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About 3,746 papers for “long-range dependent processes”
Jean‐Marc Bardet, Gabriel Lang, Georges Oppenheim et al. · HAL (Le Centre pour la Communication Scientifique Directe) · 2003
We describe methods for generating discretized sample paths of long-range dependent processes such as fractional Brownian motion, its increment fractional Gaussian noise and FARIMA processes.
Cited by 94
D. J. Daley · The Annals of Probability · 1999
A stationary renewal process $N(\cdot)$ for which the lifetime distribution has its $k$th moment finite or infinite according as $k$ is less than or greater than $\kappa$ for some $1 \lt \kappa \lt 2$, is long-range dependent and has Hurst index $\alpha=1/2(3-\kappa)$ (this is the critical index $\alpha$ for which $\lim\sup_{t \to \infty} t^{-2a}$ var $N(0,t]$ is finite or infinite according as $\alpha$ is greater than or less than $\alpha$. This identification is accomplished by delineating the growth rate properties of the difference between the renewal function and its linear asymptote, thereby extending work of Täcklind.
Cited by 17
Gennady Samorodnitsky, Yizao Wang · The Annals of Probability · 2019
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $\alpha$-Fréchet distribution and the skewed $\alpha$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and $\beta\in(0,1)$, with representations based on intersections of independent $\beta$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-\alpha$. The intriguing structure of these random sup-measures is due to intersections of independent $\beta$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $\beta$ increases to one. The results in this paper extend substantially previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have been considered.
Cited by 25Open access
Gennady Samorodnitsky, Yizao Wang · arXiv (Cornell University) · 2017
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $α$-Fréchet distribution and the skewed $α$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $α\in(0,\infty)$ and $β\in(0,1)$, with representations based on intersections of independent $β$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-α$. The intriguing structure of these random sup-measures is due to intersections of independent $β$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $β$ increases to one. The results in this paper extend substantially previous investigations where only $α\in(0,2)$ and $β\in(0,1/2)$ have been considered.
Cited by 2Open access
Guilherme Pumi, Sílvia R. C. Lopes · arXiv (Cornell University) · 2013
In this paper we propose a generalization of a class of Gaussian Semiparametric Estimators (GSE) of the fractional differencing parameter for long-range dependent multivariate time series. We generalize a known GSE-type estimator by introducing some modifications at the objective function level regarding the process' spectral density matrix estimator. We study large sample properties of the estimator without assuming Gaussianity as well as hypothesis testing. The class of models considered here satisfies simple conditions on the spectral density function, restricted to a small neighborhood of the zero frequency. This includes, but is not limited to, the class of VARFIMA models. A simulation study to assess the finite sample properties of the proposed estimator is presented and supports its competitiveness. We also present an empirical application to an exchange rate data.
Cited by 1Open access
Andrew Feutrill, Matthew Roughan · arXiv (Cornell University) · 2021
A quantity of interest to characterise continuous-valued stochastic processes is the differential entropy rate. The rate of convergence of many properties of LRD processes is slower than might be expected, based on the intuition for conventional processes, e.g. Markov processes. Is this also true of the entropy rate? In this paper we consider the properties of the differential entropy rate of stochastic processes that have an autocorrelation function that decays as a power law. We show that power law decaying processes with similar autocorrelation and spectral density functions, Fractional Gaussian Noise and ARFIMA(0,d,0), have different entropic properties, particularly for negatively correlated parameterisations. Then we provide an equivalence between the mutual information between past and future and the differential excess entropy for stationary Gaussian processes, showing the finiteness of this quantity is the boundary between long and short range dependence. Finally, we analyse the convergence of the conditional entropy to the differential entropy rate and show that for short range dependence that the rate of convergence is of the order $O(n^{-1})$, but it is slower for long range dependent processes and depends on the Hurst parameter.
