Gaussian Self-similar Processes Research Articles

Path Properties of a Generalized Fractional Brownian Motion

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power-law shape function and non-stationary noises with a power-law variance function. In this paper, we study sample path properties of the generalized fractional Brownian motion, including Hölder continuity, path differentiability/non-differentiability, and functional and local law of the iterated logarithms.

Testing of fractional Brownian motion in a noisy environment

Fractional Brownian motion (FBM) is the only Gaussian self-similar process with stationary increments. Its increment process, called fractional Gaussian noise, is ergodic and exhibits a property of power-like decaying autocorrelation function (ACF) which leads to the notion of long memory. These properties have made FBM important in modelling real-world data recorded in different experiments ranging from biology to telecommunication. These experiments are often disturbed by a noise which source can be just the instrument error. In this paper we propose a rigorous statistical test based on the ACF for FBM with added white Gaussian noise. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power and comparing with other tests existing in the literature. We also note that the introduced test procedure can be applied to an arbitrary Gaussian process.

Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.

On the Self-Similar Nature of ATM Network Traffic

Modeling multimedia traffic is an important issue in performance analysis and design of communication networks. With introduction of new applications, the characteristics of data traffic changes. In this paper, a measurement study of ATM Network traffic has been carried out and it is shown that the recorded data exhibit self-similar features. The conclusions are supported by a comprehensive analysis using one of the most popular statistical methods called Indices of Dispersion. Our results validate one of the most striking findings of the present teletraffic research: a broad range of packet network traffic has fractal-like behavior. We also investigate three popular synthetic self-similar traffic models and find out the most accurate one for the measured traffic. Keywords: ATM, Modeling, Self-similar traffic, Indices of Dispersion, Hurst parameter, Performance Testing. Introduction The characterization of real traffic is a critical issue to the success of efficient traffic engineering in ATM networks. Research in this field has resulted in numerous models and techniques over the last decade (Bjorkman, Latour-Henner, Hasson, pers, and Miah, 1995; Stamoulis, Anagostu, and Georgantas, 1994). However, the developed traffic characterization methods have been, in general, rather complicated and demand intensive computation of several statistical parameters. There is a lack of simple and accurate methods that can be of practical use to network operators. Extensive data studies indicate that traffic in high-speed communications networks has long-memory and heavy tailed (impulsive) characteristics. With the rising popularity of multimedia applications over networks, these properties of the traffic are only likely to become more dominant, posing unique new challenges to designers of network systems and protocols. Traditional teletraffic theory cannot capture these traffic characteristics. During the last few years, significant research results have been proposed on models that capture self-similarity of traffic. These models, however, are inadequate for predicting queuing performance, delays, and buffer dimensions since the implications of the combination of self-similarity and impulsiveness queuing performance can be dramatically different from that of self-similarity alone. To the best of my knowledge there are no models that have been derived based on real traffic dynamics that also capture the data impulsiveness? The modeling of self-similar traffic appeared as an emerging and challenging field of the present teletraffic research. It seems that there are different promising approaches to capture this complex fractal-like behavior. Norros (1993, 1995) used a Gaussian self-similar process known as the Fractional Brownian Motion. Willinger, Taqqu, Sherman and Wilson (2004) applied the superposition of on/off sources with heavy-tailed on and off period. Erramilli (1994) and Singh and Erramilli (1999) studied different chaotic maps. The rest of this paper is organized as follows. In section 2 we give information about the self-similar traffic characteristics and the used techniques for measuring the self-similarity level (Hurst-parameter H). In section 3 we present the analysis based on the real measurements taken from the Eastern Mediterranean University (EMU) ATM network and find out the self-similarity level. In section 4 we investigate the three promising self-similar modeling approaches to capture the observed properties, and we find out the most appropriate model for EMU ATM network traffic. Finally, section 5 summarizes the results of the paper and identifies areas for future research. The Self-Similarity Phenomena and its Testing A self-similar phenomenon represents a process displaying structural similarities across a wide range of scales of a specific dimension. In other words, the reference structure is repeating itself over a wide range of scales of diverse dimensions (geometrical, or statistical, or temporal). …

Identifying the Anisotropical Function of a d-Dimensional Gaussian Self-similar Process with Stationary Increments

We perform the estimation of the anisotropical function of a Gaussian self-similar process with stationary increments.

Can continuous-time stationary stable processes have discrete linear representations?

We show that a non-trivial continuous-time strictly α-stable, α∈(0,2), stationary process cannot be represented in distribution as a discrete linear process ∑ n=−∞ ∞ f t(n)ε n, t∈ R, where {f t} t∈ R is a collection of deterministic functions and {ε n} n∈ Z are independent strictly α-stable random variables. Analogous results hold for self-similar strictly α-stable processes and for strictly α-stable processes with stationary increments. As a consequence, the usual wavelet decomposition of Gaussian self-similar processes cannot be extended to the α-stable, α<2 case.

Rates of clustering for some Gaussian self-similar processes

The analogue of Strassen's functional law of the iterated logarithm in known for many Gaussian processes which have suitable scaling properties, and here we establish rates at which this convergence takes place. We provide a new proof of the best upper bound for the convergence toK by suitably normalized Brownian motion, and then continue with this method to get similar bounds for the Brownian sheet and other self-similar Gaussian processes. The previous method, which produced these results for Brownian motion in ℝ1, was highly dependent on many special properties unavailable when dealing with other Gaussian processes.

A representation for self-similar processes

A self-similar process Z( t) has stationary increments and is invariant in law under the transformation Z( i)→ c - H Z( ct), c⩾0. The choice 1 2 <H<1 ensures that the increments of Z( t) exhibit a long range positive correlation. Mandelbrot and Van Ness investigated the case where Z( t) is Gaussian and represented that Gaussian self-similar process as a fractional integral of Brownian motion. They called it fractional Brownian motion. This paper provides a time-indexed representation for a sequence of self- similar processes Z ̄ m(t) , m=1,2,…, whose finite-dimensional moments have been specified in an earlier paper. Z ̄ 1(t) is the Gaussian fractional Brownian motion but the process Z ̄ m(t) are not Gaussian when m⩾2. Self-similar processes are being studied in physics, in the context of the renormalization group theory for critical phenomena, and in hydrology where they account for the so-called “Hurst effect”.