion of second kind (TFBM II/TFSM II).We prove that TFBM/TFSM and TFBM II/TFSM II are different processes. Particularly, large time properties of TFBM II/TFSM II are similar to those of FBM/FSM and are in deep contrast to large time properties of TFBM/TFSM.

math.PR↗

arXiv2016

Tempered Hermite process

A tempered Hermite process modifies the power law kernel in the time domain representation of a Hermite process by multiplying an exponential tempering factor $λ>0$ such that the process is well defined for Hurst parameter $H>\frac{1}{2}$. A tempered Hermite process is the weak convergence limit of a certain discrete chaos process.

math.PR↗

arXiv2014

Invariance principle for tempered fractional time series models

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^{\fracλ{n}}_{k}\rightarrow \int_{\rr}f(u)Z^{1}_{H,λ}(du) \end{equation*} in distribution, as $n\to\infty$, where $X_{k}$ is an ARTFIMA time series and $Z^{1}_{H,λ}$ is $THP^{1}$.

math.PR↗