scholarly journals Random Variables and Stable Distributions on Fractal Cantor Sets

2019 ◽  
Vol 3 (2) ◽  
pp. 31 ◽  
Author(s):  
Alireza Khalili Golmankhaneh ◽  
Arran Fernandez
Keyword(s):  
Probability Theory ◽  
Shannon Entropy ◽  
Random Variables ◽  
Stable Distributions ◽  
Distribution Functions ◽  
Cantor Sets ◽  
Physical Models ◽  
Fractal Sets ◽  
Fractal Calculus ◽  
Related Distribution

In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results.

scholarly journals Non-local Integrals and Derivatives on Fractal Sets with Applications

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 542-548 ◽  
Author(s):  
Alireza K. Golmankhaneh ◽  
D. Baleanu
Keyword(s):  
Differential Equations ◽  
Cantor Sets ◽  
Physical Models ◽  
Scaling Properties ◽  
Fractal Sets ◽  
Fractal Differential Equations ◽  
Non Local

AbstractIn this paper, we discuss non-local derivatives on fractal Cantor sets. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared. Related physical models are also suggested.

CHARACTERIZATIONS OF THE CLASS OF FREE SELF-DECOMPOSABLE DISTRIBUTIONS AND ITS SUBCLASSES

2009 ◽  
Vol 12 (01) ◽  
pp. 51-65 ◽  
Author(s):  
NORIYOSHI SAKUMA
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Stable Distributions ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Limiting Distributions ◽  
Classical Probability ◽  
Classical Probability Theory

In this paper, we firstly characterize the class of free self-decomposable distributions as a class of limiting distributions of suitably normalized partial sums of free independent random variables. Secondly, we introduce nested classes between the class of free self-decomposable distributions and that of free stable distributions, characterize them and show that the limit of the nested classes coincides with the closure of the class of free stable distributions. All results here are analogues of the results known in classical probability theory.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Finding Maximum Clique in Stochastic Graphs Using Distributed Learning Automata

2015 ◽  
Vol 23 (01) ◽  
pp. 1-31 ◽  
Author(s):  
Alireza Rezvanian ◽  
Mohammad Reza Meybodi
Keyword(s):  
Community Structure ◽  
Dominating Set ◽  
Learning Automata ◽  
Random Variables ◽  
Graph Model ◽  
Maximum Clique ◽  
Distribution Functions ◽  
Maximum Clique Problem ◽  
Stochastic Graphs ◽  
Stochastic Graph

Because of unpredictable, uncertain and time-varying nature of real networks it seems that stochastic graphs, in which weights associated to the edges are random variables, may be a better candidate as a graph model for real world networks. Once the graph model is chosen to be a stochastic graph, every feature of the graph such as path, clique, spanning tree and dominating set, to mention a few, should be treated as a stochastic feature. For example, choosing stochastic graph as the graph model of an online social network and defining community structure in terms of clique, and the associations among the individuals within the community as random variables, the concept of stochastic clique may be used to study community structure properties. In this paper maximum clique in stochastic graph is first defined and then several learning automata-based algorithms are proposed for solving maximum clique problem in stochastic graph where the probability distribution functions of the weights associated with the edges of the graph are unknown. It is shown that by a proper choice of the parameters of the proposed algorithms, one can make the probability of finding maximum clique in stochastic graph as close to unity as possible. Experimental results show that the proposed algorithms significantly reduce the number of samples needed to be taken from the edges of the stochastic graph as compared to the number of samples needed by standard sampling method at a given confidence level.

scholarly journals Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Thabet Abdeljawad ◽  
Saima Rashid ◽  
Zakia Hammouch ◽  
İmdat İşcan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Functions ◽  
Fractional Derivatives ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Auxiliary Result ◽  
Fractal Sets ◽  
Different Types ◽  
Novel Applications ◽  
Class Of Functions ◽  
Harmonically Convex Functions

Abstract The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

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