Review Article | Open Access

# On Noise

**Academic Editor:**Carlo Cattani

#### Abstract

Due to the fact that noise gains the increasing interests in the field of biomedical signal processing and living systems, we present this introductive survey that may suffice to exhibit the elementary and the particularities of noise in comparison with conventional random functions. Three theorems are given for highlighting the particularities of noise. The first says that a random function with long-range dependence (LRD) is a noise. The secondindicates that a heavy-tailed random function is in the class of noise. The third provides a type of stochastic differential equations that produce noise.

#### 1. Introduction

The pioneering work of noise may refer to the paper by Schottky [1], where he introduced the concept of two classes of noise. One class is thermal noise, such as the random motion of molecules in the conductors. The other is the shot noise, which may be caused by randomness of the emission from the cathode and the randomness of the velocity of the emitted electrons [1, 2]. Johnson described the latter using the term of the Schottky effect [3], which may be the first paper in the sense of expressing such a type of processes by the term of noise.

Let be a random function. Let be its power spectrum density (PSD) function, where is frequency and is radian frequency. Then, by noise, one means that for . Note that the PSD of a conventional random function, such as , is convergent at . In the field, the term “ noise” is a collective noun, which implies in fact noise. In the general case, noise has the meaning of noise for , where is the set of real numbers; see, for example, [4]. However, since for , one may usually not be interested in the case of . In what follows, we discuss the noise of type for unless otherwise stated.

Since the notion of noise appeared [3], it has gained increasing interests of scientists in various fields, ranging from bioengineering to computer networks; see, for example, [4–54], simply to mention a few. That fact gives rise to a question of what noise is. The question may be roughly answered in a way that a noise is such that its PSD is divergent at as previously mentioned. Nonetheless, that answer may be never enough to describe the full picture of noise. By full picture, we mean that we should describe a set of main properties of noise, in addition to the property in frequency domain. When regarding as its first property, denoted by P1, we would like to list other three as follows.P2: What is the qualitative structure of an autocorrelation function (ACF) of noise? For this property, we will discuss the statistical dependence based on the hyperbolically decayed ACF structure.P3: What is the main property of its probability density function (PDF)? With P3, we will explain the heavy-tailed property of noise, which may produce random functions without mean or variance.P4: What is the possible structure of the differential equation to synthesize noise?

For facilitating the description of the full picture of noise, we will brief the preliminaries in Section 2. Then, P2–P4 will be discussed in Sections 3–5, respectively. After that, we will conclude the paper in Section 6.

#### 2. Preliminaries

##### 2.1. Dependence of Random Variables

A time series may also be called a random function [55]. The term random function apparently exhibits that is a random variable, implying We would like to discuss the dependence of and ) for , as well as .

###### 2.1.1. Dependence Description of Random Variables with Probability

Let be the probability of the event . Denote by the probability of the event . Then [56], and are said to be independent events if If and are dependent, on the other side, where is the conditional probability, implying that the probability of the event provided that has occurred.

*Note 1. *The dependence of and is reflected in the conditional probability . If and are independent, .

###### 2.1.2. Dependence Description of Gaussian Random Variables with Correlation

Let be a Gaussian random function. Denote by the correlation coefficient between and . Then [57, 58], and are independent if On the other hand, and are dependent provided that

The condition (2.4) or (2.5) expressed by the correlation coefficient regarding the independence or dependence may not be enough to identify the independence or dependence of a Gaussian random function completely. For example, when , that (2.4) holds for large may not imply that it is valid for small . In other words, one may encounter the situations expressed by By using the concept of probability, (2.6) corresponds to Similarly, (2.7) corresponds to (2.9)

*Note 2. *The notion of the dependence or independence of a set of random variables plays a role in the axiomatic approach of probability theory and stochastic processes; see Kolmogorov [59].

The above example exhibits an interesting fact that the dependence or independence relies on the observation scale or observation range. In conventional time series, we do not usually consider the observation scale. That is, (2.2) and (2.3) or (2.4) and (2.5) hold for all observation ranges no matter whether is small or large; see, for example, [60, 61]. Statistical properties that depend on observation ranges are briefed by Papoulis and Pillai [56] and Fuller [62] but detailed in Beran [63, 64].

*Note 3. *The Kolmogorov’s work on axiomatic approach of probability theory and stochastic processes needs the assumption that and are independent in most cases likely for the completeness of his theory. Nevertheless, he contributed a lot in random functions that are range dependent; see, for example, [65, 66].

###### 2.1.3. Dependence Description of Gaussian Random Variables with ACF

Denote by the ACF of . Denote by E the operator of mean. Then, it is given by It represents the correlation between the one point and the other apart, that is, . For facilitating the discussions, we assume that is stationary in the wide sense (stationary for short). With this assumption, only replies on the lag . Therefore, one has In the normalized case, ACF is a convenient tool of describing the dependence of a Gaussian random function . For instance, on the one hand, we say that any two different points of are uncorrelated, accordingly independent as is Gaussian, if . That is the case of Gaussian white noise. On the other hand, any two different points of are strongest dependent if . This is the case of the strongest long-range dependence. In the case of , the value of varies with the lag .

A useful measure called correlation time, which is denoted by [67, page 74], is defined in the form By correlation time, we say that the correlation can be neglected if , where is the time scale of interest [67].

*Note 4. *For a conventional Gaussian random function , its correlation can be neglected if . This implies that the statistical dependence or independence of relies on its correlation time . However, we will show in Note 8 that correlation time fails if is a noise.

##### 2.2. ACF and PSD

The PSD of is the Fourier transform of its ACF. That is, Equivalently, Letting in (2.14) yields Similarly, letting in (2.15) produces

*Note 5. *Equation (2.16) implies that an ACF is integrable if . On the other side, a PSD is integrable when as indicated in (2.17). Both are usual cases in conventional random functions.

*Note 6. *The noise of type has the property , which makes noise substantially different from conventional random functions.

##### 2.3. Mean and Variance

Denote by the PDF of . Then, the mean denoted by is given by The variance denoted by is in the form Assume that is stationary. Then, and do not rely on time . In this case, they are expressed by Without generality losing, we always assume that is stationary in what follows unless otherwise stated.

Denote by the autocovariance of . Then, Taking into account (2.19) and (2.21), one has Denote by the mean square value of . Then, Considering given by one has Therefore,

It is worth noting that the above number characteristics are crucial to the analysis of in practice; see, for example, [60–62, 65–87], just to cite a few. However, things turn to be complicated if is in the class of noise. In Section 4 below, we will show that and/ or may not exist for specific types of noise.

#### 3. Hyperbolically Decayed ACFs and Noise

The qualitative structure of noise is in the form , where is a constant. Since , as we previously mentioned several times, its ACF is nonintegrable over . That is,

Note that the above may be taken as a definition of LRD property of a random function; see, for example, [4, 63, 64, 88–90]. Thus, the above exhibits that noise is LRD. Consequently, of a noise may have the asymptotic property given by Following [91–93], we have the Fourier transform of in the form From the above, we have Thus, we have the following note.

*Note 7. *Qualitatively, the ACF of noise is in the structure of power function. It follows power law. This is the answer to P2 explained in the Introduction.

*Example 3.1. *A well-known example of noise is the fractional Gaussian noise (fGn) [93, 94]. Its ACF is in the form
where is the Hurst parameter; is the strength of fGn. Its PSD is in the form [93]

*Example 3.2. *Another example of noise is fractional Brownian motion (fBm). The PSD of the fBm of the Weyl type is in the form [95, 96]
The PSD of the fBm of the Riemann-Liouville type is given by [97–99]
where is the Bessel function of order and is the Struve function of order , respectively.