Cited by 4Open access
Jan Mielniczuk, Piotr Wojdyłło · IEEE Transactions on Information Theory · 2007
<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We consider a discrete-time stationary long-range dependent process <emphasis><formula formulatype="inline"><tex>$(X_k)_{k\in Z}$</tex></formula></emphasis> such that its spectral density equals <emphasis><formula formulatype="inline"> <tex>${\varphi}(\vert {\lambda}\vert)^{-2d}$</tex></formula></emphasis>, where <emphasis><formula formulatype="inline"><tex>${\varphi}$</tex></formula></emphasis> is a smooth function such that <emphasis><formula formulatype="inline"><tex>${\varphi}(0)={\varphi}^{\prime\prime}(0)=0$</tex> </formula></emphasis> and <emphasis><formula formulatype="inline"><tex>${\varphi}({\lambda})\geq c{\lambda}$</tex></formula></emphasis> for <emphasis><formula formulatype="inline"> <tex>${\lambda}\in [0,\pi]$</tex></formula></emphasis>. Then for any wavelet <emphasis><formula formulatype="inline"><tex>$\psi$</tex></formula></emphasis> with <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis> vanishing moments, the lag <emphasis><formula formulatype="inline"><tex>$k$</tex></formula></emphasis> within-level covariance of wavelet coefficients decays as <emphasis><formula formulatype="inline"> <tex>${\cal O}(k^{2d-2N-1})$</tex></formula></emphasis> when <emphasis><formula formulatype="inline"><tex>$k\to\infty$</tex></formula></emphasis>. The result applies to fractionally integrated autoregressive moving average (ARMA) processes as well as to fractional Gaussian noise. </para>
Cited by 16
D. J. Daley · The Annals of Probability · 1999
A stationary renewal process $N(\cdot)$ for which the lifetime distribution has its $k$th moment finite or infinite according as $k$ is less than or greater than $\kappa$ for some $1 \lt \kappa \lt 2$, is long-range dependent and has Hurst index $\alpha=1/2(3-\kappa)$ (this is the critical index $\alpha$ for which $\lim\sup_{t \to \infty} t^{-2a}$ var $N(0,t]$ is finite or infinite according as $\alpha$ is greater than or less than $\alpha$. This identification is accomplished by delineating the growth rate properties of the difference between the renewal function and its linear asymptote, thereby extending work of Täcklind.
Cited by 28Open access
Nuno Crato, Bonnie K. Ray · Journal of Forecasting · 1996
Fractionally integrated autoregressive moving-average (ARFIMA) models have proved useful tools in the analysis of time series with long-range dependence. However, little is known about various practical issues regarding model selection and estimation methods, and the impact of selection and estimation methods on forecasts. By means of a large-scale simulation study, we compare three different estimation procedures and three automatic model-selection criteria on the basis of their impact on forecast accuracy. Our results endorse the use of both the frequency-domain Whittle estimation procedure and the time-domain approximate MLE procedure of Haslett and Raftery in conjunction with the AIC and SIC selection criteria, but indicate that considerable care should be exercised when using ARFIMA models. In general, we find that simple ARMA models provide competitive forecasts. Only a large number of observations and a strongly persistent time series seem to justify the use of ARFIMA models for forecasting purposes.
Cited by 7
Hae-Duck J. Jeong · Journal of the Korean Physical Society · 2009
Cited by 0
D. J. Daley, Tomasz Rolski, Rein Vesilo · Advances in Applied Probability · 2000
For a stationary long-range dependent point process N (.) with Palm distribution P 0 , the Hurst index H ≡ sup{ h : lim sup t →∞ t -2 h var N (0, t ] = ∞} is related to the moment index κ ≡ sup{ k : E 0 ( T k ) < ∞} of a generic stationary interval T between points (E 0 denotes expectation with respect to P 0 ) by 2 H + κ ≥ 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which κ < 2 is necessarily long-range dependent with Hurst index greater than ½. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., E 0 ( T 2 ) < ∞.
Cited by 8
Ming Li, Jun‐Sheng Duan, Mohammad Heydari et al. · Frontiers in Physics · 2024
Keywords: long-range dependent processes, fractional Gaussian noise, generalized cauchy process, 1/f noise, fractal time series
Cited by 1Open access
Andrew Feutrill, Matthew Roughan · arXiv (Cornell University) · 2021
A quantity of interest to characterise continuous-valued stochastic processes\nis the differential entropy rate. The rate of convergence of many properties of\nLRD processes is slower than might be expected, based on the intuition for\nconventional processes, e.g. Markov processes. Is this also true of the entropy\nrate?\n In this paper we consider the properties of the differential entropy rate of\nstochastic processes that have an autocorrelation function that decays as a\npower law. We show that power law decaying processes with similar\nautocorrelation and spectral density functions, Fractional Gaussian Noise and\nARFIMA(0,d,0), have different entropic properties, particularly for negatively\ncorrelated parameterisations. Then we provide an equivalence between the mutual\ninformation between past and future and the differential excess entropy for\nstationary Gaussian processes, showing the finiteness of this quantity is the\nboundary between long and short range dependence. Finally, we analyse the\nconvergence of the conditional entropy to the differential entropy rate and\nshow that for short range dependence that the rate of convergence is of the\norder $O(n^{-1})$, but it is slower for long range dependent processes and\ndepends on the Hurst parameter.\n
Cited by 8Open access
Hira L. Koul, K. Mukherjee · Journal of Multivariate Analysis · 1994
Cited by 40
Yeng Nordman Miller-Chang · 2020
Cited by 0
Johannes Tewes · Electronic Journal of Statistics · 2017
We consider the change-point problem for the marginal distribution of subordinated Gaussian processes that exhibit long-range dependence. The asymptotic distributions of Kolmogorov-Smirnov- and Cramér-von Mises type statistics are investigated under local alternatives. By doing so we are able to compute the asymptotic relative efficiency of the mentioned tests and the CUSUM test. In the special case of a mean-shift in Gaussian data it is always $1$. Moreover, our theory covers the scenario where the Hermite rank of the underlying process changes. In a small simulation study, we show that the theoretical findings carry over to the finite sample performance of the tests.
