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In this paper we show that US prices can be specified in terms of a time series model that contains roots simultaneously at zero and the cyclical frequencies. Using a general procedure for testing this type of hypothesis, the results show that the secular component in the US prices is nonstationary, with an order of integration ranging between 0.7 and 1.4. However, the cyclical part seems to be stationary, with the order of integration fluctuating around 0. This implies that shocks affecting the long run component will be highly persistent, while those affecting the cyclical part will disappear quickly.

Much attention has been given to the long-range dependence and fractal properties in network traffic engineering, and these properties are also widely observed in many fields of science and technologies. Traffic time series is conventionally characterized by its fractal dimension D, which is a measure for roughness, and by the Hurst parameter H, which is a measure for long-range dependence, see for examples (Refs. 10–12). Each property has been traditionally modeled and explained by self-affine random functions, such as fractional Gaussian noise (FGN)^{1,10–13,18,22–28} and fractional Brownian motion (FBM),^6,7 where a linear relationship between D and H, say D = 2 - H for one-dimensional series, links the two properties. The limitation of single parameter models (e.g., FGN) in long-range dependent (LRD) traffic modeling has been noticed as can be seen from Refs. 1, 18 and 25. Hence, models which can provide good fitting of LRD traffic for both short-term lags and long-term ones are worth studying due to the importance of accurate models of traffic in network communications.¹³ This letter utilizes a statistical model called the Cauchy correlation model to model LRD traffic. This model characterizes D and H separately, and it allows any combination of two within the constraint of LRD condition. It is a new power-law correlation model for LRD traffic modeling with its local and global behavior decoupling. Its flexibility in data modeling in comparison with a single parameter model of FGN is briefly discussed, and applications to LRD traffic modeling demonstrated.

Over the past few years, scaling phenomena involving self-similarity and heavy-tailed distributions have attracted the interest of various researchers in telecommunications and networks. In this paper, we study the linear fractional stable noise (LFSN) which exhibits both long-range dependence and heavy tails property. LFSN can be represented as a linear process with weight coefficients and α-stable random variables. The coefficients of the linear process are determined by a kernel function and depend on five parameters. This paper focuses on estimating two unknown parameters a and b. Based on minimizing square errors, several methods for estimating these two parameters are presented. Detailed tables and graphs have been included in extensive simulations which show the methods are good estimates.

The complexity of the cardiac rhythm is demonstrated to exhibit self-affine multifractal variability. The dynamics of heartbeat interval time series was analyzed by application of the multifractal formalism based on the Cramer theory of large deviations. The continuous multifractal large deviation spectrum uncovers the nonlinear fractal properties in the dynamics of heart rate and presents a useful diagnostic framework for discrimination and classification of patients with cardiac disease, e.g. congestive heart failure. The characteristic multifractal spectral pattern in heart transplant recipients or chronic heart disease highlights the importance of neuroautonomic control mechanisms regulating the fractal dynamics of the cardiac rhythm.

A method is developed for the automatic detection of the onset of scaling for long-range dependent (LRD) time series and other asymptotically scale-invariant processes. Based on wavelet techniques, it provides the lower cutoff scale for the regression that yields the scaling exponent. The method detects the onset of scaling through the dramatic improvement of a goodness-of-fit statistic taken as a function of this lower cutoff scale. It relies on qualitative features of the goodness-of-fit statistic and on features of the wavelet analysis. The method is easy to implement, appropriate for large data sets and highly robust. It is tested against 34 time series models and found to perform very well. Examples involving telecommunications data are presented.

A stationary time series is said to be long-range dependent (LRD) if its autocovariance function decays as a power of the lag, in such a way that the sum (over all lags) of the autocovariances diverges. The asymptotic rate of decay is determined by a parameter H, called the Hurst parameter. The time series is said to be short-range dependent (H = 1/2) if the sum converges. It is commonly believed that a random permutation of a sequence maintains the marginal distribution of each element but destroys the dependence, and in particular, that a random permutation of an LRD sequence creates a new sequence whose estimate of Hurst parameter H is close to 1/2. This paper provides a theoretical basis for investigating these claims. In reality, a complete random permutation does not destroy the covariances, but merely equalizes them. The common value of the equalized covariances depends on the length N of the original sequence and it decreases to 0 as N → ∞. Using the periodogram method, we explain why one is led to think, mistakenly, that the randomized sequence yields an "estimated H close to 1/2.".

In some applications, for instance, finance, biomechanics, turbulence or internet traffic, it is relevant to model data with a generalization of a fractional Brownian motion for which the Hurst parameter H is dependent on the frequency. In this contribution, we describe the multiscale fractional Brownian motions which present a parameter H as a piecewise constant function of the frequency. We provide the main properties of these processes: long-memory and smoothness of the paths. Then we propose a statistical method based on wavelet analysis to estimate the different parameters and prove a functional Central Limit Theorem satisfied by the empirical variance of the wavelet coefficients.

Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

The FARIMA models, which have long-range-dependence (LRD), are widely used in many areas. Through the derivation of a precise characterization of the spectrum and variance time function, we show that this family is very atypical among LRD processes, being extremely close to the fractional Gaussian noise in a precise sense which results in ultra-fast convergence to fGn under rescaling. Furthermore, we show that this closeness property is not robust to additive noise. We argue that the use of FARIMA, and more generally fractionally differenced time series, should be reassessed in some contexts, in particular when convergence rate under rescaling is important and noise is expected.

We study the asymptotic behavior of the aggregated fractional Ornstein–Uhlenbeck processes with random coefficients that follows a Gamma-type distribution. We also analyze the properties of the limit process.