*Example 3.3. *The Cauchy-class process with LRD discussed in [100, 101] is a case of noise. Its ACF is in the form
Its PSD is given by
where is the modified Bessel function of the second kind. It has the asymptotic expression for in the form, being noise,

*Example 3.4. *The generalized Cauchy process with LRD reported in [102–105] is an instance of noise. Its ACF is given by
where and . Its PSD in the complete form refers to [92]. The following may suffice to exhibit its noise behavior [102]:
It may be worthwhile for us to write a theorem and a note when this section will soon finish.

Theorem 3.5. *If a random function is LRD, it belongs to the class of noise and vice versa.*

*Proof. *LRD implies that the right side of (2.16) is divergent, which implies that is a noise. On the other side, when is a noise, , which means that its ACF is nonintegrable, hence, LRD.

Since any random function with LRD is in the class of noise, one may observe other types of random functions that belong to noise from a view of LRD processes, for example, those described in [106–112].

*Note 8. *As noise is of LRD, its ACF is nonintegrable over . Thus, in (2.13), its correlation time . That implies that correlations at any time scale cannot be neglected. Consequently, the measure of correlation time fails to characterize the statistical dependence of noise.

#### 4. Heavy-Tailed PDFs and Noise

Heavy-tailed PDFs are widely observed in various fields of sciences and technologies, including life science and bioengineering; see, for example, [113–146]. Typical heavy-tailed PDFs are the Pareto distribution, the log-Weibull distribution, the stretched exponential distribution, the Zipfian distribution, Lévy distribution, and the Cauchy distribution; see, for example [51, 96, 125, 143–169], merely to cite a few.

By heavy tails, we mean that the tail of a PDF decays slower than the tails of PDFs in the form of exponential functions. More precisely, the term of heavy tail implies that of a random function decays slowly such that in (2.21), or in (2.24), decays hyperbolically such that it is of LRD. Thus, we may have the theorem below.

Theorem 4.1. *Let be a heavy-tailed random function. Then, it is in the class of noise and vice versa.*

*Proof. *Considering that is heavy tailed, we may assume its ACF hyperbolically decays, that is, for and . According to (3.3), therefore, we see that is noise. On the other side, if is a noise, it is LRD and accordingly heavy tailed [54, 146]. This completes the proof.

*Note 9. *Theorem 4.1 may be taken as an answer to P3 stated in the Introduction. Since in Theorem 4.1 is restricted to (0, 1), Theorem 4.1 is consistent with the result of the Taqqiu’s law, referring [53, 54, 89] for the details of the Taqqiu’s law.

*Note 10. *The tail of may be so heavy that the mean or variance of does not exist.

A commonly used instance to clarify Note 10 is the Pareto distribution; see, for example, [96, 134, 147, 148, 161]. To clarify it further, we would like to write more. Denote by the Cauchy distribution. Then, one has where is the half width at half maximum and is the statistical median [170]. Its th moment denoted by is computed by Since the above integral is divergent for , the mean and variance of obeying do not exist.

*Note 11. *Application of the Cauchy distribution to network traffic modeling refers to [169]. The generalized Cauchy distribution is reported in [171–173].

Another type of heavy-tailed random functions without mean and variance is the Lévy distribution; see, for example, [162–166, 174]. Suppose that follows the Lévy distribution that is denoted by . Then, for is given by where is the location parameter and the scale parameter. The th moment of a Lévy distributed random function, for the simplicity by letting , is given by The integral in (4.4) is divergent for . Thus, its mean and variance do not exist because they approach .

*Note 12. *Application of the Lévy distribution to network traffic modeling is discussed in [175]. Tools in Matlab for simulating Lévy distributed random data are described in [174].

The previously discussed heavy-tailed distributions, such as the Cauchy distribution and the Lévy distribution, are special cases of stable distributions, which are detailed in [89, 120, 176–179]. Denote by the PDF of a stable-distributed random function . Then, it is indirectly defined by its characteristic function denoted by . It is in the form where is the stability parameter, is the skewness parameter, is the scale parameter, the location parameter, and It may be easy to see that has mean when . However, its mean is undefined otherwise. In addition, its variance equals to if . Otherwise, its variance is infinite.

*Note 13. *A stable distribution is characterized by 4 parameters. In general, the analytical expression of is unavailable except for some specific values of parameters, due to the difficulties in performing the inverse Fourier transform of the right side of (4.5).

*Note 14. *A stable distribution is generally non-Gaussian except for some specific values of parameters. When , it is Gaussian with the mean and the variance of . Generally speaking, is heavy tailed.

*Note 15. * reduces to the Landau distribution when . Denote by Landau the Landau distribution [180]. Then, its characteristic function is given by
where is the location parameter and the scaled parameter. Its PDF is given by
Applications of the Landau distribution can be found in nuclear physics [181, 182].

*Note 16. * reduces to the Holtsmark distribution if and . Accordingly,
Its PDF is given by
This type of random functions has mean but its variance is infinite [183–185].

The literature regarding applications of stable distributions is rich. Their applications to network traffic modeling can also be found in [186–188].

It is worth noting that the observation of random functions without mean and variance may be traced back to the work of the famous statistician Daniel Bernoulli’s cousin, Nicolas Bernoulli in 1713 [189, 190]. Nicolas Bernoulli studied a casino game as follows. A player bets on how many tosses of a coin will be needed before it first turns up heads. If it falls heads on the first toss the player wins $2; if it falls tails, heads, the player wins $4; if it falls tails, tails, heads, the player wins $8, and so on. According to this game rule, if the probability of an outcome is , the player wins $. Thus, the mean for is given by The above may be used to express the game that is now termed “Petersburg Paradox.” That paradox is now named after Daniel Bernoulli due to his presentation of the problem and his solution in the Commentaries of the Imperial Academy of Science of Saint Petersburg [191].

#### 5. Fractionally Generalized Langevin Equation and Noise

The standard Langevin equation is in the form where [56, 192] and is a standard white noise. By standard white noise, we mean that its PSD is in the form The solution to (5.1) in frequency domain is given by

The standard Langevin equation may not attract people much in the field of noise. People are usually interested in fractionally generalized Langevin equations. There are two types of fractionally generalized Langevin equations. One is in the form [193–208] The other is expressed by [209–215] Two are consistent when . We now adopt the one expressed by (5.5).

Theorem 5.1. *A solution to the stochastically fractional differential equation below belongs to noise:
*

*Proof . *Denote by the impulse response function of (5.5). That is,
where is the Dirac- function defined by, for being continuous at ,
The function is called the impulse function in linear systems [61, 216]. According to the theory of linear systems, is the solution to (5.7) under the zero initial condition, which is usually called the impulse response function in linear systems [61, 68, 69, 76, 77, 216–222]. Denote by the Fourier transform of . Then, with the techniques in fractional calculus [213, 215, 223], doing the Fourier transforms on (5.7) yields
Therefore, we have, by taking into account (5.2),
If in (5.10), we have noise expressed by
This finishes the proof.

From the above, we see that belongs to noise. As a matter of fact, by using fractional integral in (5.6), we have

The above expression implies that a noise may be taken as a solution to a stochastically fractional differential equation, being an answer to P4 described in the Introduction. The following example will soon refine this point of view.

*Example 5.2. *Let be the Wiener Brownian motion for ; see [224] for the details of the Brownian motion. Then, it is nondifferentiable in the domain of ordinary functions. It is differentiable, however, in the domain of generalized functions over the Schwartz space of test functions [91, 225]. Therefore, in the domain of generalized functions, we write the stationary Gaussian white noise by
Based on the definitions of the fractional integrals of the Riemann-Liouville’s and the Weyl’s [223], on the one hand, when using the Riemann-Liouville integral operator, we express the fBm of the Riemann-Liouville type, which is denoted by , in the form
where is the Riemann-Liouville integral operator of order for . On the other hand, the fBm of the Weyl type by using the Weyl fractional integral, which we denote by , is given by
where is the Weyl integral operator of order . Expressions (5.14) and (5.15) are the fBms introduced by Mandelbrot and van Ness in [94] but we provide a new outlook of describing them from the point of view of the fractional generalized Langevin equation with the topic of noise.

