A Closed-Form Approximation to the Distribution for the Sum of Independent Non-identically Generalized Gamma Variates and Applications

Evaluating the sum of independent and not necessarily identically distributed (i.n.i.d) random variables (RVs) is essential to study different variables linked to various scientific fields, particularly, in wireless communication channels. However, it is difficult to evaluate the distribution of this sum when the number of RVs increases. Consequently, the complex contour integral will be difficult to determine. Considering this issue, a more accurate approximation of the distribution function is required. By assuming the probability density function (PDF) of a generalized gamma (GG) RV evaluated in terms of a proper subset H1,0 1,1 class of Fox’s H-function (FHF) and the moment-based approximation to estimate the FHF parameters, a closed-form tight approximate expression for the distribution of the sum of i.n.i.d GG RVs and a sufficient condition for the convergence are investigated. The proposed approximate may be an analytical useful tool for analyzing the performance of certain numbers branch maximal-ratio combining receivers subject to GG fading channels. Hence, various closed-form performance metrics are derived and examined in terms of FHF. Numerical simulations are carried out to illustrate the theoretical results.

CFAR Strategy Formulation and Evaluation Based on Fox’s H-function in Positive Alpha-Stable Sea Clutter

The problem of target detection in impulsive non-Gaussian sea clutter has attracted a lot of attention in recent years. The positive alpha-stable (PαS) distribution has been validated as a suitable model for the impulsive non-Gaussian sea clutter. Since the probability density function (PDF) of the PαS variable cannot be expressed as a closed-form expression, the research into constant false alarm rate (CFAR) detectors in PαS distributed sea clutter is limited. This paper formulates and evaluates some CFAR detectors, such as Greatest Of-CFAR (GO-CFAR), Smallest Of-CFAR (SO-CFAR), Order Statistic-CFAR (OS-CFAR) and censored mean level (CML) detectors, in PαS distributed sea clutter. Firstly, the Fox’s H-function is adopted to express the PDF of the PαS variable, and the cumulative density function based on Fox’s H-function is derived in this paper. Then, by use of the properties of the H-function and PαS distribution, exact expressions of the probabilities of false alarm and detection for CFAR detectors in the PαS background are derived. Some CFAR properties of these detectors in the PαS background are also explored. Numerical results based on derived expressions are given and verified by Monte Carlo simulation. Some analyses of detection performance from a practical perspective are also given.

Generalized Fractional Calculus Operators Associated with K-function

The aim of this paper is to study some properties of K-function introduced by Sharma. Here we establish two theorems which give the image of this K-function under the generalized fractional integral operators involving Fox’s H-function as kernel. Corresponding assertions in term of Euler, Whittaker and K-transforms are also presented. On account of general nature of H-function and K-function a number of results involving special functions can be obtained merely by giving particular values for the parameters.

Certain Unified Integrals Associated with Product of M-Series and Incomplete H-functions

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions (for example, Fox’s H-Function, Incomplete Fox Wright functions, Fox Wright functions and Incomplete generalized hypergeometric functions) and also listed few known results. The results obtained in this work are general in nature and very useful in science, engineering and finance.

A Versatile Integral in Physics and Astronomy and Fox’s H-Function

This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in physics and astronomy. Nuclear reaction rate probability integrals in nuclear physics, Krätzel integrals in applied mathematical analysis, inverse Gaussian distributions, generalized type-1, type-2, and gamma families of distributions in statistical distribution theory, Tsallis statistics and Beck–Cohen superstatistics in statistical mechanics, and Mathai’s pathway model are all shown to be connected to the integral under consideration. Representations of the integral in terms of Fox’s H-function are pointed out.

A Study on Generalized Multivariable Mittag-Leffler Function via Generalized Fractional Calculus Operators

We study some properties of generalized multivariable Mittag-Leffler function. Also we establish two theorems, which give the images of this function under the generalized fractional integral operators involving Fox’s H-function as kernel. Relating affirmations in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some known special cases have also been mentioned in the concluding section.