^{1,2}

^{1}

^{1}

^{2}

Due to the fact that

The pioneering work of

Let

Since the notion of

What is the qualitative structure of an autocorrelation function (ACF) of

What is the main property of its probability density function (PDF)? With P3, we will explain the heavy-tailed property of

What is the possible structure of the differential equation to synthesize

For facilitating the description of the full picture of

A time series

Let

The dependence of

Let

The condition (

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 [

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, (

The Kolmogorov’s work on axiomatic approach of probability theory and stochastic processes needs the assumption that

Denote by

A useful measure called correlation time, which is denoted by

For a conventional Gaussian random function

The PSD of

Equation (

The noise of

Denote by

Denote by

It is worth noting that the above number characteristics are crucial to the analysis of

The qualitative structure of

Note that the above may be taken as a definition of LRD property of a random function; see, for example, [

Qualitatively, the ACF of

A well-known example of

Another example of

The Cauchy-class process with LRD discussed in [

The generalized Cauchy process with LRD reported in [

If a random function

LRD implies that the right side of (

Since any random function with LRD is in the class of

As

Heavy-tailed PDFs are widely observed in various fields of sciences and technologies, including life science and bioengineering; see, for example, [

By heavy tails, we mean that the tail of a PDF

Let

Considering that

Theorem

The tail of

A commonly used instance to clarify Note

Application of the Cauchy distribution to network traffic modeling refers to [

Another type of heavy-tailed random functions without mean and variance is the Lévy distribution; see, for example, [

Application of the Lévy distribution to network traffic modeling is discussed in [

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 [

A stable distribution is characterized by 4 parameters. In general, the analytical expression of

A stable distribution is generally non-Gaussian except for some specific values of parameters. When

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

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 [

The standard Langevin equation is in the form

The standard Langevin equation may not attract people much in the field of

A solution to the stochastically fractional differential equation below belongs to

Denote by

From the above, we see that

The above expression implies that a

Let

We have explained the main properties of

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.

^{α}noise in human cognition

^{α}scaling exponents from short time-series

^{α}power law noise generation

^{γ}power spectrum noise sequence generator

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