#### 6. Conclusions

We have explained the main properties of noise as follows. First, it is LRD and its ACF is hyperbolically decayed. Second, its PDF obeys power laws and it is heavy tailed. Finally, it may be taken as a solution to a stochastic differential equation. Fractal time series, such as fGn, fBm, the generalized Cauchy process, and the Lévy flights, -stable processes, are generally in the class of noise.

#### Acknowledgments

This work was supported in part by the 973 Plan under the Project Grant no. 2011CB302800 and by the National Natural Science Foundation of China under the Project Grant nos. 61272402, 61070214, and 60873264.

#### References

- W. Schottky, “Uber spontane Stromschwankungen in verschiedenen Elektri-zittsleitern,”
*Annalen der Physik*, vol. 362, no. 23, pp. 541–567, 1918. View at: Publisher Site | Google Scholar - W. Schottky, “Zur Berechnung und Beurteilung des Schroteffektes,”
*Annalen der Physik*, vol. 373, no. 10, pp. 157–176, 1922. View at: Publisher Site | Google Scholar - J. B. Johnson, “The Schottky effect in low frequency circuits,”
*Physical Review*, vol. 26, no. 1, pp. 71–85, 1925. View at: Publisher Site | Google Scholar - B. B. Mandelbrot,
*Multifractals and 1/f Noise*, Springer, New York, NY, USA, 1998. View at: Zentralblatt MATH - K. Fraedrich, U. Luksch, and R. Blender, “1/
*f*model for long-time memory of the ocean surface temperature,”*Physical Review E*, vol. 70, no. 3, Article ID 037301, 4 pages, 2004. View at: Publisher Site | Google Scholar - E. J. Wagenmakers, S. Farrell, and R. Ratcliff, “Estimation and interpretation of 1/
*f*^{α}noise in human cognition,”*Psychonomic Bulletin and Review*, vol. 11, no. 4, pp. 579–615, 2004. View at: Publisher Site | Google Scholar - F. Principato and G. Ferrante, “1/
*f*noise decomposition in random telegraph signals using the wavelet transform,”*Physica A*, vol. 380, no. 1-2, pp. 75–97, 2007. View at: Publisher Site | Google Scholar - V. P. Koverda and V. N. Skokov, “Maximum entropy in a nonlinear system with a 1/
*f*power spectrum,”*Physica A*, vol. 391, no. 1-2, pp. 21–28, 2012. View at: Publisher Site | Google Scholar - Y. Nemirovsky, D. Corcos, I. Brouk, A. Nemirovsky, and S. Chaudhry, “1/
*f*noise in advanced CMOS transistors,”*IEEE Instrumentation and Measurement Magazine*, vol. 14, no. 1, pp. 14–22, 2011. View at: Publisher Site | Google Scholar - O. Miramontes and P. Rohani, “Estimating 1/
*f*^{α}scaling exponents from short time-series,”*Physica D*, vol. 166, no. 3-4, pp. 147–154, 2002. View at: Publisher Site | Google Scholar - C. M. van Vliet, “Random walk and 1/
*f*noise,”*Physica A*, vol. 303, no. 3-4, pp. 421–426, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. S. Kim, Y. S. Kim, H. S. Min, and Y. J. Park, “Theory of 1/
*f*noise currents in semiconductor devices with one-dimensional geometry and its application to Si Schottky barrier diodes,”*IEEE Transactions on Electron Devices*, vol. 48, no. 12, pp. 2875–2883, 2001. View at: Publisher Site | Google Scholar - T. Antal, M. Droz, G. Györgyi, and Z. Rácz, “1/
*f*noise and extreme value statistics,”*Physical Review Letters*, vol. 87, no. 24, Article ID 240601, 4 pages, 2001. View at: Google Scholar - B. Pilgram and D. T. Kaplan, “A comparison of estimators for 1/
*f*noise,”*Physica D*, vol. 114, no. 1-2, pp. 108–122, 1998. View at: Publisher Site | Google Scholar - H. J. Jensen, “Lattice gas as a model of 1/
*f*noise,”*Physical Review Letters*, vol. 64, no. 26, pp. 3103–3106, 1990. View at: Publisher Site | Google Scholar - E. Marinari, G. Parisi, D. Ruelle, and P. Windey, “Random walk in a random environment and 1/
*f*noise,”*Physical Review Letters*, vol. 50, no. 17, pp. 1223–1225, 1983. View at: Publisher Site | Google Scholar - F. N. Hooge, “1/
*f*noise,”*Physica B*, vol. 83, no. 1, pp. 14–23, 1976. View at: Publisher Site | Google Scholar - M. B. Weissman, “Simple model for 1/
*f*noise,”*Physical Review Letters*, vol. 35, no. 11, pp. 689–692, 1975. View at: Publisher Site | Google Scholar - F. N. Hooge, “Discussion of recent experiments on 1/
*f*noise,”*Physica*, vol. 60, no. 1, pp. 130–144, 1972. View at: Publisher Site | Google Scholar - C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 207–217, 2010. View at: Publisher Site | Google Scholar - C. Cattani, “On the existence of wavelet symmetries in archaea DNA,”
*Computational and Mathematical Methods in Medicine*, vol. 2012, Article ID 673934, 21 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. Cattani, E. Laserra, and I. Bochicchio, “Simplicial approach to fractal structures,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 958101, 21 pages, 2012. View at: Publisher Site | Google Scholar - C. Cattani, “Fractional calculus and Shannon wavelet,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 502812, 26 pages, 2012. View at: Publisher Site | Google Scholar - C. Cattani, G. Pierro, and G. Altieri, “Entropy and multifractality for the mye-loma multiple TET 2 gene,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 193761, 14 pages, 2012. View at: Publisher Site | Google Scholar - M. S. Keshner, “1/
*f*noise,”*Proceedings of the IEEE*, vol. 70, no. 3, pp. 212–218, 1982. View at: Publisher Site | Google Scholar - B. Ninness, “Estimation of 1/
*f*Noise,”*IEEE Transactions on Information Theory*, vol. 44, no. 1, pp. 32–46, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Yazici and R. L. Kashyap, “A class of second-order stationary self-similar processes for 1/
*f*phenomena,”*IEEE Transactions on Signal Processing*, vol. 45, no. 2, pp. 396–410, 1997. View at: Publisher Site | Google Scholar - G. W. Wornell, “Wavelet-based representations for the 1/
*f*family of fractal processes,”*Proceedings of the IEEE*, vol. 81, no. 10, pp. 1428–1450, 1993. View at: Publisher Site | Google Scholar - B. B. Mandelbrot, “Some noises with 1/
*f*spectrum, a bridge between direct current and white noise,”*IEEE Transactions on Information Theory*, vol. 13, no. 2, pp. 289–298, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/
*f*^{α}power law noise generation,”*Proceedings of the IEEE*, vol. 83, no. 5, pp. 802–827, 1995. View at: Publisher Site | Google Scholar - G. Corsini and R. Saletti, “1/
*f*^{γ}power spectrum noise sequence generator,”*IEEE Transactions on Instrumentation and Measurement*, vol. 37, no. 4, pp. 615–619, 1988. View at: Publisher Site | Google Scholar - W. T. Li and D. Holste, “Universal 1/
*f*noise, crossovers of scaling exponents, and chromosome-specific patterns of guanine-cytosine content in DNA sequences of the human genome,”*Physical Review E*, vol. 71, no. 4, Article ID 041910, 9 pages, 2005. View at: Publisher Site | Google Scholar - W. T. Li, G. Stolovitzky, P. Bernaola-Galván, and J. L. Oliver, “Compositional heterogeneity within, and uniformity between, DNA sequences of yeast chromosomes,”
*Genome Research*, vol. 8, no. 9, pp. 916–928, 1998. View at: Google Scholar - W. T. Li and K. Kaneko, “Long-range correlation and partial spectrum in a noncoding DNA sequence,”
*Europhysics Letters*, vol. 17, no. 7, pp. 655–660, 1992. View at: Publisher Site | Google Scholar - P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger et al., “From 1/