Detrended fluctuation analysis (DFA) is used to examine long-range dependence in variations and volatilities of American treasury bills (TB) during periods of low and high movements in TB rates. Volatility series are estimated by generalized autoregressive conditional heteroskedasticity (GARCH) model under Gaussian, Student, and the generalized error distribution (GED) assumptions. The DFA-based Hurst exponents from 3-month, 6-month, and 1-year TB data indicates that in general the dynamics of the TB variations process is characterized by persistence during stable time period (before 2008 international financial crisis) and anti-persistence during unstable time period (post-2008 international financial crisis). For volatility series, it is found that; for stable period; 3-month volatility process is more likely random, 6-month volatility process is anti-persistent, and 1-year volatility process is persistent. For unstable period, estimation results show that the generating process is persistent for all maturities and for all distributional assumptions.

Financial markets are characterized by complex and often unpredictable dynamics, presenting significant challenges for investors, analysts, and policymakers. In recent years, fractal theory has emerged as a powerful tool for understanding the intricate patterns and behaviors exhibited by financial time series data. This paper provides a comprehensive review of the application of fractal theory in financial time series analysis, examining its theoretical foundations, empirical applications, and practical implications. Through a synthesis of relevant literature, we explore the utility of fractal techniques such as fractal dimension estimation, detrended fluctuation analysis (DFA), and multifractal analysis in quantifying the long-range dependence, self-similarity, and scaling properties of financial time series. Additionally, we discuss the implications of fractal dynamics for risk management, portfolio optimization, and market microstructure analysis, highlighting opportunities for future research and innovation in this evolving field.

China has taken important steps to reform its economy and capital markets in the past 20 years. Despite these efforts there is a lack of quantitative evidence on how these measures have impacted price returns in the stock exchanges. The purpose of this research was to determine the randomness of Chinese equity returns and to measure the scaling property of volatility over time. The main assumption of the Efficient Market Hypothesis is that security returns follow the path of a random walk and that volatility scales with the square root of time. This notion was tested by analyzing daily stock returns of the Shanghai- and Shenzhen Composite Indexes between 1990 and 2013. The Kolmogorov–Smirnov (KS) test rejected the log-normal distribution and random walk hypothesis. The measured Hurst exponents revealed a multiscaling property of fractal Brownian motion and indicated the presence of long-range dependence. The findings also showed that the degree of persistence and cycle length has reduced over time.

The long-range dependence of Internet traffic has been experimentally observed. One issue in handling long-range dependent traffic is how to simulate random traffic data with long-range dependence. The authors discuss a correlation-based simulator with a white noise input for generating long-range dependent traffic data. With the real TCP traffic traces, a simulation model of TCP arrival traffic is empirically developed and the experimental results are satisfactory.

This study investigates the volatility of the Euro-to-US Dollar exchange rate, specifically focusing on identifying long-memory characteristics. Through the analysis of daily data spanning from January 1, 2018, to January 10, 2023, the study uncovers a robust long-memory feature. Supporting this exploration, the study endorses the use of sophisticated models such as Fractionally Integrated Generalized Autoregressive Conditionally Heteroskedastic (FIGARCH) and Hyperbolic Generalized Autoregressive Conditionally Heteroskedastic (HYGARCH), incorporating both student and skewed student innovation distributions. The results underscore the superior performance of FIGARCH and HYGARCH models, particularly when coupled with a skewed student distribution. This collaborative approach enhances the predictability of crucial financial metrics, including Value at Risk (VaR) and Expected Shortfall (ESF), for both long and short trading positions. Significantly, the FIGARCH model, when utilizing a skewed student distribution, demonstrates exceptional predictive power. This outcome challenges the efficient market hypothesis and suggests the potential for generating outstanding returns. In light of these findings, this research contributes valuable insights for comprehending and navigating the intricacies of the Euro-to-US Dollar exchange rate, providing a forward-looking perspective for financial practitioners and researchers alike.

Fractional Brownian motion has become a standard tool to address long-range dependence in financial time series. However, a constant memory parameter is too restrictive to address different market conditions. Here, we model the price fluctuations using a multifractional Brownian motion assuming that the Hurst exponent is a time-deterministic function. Through the multifractional Itô calculus, both the related transition density function and the analytical European Call option pricing formula are obtained. The empirical performance of the multifractional Black–Scholes model is tested by calibration of option market quotes for the SPX index and offers best fit than its counterparts based on standard and fractional Brownian motions.

Let w (x, t) := (u, v)(x, t), x ∈ ℝ³, t > 0, be the ℝ²-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), u_t(x,0), and v(x,0), v_t(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

Starting from the moving average representation of fractional Brownian motion, there are two different approaches to constructing fractional Lévy processes in the literature. Applying L²-integration theory, one can keep the same moving average kernel and replace the driving Brownian motion by a pure jump Lévy process with finite second moments. Alternatively, in the framework of alpha-stable random measures, the Brownian motion is replaced by an alpha-stable Lévy process and the exponent in the kernel is reparametrized by H - 1/α. We now provide a unified approach taking kernels of the form , where γ can be chosen according to the existing moments and the Blumenthal–Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g., regularity of the sample paths and the semimartingale property.

We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square integrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér–von-Mises statistic.

We use a wavelet based estimator for the parameter of long-range dependent process to analyze the scaling nature of internet traffic. Wherever and whenever these internet traffic are collected from, they exhibit long-range dependent properties. And the range of Hurst parameter is [0.5,1].