Fractional calculus approach to the statistical characterization of random variables and vectors

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Abstract

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.

Introduction

Fractional moments are very useful in dealing with random variables with power-law distributions, F(x)|x|μ, μ>0, where F(x) is the distribution function. Indeed, in such cases, moments |X|q exist only if q<μ and integer order moments greater than μ diverge. Distributions of this type are encountered in a wide variety of contexts; see the extensive literature in Refs. [1], [2] where power-law statistics appear in the framework of anomalous diffusion in many fields of applied science. To name but a few examples, we mention the travel length distribution in human motion patterns and animal search processes [3], [4], [5], [6], fluctuations in plasma devices [7], [8], scaling laws in polymer physics and its applications to gene regulation models [9], [10], as well as the distribution of time scales in processes such as the motion of charge carriers in amorphous semiconductors [11], tracer dispersion in groundwater [12], or sticking times in turbulent flows [13].

The lack of moments is of course a great limitation for the characterization of such distributions, because many methods of statistical analysis fail, like for example, the representation of the characteristic function or the log-characteristic function by moments or by cumulants, respectively. Then, it is necessary to resort to other distribution properties and a good option is given by the so-called fractional moments which have been extensively studied in the seventies and have recently attracted new interest both in theoretical [14], [15], [16] and in experimental settings [17]. In particular, in Refs. [15], [16] the moment problem originally stated in terms of integer moments, has been extended to fractional moments where it has been shown that the knowledge of some fractional moments Xq improves significantly the convergence speed of the maximum entropy method for non-negative densities–which is defined in [0,[–if a proper optimization procedure is used in selecting the order q. This can be explained by the non-local nature of fractional moments that will be better highlighted in the following.

Fractional moments of a non-negative random variable are expressible by the Mellin transform of the density and this fact has been widely used in literature principally in the field of the algebra of random variables. That is, the Mellin transform is the principal mathematical tool to handle problems involving products and quotients of independent random variables. As the approach presented in this paper is based on the Mellin transform, a brief outline of previous results is given in the following.

The first author who developed a systematic method to express the density of the product of independent random variables by the Mellin transform was Epstein. As consequence, in his pioneering work [18] he recognizes that any density function can be obtained as Mellin convolution of two independent densities. It must be remarked that such a result was already obtained by Ref. [19] outside the framework of integral transforms. Many other studies on the use of Mellin transform extending Epstein’s work have been reported providing explicit analytic forms for products of independent random variables with assigned densities. In Ref. [20] examples on product and quotient of independent random variables with Rayleigh distribution and moments of Rice distribution are obtained by Mellin transform tables. An extensive use of such concepts and the state-of-art on the algebra of random variables may be found in Ref. [21].

Another research direction on Mellin applications in probability is represented by the use of special functions like Fox’ H-functions and the Meijer’s G-functions, due principally to Mathai and co-workers. Such functions are indeed representable as Mellin–Barnes integrals of the product of gamma functions and are therefore suited to represent statistics of products and quotients of independent random variables whose fractional moments are expressible as gamma or gamma related functions. Applications of special functions to statistics and probability theory can be found in Refs. [22], [23], [24], [25], [26], [27] and references therein.

Whether fractional moments exists and how they are correlated to local properties of the characteristic function has been investigated in Refs. [28], [29], [30], [31]. In particular, in Ref. [28] a relation is derived between the absolute moment of real order and integrals involving the Marchaud fractional derivative of the characteristic function. Other relevant studies on the existence of fractional (absolute) moments can also be found in Refs. [32], [33], [34].

All the cited works share the common issue of using the Mellin transform because it naturally coincides with moments of the type Xγ1 if the random variable X has a density defined only in the positive domain. In this paper, following the previous results in Ref. [35], it is shown that applying the Mellin transform to the characteristic function, and not to the density, a sound representation of the statistics of general multivariate random variables is possible. In this way, exploiting the Hermitian nature of the characteristic function, a unique representation valid for every random variable and vectors, defined in the whole real range is presented. Further, by means of simple properties of fractional operators, a generalization of the well-known link between moments and the derivative of the characteristic function evaluated in zero is provided. Such a link is useful because it allows one to find existence criteria for fractional moments that can be derived in a simpler way with respect to Ref. [28] and clarifies the non-local nature of fractional moments. Moreover, this approach that starts from the characteristic function plainly applies to random variables defined in the whole domain without the need of partitioning. Such theoretical aspects go along with a better numerical treatment of the inverse Mellin transform involved in this approach as it will be stressed in the following.

In the application section, it will be shown that the proposed approach is extremely useful in the statistical characterization of raw data and in the representation of both random variables and vectors with small computational effort. Thus, the good numerical convergence makes the proposed approach a good and reliable tool also for data analysis. We want to stress that the representation proposed in this paper can be applied to any Fourier pair functions used in statistical analysis and, very recently, applications to the representation of stochastic processes have been proposed in Refs. [36], [37].

Section snippets

Probabilistic characterization of random variables

A brief outline of the basic tools of probability will serve to frame the problem in hand and to introduce symbols. Let (S,A,P) be a probability space, where A is a collection of events defining a σ-algebra of subsets on the sample space S, and P a probability measure that assigns a number to each event on A and X:(S,A)(R,B) a random variable, where B is a Borel set. F(x)=Pr(X<x) is the cumulative distribution and p(x)=dF/dx is the probability density function (PDF). The Fourier transform of p(

Extension to multivariate random vectors

In this section, the concepts briefly outlined in the previous section will be derived in case of multivariate random vectors. Multivariate random vectors are fully characterized in probabilistic setting by the joint probability density function, or by its spectral counterpart, namely the joint characteristic function; alternatively also by joint moments or by joint cumulants of every order. Let (S,A,P) be a probability space, X:(S,A)(Rd,Bd), that is XT=(X1,,Xd) denotes a d-dimensional random

Applications

In this section it will be shown how the previous results can be used to characterize probability distributions. The integral forms in the monovariate case, Eqs. , (14), and in the multivariate case, Eqs. (29), (31), will be discretized in a suitable form. A remarkable effect of having applied the Mellin transform to the characteristic function rather than to the density is the presence of the gamma function in the integrand. Indeed, by its fast decay with increasing imaginary part of the

Conclusions

In this paper, we provide new insights into the problem of the statistical representation of random variables by means of the Mellin transform and fractional calculus. We define a class of complex moments (comprehending fractional moments) that can represent every kind of joint PDF and joint characteristic function, including joint characteristic functions that are non-differentiable in zero, and densities with heavy tails, like the bivariate Cauchy distribution or other power-law

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