*f*noise to multifractal cascades in heartbeat dynamics,”*Chaos*, vol. 11, no. 3, pp. 641–652, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. N. Mantegna and H. E. Stanley,
*An Introduction to Econophysics: Correlations and Complexity in Finance*, Cambridge University Press, Cambridge, UK, 2000. - W. Q. Duan and H. E. Stanley, “Cross-correlation and the predictability of financial return series,”
*Physica A*, vol. 390, no. 2, pp. 290–296, 2010. View at: Publisher Site | Google Scholar - B. Podobnik, D. Horvatic, A. Lam Ng, H. E. Stanley, and P. C. Ivanov, “Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes,”
*Physica A*, vol. 387, no. 15, pp. 3954–3959, 2008. View at: Publisher Site | Google Scholar - G. Aquino, M. Bologna, P. Grigolini, and B. J. West, “Beyond the death of linear response: 1/
*f*optimal information transport,”*Physical Review Letters*, vol. 105, no. 4, Article ID 040601, 4 pages, 2010. View at: Google Scholar - B. J. West and P. Grigolini, “Chipping away at memory,”
*Biological Cybernetics*, vol. 103, no. 2, pp. 167–174, 2010. View at: Publisher Site | Google Scholar - B. J. West and M. F. Shlesinger, “On the ubiquity of $1/f$ noise,”
*International Journal of Modern Physics B*, vol. 3, no. 6, pp. 795–819, 1989. View at: Publisher Site | Google Scholar - A. L. Goldberger, V. Bhargava, B. J. West, and A. J. Mandell, “On the mechanism of cardiac electrical stability. The fractal hypothesis,”
*Biophysical Journal*, vol. 48, no. 3, pp. 525–528, 1985. View at: Publisher Site | Google Scholar - T. Musha, H. Takeuchi, and T. Inoue, “1/
*f*fluctuations in the spontaneous spike discharge intervals of a giant snail neuron,”*IEEE Transactions on Biomedical Engineering*, vol. 30, no. 3, pp. 194–197, 1983. View at: Publisher Site | Google Scholar - M. Kobayashi and T. Musha, “1/
*f*fluctuation of heartbeat period,”*IEEE Transactions on Biomedical Engineering*, vol. 29, no. 6, pp. 456–457, 1982. View at: Publisher Site | Google Scholar - B. Neumcke, “1/
*f*noise in membranes,”*Biophysics of Structure and Mechanism*, vol. 4, no. 3, pp. 179–199, 1978. View at: Publisher Site | Google Scholar - J. R. Clay and M. F. Shlesinger, “Unified theory of 1/
*f*and conductance noise in nerve membrane,”*Journal of Theoretical Biology*, vol. 66, no. 4, pp. 763–773, 1977. View at: Publisher Site | Google Scholar - E. Frehland, “Diffusion as a source of 1/
*f*noise,”*The Journal of Membrane Biology*, vol. 32, no. 1, pp. 195–196, 1977. View at: Publisher Site | Google Scholar - M. E. Green, “Diffusion and 1/
*f*noise,”*The Journal of Membrane Biology*, vol. 28, no. 1, pp. 181–186, 1976. View at: Publisher Site | Google Scholar - I. Csabai, “1/
*f*noise in computer network traffic,”*Journal of Physics A*, vol. 27, no. 12, pp. L417–L421, 1994. View at: Publisher Site | Google Scholar - M. Takayasu, H. Takayasu, and T. Sato, “Critical behaviors and 1/
*f*noise in information traffic,”*Physica A*, vol. 233, no. 3-4, pp. 824–834, 1996. View at: Publisher Site | Google Scholar - V. Paxson and S. Floyd, “Wide area traffic: the failure of Poisson modeling,”
*IEEE/ACM Transactions on Networking*, vol. 3, no. 3, pp. 226–244, 1995. View at: Publisher Site | Google Scholar - W. Willinger, R. Govindan, S. Jamin, V. Paxson, and S. Shenker, “Scaling phenomena in the internet: critically examining criticality,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 99, supplement 1, pp. 2573–2580, 2002. View at: Publisher Site | Google Scholar - P. Loiseau, P. Gonçalves, G. Dewaele, P. Borgnat, P. Abry, and P. V. B. Primet, “Investigating self-similarity and heavy-tailed distributions on a large-scale experimental facility,”
*IEEE/ACM Transactions on Networking*, vol. 18, no. 4, pp. 1261–1274, 2010. View at: Publisher Site | Google Scholar - P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch, “Revisiting an old friend: on the observability of the relation between long range dependence and heavy tail,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 147–165, 2010. View at: Publisher Site | Google Scholar - A. M. Yaglom,
*Correlation Theory of Stationary and Related Random Functions*, vol. 1, Springer, New York, NY, USA, 1987. - A. Papoulis and S. U. Pillai,
*Probability, Random Variables, and Stochastic Processes*, McGraw-Hill, New York, NY, USA, 1997. - B. W. Lindgren and G. W. McElrath,
*Introduction to Probability and Statistics*, The Macmillan, New York, NY, USA, 1959. - J. L. Doob, “The elementary Gaussian processes,”
*Annals of Mathematical Statistics*, vol. 15, pp. 229–282, 1944. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. N. Kolmogorov,
*Fundamental of Probability*, Business Press, Shanghai, China, 1954, Translated from Russian by S.-T. Ding. - G. E. P. Box, G. M. Jenkins, and G. C. Reinsel,
*Time Series Analysis: Forecasting and Control*, Prentice Hall, Englewood Cliffs, NJ, USA, 1994. - J. S. Bendat and A. G. Piersol,
*Random Data: Analysis and Measurement Procedure*, Wiley Series in Probability and Statistics, John Wiley & Sons, Hoboken, NJ, USA, 3rd edition, 2000. - W. A. Fuller,
*Introduction to Statistical Time Series*, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1996. - J. Beran,
*Statistics for Long-Memory Processes*, vol. 61 of*Monographs on Statistics and Applied Probability*, Chapman & Hall, New York, NY, USA, 1994. - J. Beran, “Statistical methods for data with long-range dependence,”
*Statistical Science*, vol. 7, no. 4, pp. 404–416, 1992. View at: Publisher Site | Google Scholar - A. S. Monin and A. M. Yaglom,
*Statistical Fluid Mechanics: Mechanics of Turbulence*, vol. 2, MIT Press, Cambridge, Mass, USA, 1971. - A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers,”
*Soviet Physics Uspekhi*, vol. 10, no. 6, pp. 734–736, 1968. View at: Publisher Site | Google Scholar - N. C. Nigam,
*Introduction to Random Vibrations*, MIT Press, Cambridge, Mass, USA, 1983. - T. T. Song and M. Grigoriu,
*Random Vibration of Mechanical and Structural Systems*, Prentice Hall, New York, NY, USA, 1993. - C. M. Harris,
*Shock and Vibration Handbook*, McGraw-Hill, New York, NY, USA, 4th edition, 1995. - H. Czichos, T. Saito, and L. Smith,
*Springer Handbook of Metrology and Testing*, Springer, New York, NY, USA, 2011. - W. N. Sharpe Jr.,
*Springer Handbook of Experimental Solid Mechanics*, Springer, New York, NY, USA, 2008. - C. Tropea, A. L. Yarin, and J. F. Foss, Eds.,
*Springer Handbook of Experimental Fluid Mechanics*, Springer, New York, NY, USA, 2007. - K. H. Grote and E. K. Antonsson, Eds.,
*Springer Handbook of Mechanical Engineering*, Springer, New York, NY, USA, 2009. - R. Kramme, K. P. Hoffmann, and R. S. Pozos,
*Springer Handbook of Medical Technology*, Springer, New York, NY, USA, 2012. - W. Kresse and D. M. Danko,
*Springer Handbook of Geographic Information*, Springer, New York, NY, USA, 2012. - S. S. Bhattacharyya, F. Deprettere, R. Leupers, and J. Takala, Eds.,
*Handbook of Signal Processing Systems*, Springer, New York, NY, USA, 2010. - S. K. Mitra and J. F. Kaiser,
*Handbook for Digital Signal Processing*, John Wiley & Sons, New York, NY, USA, 1993. - W. A. Woyczyński,
*A First Course in Statistics for Signal Analysis*, Birkhäuser, Boston, Mass, USA, 2006. - R. A. Bailey,