Cited by 1Open access
Richard G. Clegg · arXiv (Cornell University) · 2006
The aim of this paper is to use a very simple queuing model to compare a number of models from the literature which have been used to replicate the statistical nature of internet traffic and, in particular, the long-range dependence of this traffic. The four models all have the form of discrete time Markov-modulated processes (two other models are introduced for comparison purposes). While it is often stated that long-range dependence has a critical effect on queuing performance, it appears that the models used here do not well replicated the queuing performance of real internet traffic. In particular, they fail to replicate the mean queue length (and hence the mean delay) and the probability of the queue length exceeding a given level.
Cited by 10Open access
É. Youndjé, Philippe Vieu · SPIRE - Sciences Po Institutional REpository · 2006
This note investigates the consistency properties of the kernel-type estimator of a quantile, in the setting of a long memory stationary stochastic process. Under a general long-range dependence situation (without any restriction of gaussian type) we give consistency results, and rates of convergence. An interesting by-product of this paper is a new consistency result for kernel-type estimator of a smooth distribution function (with rates) over the whole real line.
Cited by 0
Crouse, Matthew, Baraniuk, Richard G., Crouse, Matthew et al. · 1999
1/f noise and statistically self-similar processes such as fractional Brownian motion (fBm) are vital for modeling numerous real-world phenomena, from network traffic to DNA to the stock market. Although several algorithms exist for synthesizing discrete-time samples of a 1/f process, these algorithms are inexact, meaning that the covariance of the synthesized processes can deviate significantly from that of a true 1/f process. However, the Fast Fourier Transform (FFT) can be used to exactly and efficiently synthesize such processes in O(N log N) operations for a length-N signal. Strangely enough, the key is to apply the FFT to match the target process&apos;s covariance structure, not its frequency spectrum. In this paper, we prove that this FFT-based synthesis is exact not only for 1=f processes such as fBm, but also for a wide class of long-range dependent processes. Leveraging the flexibility of the FFT approach, we develop new models for processes that exhibit one type of fBm scaling be...
Cited by 4
Feutrill, Andrew Robert · Adelaide Research & Scholarship (AR&S) (University of Adelaide) · 2023
Much of the theory of random processes has been developed with the assumption that distant time periods are weakly correlated. However, it has been discovered in many real-world phenomena that this assumption is not valid. These findings have resulted in extensive research interest into stochastic processes that have strong correlations that persist over long time periods. This phenomenon is called long range dependence. This phenomena has been defined in the time and frequency domains by the slow decay of their autocorrelation function and the existence of a pole at the origin of the spectral density function, respectively. Information theory has proved very useful in statistics and probability theory. However, there has not been much research into the information theoretic properties and characterisations of this phenomena.This thesis characterises the phenomena of long range dependence, for discrete and continuous-valued stochastic processes in discrete time, by an information theoretic measure, the entropy rate. The entropy rate measures the amount of information contained in a stochastic process on average, per random variable. Common characterisations of long range dependence in the time and frequency domains are given by the slow convergence to quantities of interest, such as the sample mean. We show that this type of behaviour is present in the entropy rate function, by showing that long range dependence also has slow convergence of the conditional entropy to the entropy rate, due to some entropic quantities diverging to infinity. As an extension we show for classes of Gaussian processes and Markov chains that long range dependence by an infinite amount of shared information between the past and future of a stochastic process. The slow convergence has the impact of making accurate estimation of the differential entropy rate on data from long range dependent processes difficult, to the extent that existing techniques either are not accurate or are computationally intensive. We introduce a new estimation technique, that is able to balance these two concerns and make quick and accurate estimates of the differential entropy rate from continuous-valued data. We develop and utilise a connection between the differential entropy rate and the Shannon entropy rate of its quantised process as the basis of the estimation technique. This allows us to draw on the extensive research into Shannon entropy rate estimation on discrete-valued data, and we show that properties for the differential entropy rate estimator can be inherited from the choice of Shannon entropy rate estimator.
Cited by 0