*Design of Comparative Experiments*, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, UK, 2008. View at: Publisher Site - ASME,
*Measurement Uncertainty Part 1, Instruments and Apparatus*, Sup-plement to ASME, Performance Test Codes, ASME, New York, NY, USA, 1986. - D. Sheskin,
*Statistical Tests and Experimental Design: A Guidebook*, Gardner Press, New York, NY, USA, 1984. - T. W. MacFarland,
*Two-Way Analysis of Variance*, Springer, New York, NY, USA, 2012. - A. K. Gupta, W. B. Zeng, and Y. Wu,
*Probability and Statistical Models: Foundations for Problems in Reliability and Financial Mathematics*, Birkhäuser, Boston, Mass, USA, 2010. View at: Publisher Site - P. Fieguth,
*Statistical Image Processing and Multidimensional Modeling*, Information Science and Statistics, Springer, New York, NY, USA, 2011. View at: Publisher Site - J. Nauta,
*Statistics in Clinical Vaccine Trials*, Springer, New York, NY, USA, 2011. - A. Gelman, “Analysis of variance—why it is more important than ever,”
*The Annals of Statistics*, vol. 33, no. 1, pp. 1–53, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. G. Pendse, “A note on mathematical expectation,”
*The Mathematical Gazette*, vol. 22, no. 251, pp. 399–402, 1938. View at: Google Scholar - B. B. Mandelbrot,
*Gaussian Self-Affinity and Fractals*, Springer, New York, NY, USA, 2001. - G. Samorodnitsky and M. S. Taqqu,
*Stable Non-Gaussian Random Processes*, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994. - J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger, “Long-range dependence in variable-bit-rate video traffic,”
*IEEE Transactions on Communications*, vol. 43, no. 234, pp. 1566–1579, 1995. View at: Publisher Site | Google Scholar - I. M. Gelfand and K. Vilenkin,
*Generalized Functions*, vol. 1, Academic Press, New York, NY, USA, 1964. - M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 219–222, 2010. View at: Publisher Site | Google Scholar - M. Li and S. C. Lim, “A rigorous derivation of power spectrum of fractional Gaussian noise,”
*Fluctuation and Noise Letters*, vol. 6, no. 4, pp. C33–C36, 2006. View at: Publisher Site | Google Scholar - B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,”
*SIAM Review*, vol. 10, pp. 422–437, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Flandrin, “On the spectrum of fractional Brownian motions,”
*IEEE Transactions on Information Theory*, vol. 35, no. 1, pp. 197–199, 1989. View at: Publisher Site | Google Scholar - M. Li, “Fractal time series—a tutorial review,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 157264, 26 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. V. Muniandy and S. C. Lim, “Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type,”
*Physical Review E*, vol. 63, no. 4, Article ID 046104, 7 pages, 2001. View at: Google Scholar - V. M. Sithi and S. C. Lim, “On the spectra of Riemann-Liouville fractional Brownian motion,”
*Journal of Physics A*, vol. 28, no. 11, pp. 2995–3003, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. C. Lim and S. V. Muniandy, “On some possible generalizations of fractional Brownian motion,”
*Physics Letters A*, vol. 266, no. 2-3, pp. 140–145, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. P. Chilès and P. Delfiner,
*Geostatistics, Modeling Spatial Uncertainty*, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1999. View at: Publisher Site - M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 654284, 13 pages, 2011. View at: Publisher Site | Google Scholar - S. C. Lim and M. Li, “A generalized Cauchy process and its application to relaxation phenomena,”
*Journal of Physics A*, vol. 39, no. 12, pp. 2935–2951, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. C. Lim and L. P. Teo, “Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure,”
*Stochastic Processes and Their Applications*, vol. 119, no. 4, pp. 1325–1356, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,”
*Physica A*, vol. 387, no. 11, pp. 2584–2594, 2008. View at: Publisher Site | Google Scholar - P. Vengadesh, S. V. Muniandy, and W. H. A. Majid, “Fractal morphological analysis of bacteriorhodopsin (bR) layers deposited onto indium tin oxide (ITO) electrodes,”
*Materials Science and Engineering C*, vol. 29, no. 5, pp. 1621–1626, 2009. View at: Publisher Site | Google Scholar - R. J. Martin and A. M. Walker, “A power-law model and other models for long-range dependence,”
*Journal of Applied Probability*, vol. 34, no. 3, pp. 657–670, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. J. Martin and J. A. Eccleston, “A new model for slowly-decaying correlations,”
*Statistics and Probability Letters*, vol. 13, no. 2, pp. 139–145, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. A. Woodward, Q. C. Cheng, and H. L. Gray, “A $k$-factor GARMA long-memory model,”
*Journal of Time Series Analysis*, vol. 19, no. 4, pp. 485–504, 1998. View at: Publisher Site | Google Scholar - C. Ma, “Power-law correlations and other models with long-range dependence on a lattice,”
*Journal of Applied Probability*, vol. 40, no. 3, pp. 690–703, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. Ma, “A class of stationary random fields with a simple correlation structure,”
*Journal of Multivariate Analysis*, vol. 94, no. 2, pp. 313–327, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Li, W. Jia, and W. Zhao, “Correlation form of timestamp increment sequences of self-similar traffic on Ethernet,”
*Electronics Letters*, vol. 36, no. 19, pp. 1668–1669, 2000. View at: Publisher Site | Google Scholar - M. Li and W. Zhao, “Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise,”
*Physica A*, vol. 391, no. 24, pp. 6268–6278, 2012. View at: Publisher Site | Google Scholar - E. G. Tsionas, “Estimating multivariate heavy tails and principal directions easily, with an application to international exchange rates,”
*Statistics and Probability Letters*, vol. 82, no. 11, pp. 1986–1989, 2012. View at: Publisher Site | Google Scholar - J. Lin, “Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims,”
*Insurance: Mathematics and Economics*, vol. 51, no. 2, pp. 422–429, 2012. View at: Publisher Site | Google Scholar - K. Yu, M. L. Huang, and P. H. Brill, “An algorithm for fitting heavy-tailed distributions via generalized hyperexponentials,”
*INFORMS Journal on Computing*, vol. 24, no. 1, pp. 42–52, 2012. View at: Publisher Site | Google Scholar - R. Luger, “Finite-sample bootstrap inference in GARCH models with heavy-tailed innovations,”
*Computational Statistics and Data Analysis*, vol. 56, no. 11, pp. 3198–3211, 2012. View at: Publisher Site | Google Scholar - T. Ishihara and Y. Omori, “Efficient Bayesian estimation of a multivariate stochastic volatility model with cross leverage and heavy-tailed errors,”
*Computational Statistics and Data Analysis*, 2010. View at: Publisher Site | Google Scholar - J. Diebolt, L. Gardes, S. Girard, and A. Guillou, “Bias-reduced extreme quantile estimators of Weibull tail-distributions,”
*Journal of Statistical Planning and Inference*, vol. 138, no. 5, pp. 1389–1401, 2008. View at: Publisher Site | Google Scholar - J. Beran, B. Das, and D. Schell, “On robust tail index estimation for linear long-memory processes,”
*Journal of Time Series Analysis*, vol. 33, no. 3, pp. 406–423, 2012. View at: Publisher Site | Google Scholar - P. Barbe and W. P. McCormick, “Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution,”
*Stochastic Processes and Their Applications*, vol. 122, no. 4, pp. 1276–1303, 2012. View at: Publisher Site | Google Scholar - C. Weng and Y. Zhang, “Characterization of multivariate heavy-tailed distribution families via copula,”
*Journal of Multivariate Analysis*, vol. 106, pp. 178–186, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. B. García, J. García Pérez, and J. R. van Dorp, “Modeling heavy-tailed, skewed and peaked uncertainty phenomena with bounded support,”
*Statistical Methods and Applications*, vol. 20, no. 4, pp. 146–166, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - V. H. Lachos, T. Angolini, and C. A. Abanto-Valle, “On estimation and local influence analysis for measurement errors models under heavy-tailed distributions,”
*Statistical Papers*, vol. 52, no. 3, pp. 567–590, 2011. View at: Publisher Site | Google Scholar - V. Ganti, K. M. Straub, E. Foufoula-Georgiou, and C. Paola, “Space-time dynamics of depositional systems: experimental evidence and theoretical modeling of heavy-tailed statistics,”
*Journal of Geophysical Research F: Earth Surface*, vol. 116, no. 2, Article ID F02011, 17 pages, 2011. View at: Publisher Site | Google Scholar - U. J. Dixit and M. J. Nooghabi, “Efficient estimation in the Pareto distribution with the presence of outliers,”
*Statistical Methodology*, vol. 8, no. 4, pp. 340–355, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Nándori, “Recurrence properties of a special type of heavy-tailed random walk,”
*Journal of Statistical Physics*, vol. 142, no. 2, pp. 342–355, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Ceresetti, G. Molinié, and J. D. Creutin, “Scaling properties of heavy rainfall at short duration: a regional analysis,”
*Water Resources Research*, vol. 46, no. 9, Article ID W09531, 12 pages, 2010. View at: Publisher Site | Google Scholar - A. Charpentier and A. Oulidi, “Beta kernel quantile estimators of heavy-tailed loss distributions,”
*Statistics and Computing*, vol. 20, no. 1, pp. 35–55, 2010. View at: Publisher Site | Google Scholar - P. Embrechts, J. Nešlehová, and M. V. Wüthrich, “Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness,”
*Insurance: Mathematics and Economics*, vol. 44, no. 2, pp. 164–169, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. F. Alves, L. de Haan, and C. Neves, “A test procedure for detecting super-heavy tails,”
*Journal of Statistical Planning and Inference*, vol. 139, no. 2, pp. 213–227, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Beirlant, E. Joossens, and J. Segers, “Second-order refined peaks-over-threshold modelling for heavy-tailed distributions,”
*Journal of Statistical Planning and Inference*, vol. 139, no. 8, pp. 2800–2815, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Ibragimov, “Heavy-tailedness and threshold sex determination,”
*Statistics and Probability Letters*, vol. 78, no. 16, pp. 2804–2810, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Delgado, “A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic,”
*Stochastic Processes and Their Applications*, vol. 117, no. 2, pp. 188–201, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. S. Taqqu, “The modelling of ethernet data and of signals that are heavy-tailed with infinite variance,”
*Scandinavian Journal of Statistics*, vol. 29, no. 2, pp. 273–295, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. G. Lindsay, J. Kettenring, and D. O. Siegmund, “A report on the future of statistics,”
*Statistical Science*, vol. 19, no. 3, pp. 387–413, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Resnick, “On the foundations of multivariate heavy-tail analysis,”
*Journal of Applied Probability*, vol. 41, pp. 191–212, 2004. View at: Google Scholar | Zentralblatt MATH - S. Resnick and H. Rootzén, “Self-similar communication models and very heavy tails,”
*The Annals of Applied Probability*, vol. 10, no. 3, pp. 753–778, 2000. View at: Google Scholar | Zentralblatt MATH - J. Cai and Q. Tang, “On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications,”
*Journal of Applied Probability*, vol. 41, no. 1, pp. 117–130, 2004. View at: Publisher Site | Google Scholar - V. Limic, “A LIFO queue in heavy traffic,”
*The Annals of Applied Probability*, vol. 11, no. 2, pp. 301–331, 2001. View at: Google Scholar | Zentralblatt MATH - H. Le and A. O. 'Hagan, “A class of bivariate heavy-tailed distributions,”
*San-Khyā: The Indian Journal of Statistics, Series B*, vol. 60, no. 1, pp. 82–100, 1998. View at: Google Scholar | Zentralblatt MATH - M. C. Bryson, “Heavy-tailed distributions: properties and tests,”
*Technometrics*, vol. 16, no. 1, pp. 61–68, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Beran, “Discussion: heavy tail modeling and teletraffic data,”
*The Annals of Statistics*, vol. 25, no. 5, pp. 1852–1856, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Ahn, J. H. T. Kim, and V. Ramaswami, “A new class of models for heavy tailed distributions in finance and insurance risk,”
*Insurance: Mathematics and Economics*, vol. 51, no. 1, pp. 43–52, 2012. View at: Publisher Site | Google Scholar - V. Pisarenko and M. Rodkin,
*Heavy-Tailed Distributions in Disaster Analysis*, Springer, New York, NY, USA, 2010. - S. I. Resnick,
*Heavy-Tail Phenomena Probabilistic and Statistical Modeling*, Springer, New York, NY, USA, 2007, Probabilistic and statistical modeling. - R. J. Adler, R. E. Feldman, and M. S. Taqqu, Eds.,
*A Practical Guide to Heavy Tails: Statistical Techniques and Applications,*, Birkhäuser, Boston, Mass, USA, 1998. - M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 302786, 13 pages, 2012. View at: Publisher Site | Google Scholar - L. Xu, P. C. Ivanov, K. Hu, Z. Chen, A. Carbone, and H. E. Stanley, “Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques,”
*Physical Review E*, vol. 71, no. 5, Article ID 051101, 14 pages, 2005. View at: Publisher Site | Google Scholar - M. Li and J. Y. Li, “On the predictability of long-range dependent series,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 397454, 9 pages, 2010. View at: Publisher Site | Google Scholar - W. Hürlimann, “From the general affine transform family to a Pareto type IV model,”
*Journal of Probability and Statistics*, vol. 2009, Article ID 364901, 10 pages, 2009. View at: Publisher Site | Google Scholar - A. André, “Limit theorems for randomly selected adjacent order statistics from a Pareto distribution,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 21, pp. 3427–3441, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. E. Stanley, “Power laws and universality,”
*Nature*, vol. 378, no. 6557, p. 554, 1995. View at: Google Scholar - I. Eliazar and J. Klafter, “A probabilistic walk up power laws,”
*Physics Reports*, vol. 511, no. 3, pp. 143–175, 2012. View at: Publisher Site | Google Scholar - A. R. Bansal, G. Gabriel, and V. P. Dimri, “Power law distribution of susceptibility and density and its relation to seismic properties: an example from the German Continental Deep Drilling Program (KTB),”
*Journal of Applied Geophysics*, vol. 72, no. 2, pp. 123–128, 2010. View at: Publisher Site | Google Scholar - S. Milojević, “Power law distributions in information science: making the case for logarithmic binning,”
*Journal of the American Society for Information Science and Technology*, vol. 61, no. 12, pp. 2417–2425, 2010. View at: Publisher Site | Google Scholar - Y. Wu, Q. Ye, J. Xiao, and L. X. Li, “Modeling and statistical properties of human view and reply behavior in on-line society,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 969087, 7 pages, 2012. View at: Publisher Site | Google Scholar - A. Fujihara, M. Uchida, and H. Miwa, “Universal power laws in the threshold network model: a theoretical analysis based on extreme value theory,”
*Physica A*, vol. 389, no. 5, pp. 1124–1130, 2010. View at: Publisher Site | Google Scholar - A. Saiz, “Boltzmann power laws,”
*Physica A*, vol. 389, no. 2, pp. 225–236, 2010. View at: Publisher Site | Google Scholar - A. Jaishankar and G. H. McKinley, “Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations,”
*Proceedings of the Royal Society of London Series A*, vol. 469, no. 2149, Article ID 20120284, 2013. View at: Publisher Site | Google Scholar - X. Zhao, P. J. Shang, and Y. L. Pang, “Power law and stretched exponential effects of extreme events in Chinese stock markets,”
*Fluctuation and Noise Letters*, vol. 9, no. 2, pp. 203–217, 2010. View at: Publisher Site | Google Scholar - P. Kokoszka and T. Mikosch, “The integrated periodogram for long-memory processes with finite or infinite variance,”
*Stochastic Processes and Their Applications*, vol. 66, no. 1, pp. 55–78, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Belomestny, “Spectral estimation of the Lévy density in partially observed affine models,”
*Stochastic Processes and Their Applications*, vol. 121, no. 6, pp. 1217–1244, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Simon, “Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs,”
*Expositiones Mathematicae*, vol. 28, no. 3, pp. 290–298, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Lambiotte and L. Brenig, “Truncated Lévy distributions in an inelastic gas,”
*Physics Letters A*, vol. 345, no. 4–6, pp. 309–313, 2005. View at: Publisher Site | Google Scholar - G. Terdik, W. A. Woyczynski, and A. Piryatinska, “Fractional- and integer-order moments, and multiscaling for smoothly truncated Lévy flights,”
*Physics Letters A*, vol. 348, no. 3–6, pp. 94–109, 2006. View at: Publisher Site | Google Scholar - I. Koponen, “Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process,”
*Physical Review E*, vol. 52, no. 1, pp. 1197–1199, 1995. View at: Publisher Site | Google Scholar - J. Behboodian, A. Jamalizadeh, and N. Balakrishnan, “A new class of skew-Cauchydistributions,”
*Statistics and Probability Letters*, vol. 76, no. 14, pp. 1488–1493, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Garbaczewski, “Cauchy flights in confining potentials,”
*Physica A*, vol. 389, no. 5, pp. 936–944, 2010. View at: Publisher Site | Google Scholar - A. J. Field, U. Harder, and P. G. Harrison, “Measurement and modelling of self-similar traffic in computer networks,”
*IEE Proceedings-Communications*, vol. 151, no. 4, pp. 355–363, 2004. View at: Publisher Site | Google Scholar - G. A. Korn and T. M. Korn,
*Mathematical Handbook for Scientists and Engineers*, McGraw-Hill, New York, NY, USA, 1961. - H. Konno and Y. Tamura, “A generalized Cauchy process having cubic non-linearity,”
*Reports on Mathematical Physics*, vol. 67, no. 2, pp. 179–195, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Konno and F. Watanabe, “Maximum likelihood estimators for generalized Cauchy processes,”
*Journal of Mathematical Physics*, vol. 48, no. 10, Article ID 103303, 19 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. A. Lubashevsky, “Truncated Lévy flights and generalized Cauchy processes,”
*European Physical Journal B*, vol. 82, no. 2, pp. 189–195, 2011. View at: Publisher Site | Google Scholar - Y. Liang and W. Chen, “A survey on computing Lévy stable distributions and a new MATLAB toolbox,”
*Signal Processing*, vol. 93, no. 1, pp. 244–251, 2013. View at: Publisher Site | Google Scholar - G. Terdik and T. Gyires, “Lévy flights and fractal modeling of internet traffic,”
*IEEE/ACM Transactions on Networking*, vol. 17, no. 1, pp. 120–129, 2009. View at: Publisher Site | Google Scholar - E. E. Kuruoǧlu, “Density parameter estimation of skewed
*α*-stable distributions,”*IEEE Transactions on Signal Processing*, vol. 49, no. 10, pp. 2192–2201, 2001. View at: Publisher Site | Google Scholar - A. P. Petropulu, J. C. Pesquet, X. Yang, and J. J. Yin, “Power-law shot noise and its relationship to long-memory
*α*-stable processes,”*IEEE Transactions on Signal Processing*, vol. 48, no. 7, pp. 1883–1892, 2000. View at: Publisher Site | Google Scholar - S. Cohen and G. Samorodnitsky, “Random rewards, fractional brownian local times and stable self-similar processes,”
*The Annals of Applied Probability*, vol. 16, no. 3, pp. 1442–1461, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Shao and C. L. Nikias, “Signal processing with fractional lower order moments: stable processes and their applications,”
*Proceedings of the IEEE*, vol. 81, no. 7, pp. 986–1010, 1993. View at: Publisher Site | Google Scholar - L. Landau, “On the energy loss of fast particles by ionization,”
*Journal of Physics*, vol. 8, pp. 201–205, 1944. View at: Google Scholar - D. H. Wilkinson, “Ionization energy loss by charged particles part I. The Landau distribution,”
*Nuclear Instruments and Methods in Physics Research A*, vol. 383, no. 2-3, pp. 513–515, 1996. View at: Publisher Site | Google Scholar - T. Tabata and R. Ito, “Approximations to Landau's distribution functions for the ionization energy loss of fast electrons,”
*Nuclear Instruments and Methods*, vol. 158, pp. 521–523, 1979. View at: Publisher Site | Google Scholar - J. Holtsmark, “Uber die Verbreiterung von Spektrallinien,”
*Annalen der Physik*, vol. 363, no. 7, pp. 577–630, 1919. View at: Publisher Site | Google Scholar - B. Pittel, W. A. Woyczynski, and J. A. Mann, “Random tree-type partitions as a model for acyclic polymerization: holtsmark (3/2-stable) distribution of the supercritical gel,”
*The Annals of Probability*, vol. 18, no. 1, pp. 319–341, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. G. Hummer, “Rational approximations for the holtsmark distribution, its cumulative and derivative,”
*Journal of Quantitative Spectroscopy and Radiative Transfer*, vol. 36, no. 1, pp. 1–5, 1986. View at: Publisher Site | Google Scholar - R. G. Garroppo, S. Giordano, M. Pagano, and G. Procissi, “Testing
*α*-stable processes in capturing the queuing behavior of broadband teletraffic,”*Signal Processing*, vol. 82, no. 12, pp. 1861–1872, 2002. View at: Publisher Site | Google Scholar - J. R. Gallardo, D. Makrakis, and L. Orozco-Barbosa, “Use of
*α*-stable self-similar stochastic processes for modeling traffic in broadband networks,”*Performance Evaluation*, vol. 40, no. 1, pp. 71–98, 2000. View at: Publisher Site | Google Scholar - A. Karasaridis and D. Hatzinakos, “Network heavy traffic modeling using
*α*-stable self-similar processes,”*IEEE Transactions on Communications*, vol. 49, no. 7, pp. 1203–1214, 2001. View at: Publisher Site | Google Scholar - P. R. de Montmort, “Essay d'analyse sur les jeux de hazard,” 1713. View at: Google Scholar
- P. R. de Montmort, “Essay d'analyse sur les jeux de hazard,” American Mathematical Society, 1980. View at: Google Scholar
- D. Bernoulli, “Exposi-tion of a new theory on the measurement of risk,”
*Econometrica*, vol. 22, no. 1, pp. 22–36, 1954. View at: Google Scholar - W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron,
*The Langevin Equation*, World Scientific, Singapore, 2nd edition, 2004. - S. F. Kwok, “Langevin equation with multiplicative white noise: transfor-mation of diffusion processes into the Wiener process in different prescrip-tions,”
*Annals of Physics*, vol. 327, no. 8, pp. 1989–1997, 2012. View at: Publisher Site | Google Scholar - A. V. Medino, S. R. C. Lopes, R. Morgado, and C. C. Y. Dorea, “Generalized Langevin equation driven by Lévy processes: a probabilistic, numerical and time series based approach,”
*Physica A*, vol. 391, no. 3, pp. 572–581, 2012. View at: Publisher Site | Google Scholar - D. Panja, “Generalized langevin equation formulation for anomalous polymer dynamics,”
*Journal of Statistical Mechanics*, vol. 2010, no. 2, Article ID L02001, 2010. View at: Publisher Site | Google Scholar - A. Bazzani, G. Bassi, and G. Turchetti, “Diffusion and memory effects for stochastic processes and fractional Langevin equations,”
*Physica A*, vol. 324, no. 3-4, pp. 530–550, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - E. Lutz, “Fractional Langevin equation,”
*Physical Review E*, vol. 64, no. 5, Article ID 051106, 4 pages, 2001. View at: Google Scholar - M. G. McPhie, P. J. Daivis, I. K. Snook, J. Ennis, and D. J. Evans, “Generalized Langevin equation for nonequilibrium systems,”
*Physica A*, vol. 299, no. 3-4, pp. 412–426, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. S. Fa, “Fractional Langevin equation and Riemann-Liouville fractional derivative,”
*European Physical Journal E*, vol. 24, no. 2, pp. 139–143, 2007. View at: Publisher Site | Google Scholar - B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,”
*Non-Linear Analysis: Real World Applications*, vol. 13, no. 2, pp. 599–606, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions,”
*International Journal of Differential Equations*, vol. 2012, Article ID 649486, 10 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. C. Kou and X. S. Xie, “Generalized langevin equation with fractional gaussian noise: subdiffusion within a single protein molecule,”
*Physical Review Letters*, vol. 93, no. 18, Article ID 180603, 4 pages, 2004. View at: Publisher Site | Google Scholar - H. C. Fogedby, “Langevin equations for continuous time Lévy flights,”
*Physical Review E*, vol. 50, no. 2, pp. 1657–1660, 1994. View at: Publisher Site | Google Scholar - Y. Fukui and T. Morita, “Derivation of the stationary generalized Langevin equation,”
*Journal of Physics A*, vol. 4, no. 4, pp. 477–490, 1971. View at: Publisher Site | Google Scholar - S. C. Kou, “Stochastic modeling in nanoscale biophysics: subdiffusion within proteins,”
*The Annals of Applied Statistics*, vol. 2, no. 2, pp. 501–535, 2008. View at: Publisher Site | Google Scholar - V. V. Anh, C. C. Heyde, and N. N. Leonenko, “Dynamic models of long-memory processes driven by Lévy noise,”
*Journal of Applied Probability*, vol. 39, no. 4, pp. 730–747, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Damping characteristics of a fractional oscillator,”
*Physica A*, vol. 339, no. 3-4, pp. 311–319, 2004. View at: Publisher Site | Google Scholar - B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Response characteristics of a fractional oscillator,”
*Physica A*, vol. 309, no. 3-4, pp. 275–288, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. H. Eab and S. C. Lim, “Fractional generalized Langevin equation approach to single-file diffusion,”
*Physica A*, vol. 389, no. 13, pp. 2510–2521, 2010. View at: Publisher Site | Google Scholar - C. H. Eab and S. C. Lim, “Fractional Langevin equations of distributed order,”
*Physical Review E*, vol. 83, no. 3, Article ID 031136, 10 pages, 2011. View at: Publisher Site | Google Scholar - S. C. Lim and L. P. Teo, “Modeling single-file diffusion with step fractional Brownian motion and a generalized fractional Langevin equation,”
*Journal of Statistical Mechanics*, vol. 2009, no. 8, Article ID P08015, 2009. View at: Publisher Site | Google Scholar - S. C. Lim, L. Ming, and L. P. Teo, “Locally self-similar fractional oscillator processes,”
*Fluctuation and Noise Letters*, vol. 7, no. 2, pp. L169–L179, 2007. View at: Publisher Site | Google Scholar - M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 657839, 9 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. C. Lim, C. H. Eab, K. H. Mak, M. Li, and S. Chen, “Solving linear coupled fractional differential equations and their applications,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 653939, 28 pages, 2012. View at: Publisher Site | Google Scholar - S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,”
*Physics Letters A*, vol. 372, no. 42, pp. 6309–6320, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. A. Gabel and R. A. Roberts,
*Signals and Linear Systems*, John Wiley & Sons, New York, NY, USA, 1973. - M. Carlini, T. Honorati, and S. Castellucci, “Photovoltaic greenhouses: comparison of optical and thermal behaviour for energy savings,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 743764, 10 pages, 2012. View at: Publisher Site | Google Scholar - M. Carlini and S. Castellucci, “Modelling the vertical heat exchanger in thermal basin,” in
*Proceedings of the ICCSA, 2011, Part 4*, vol. 6785 of*Springer Lecture Notes in Computer Science*, pp. 277–286, Springer, New York, NY, USA, 2011. View at: Google Scholar - M. Carlini, C. Cattani, and A. Tucci, “Optical modelling of square solar con-centrator,” in
*Proceedins of the ICCSA, 2011, Part 4*, vol. 6785 of*Springer Lecture Notes in Computer Science*, pp. 287–295, Springer, New York, NY, USA, 2011. View at: Google Scholar - L. Qiu, B. G. Xu, and S. B. Li, “H
_{2}/${H}_{\infty}$ control of networked control system with random time delays,”*Science China Information Sciences*, vol. 54, no. 12, pp. 2615–2630, 2011. View at: Publisher Site | Google Scholar - J. Li, J. Z. Wang, S. K. Wang, L. L. Ma, and W. Shen, “Dynamic image stabilization precision test system based on the Hessian matrix,”
*Science China Information Sciences*, vol. 55, no. 9, pp. 2056–2074, 2012. View at: Publisher Site | Google Scholar - W. X. Zhao and H. F. Chen, “Markov chain approach to identifying Wiener systems,”
*Science China Information Sciences*, vol. 55, no. 5, pp. 1201–1217, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & sons, NewYork, NY, USA, 1993. - T. Hida,
*Brownian Motion*, Springer, New York, NY, USA, 1980. - A. H. Zemanian, “An introduction to generalized functions and the generalized Laplace and Legendre transformations,”
*SIAM Review*, vol. 10, no. 1, pp. 1–24, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH

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Copyright © 2012 Ming Li and Wei Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.