Several Limit Theorems on Fuzzy Quantum Space

Viliam Ďuriš,Renáta Bartková,Anna Tirpáková

doi:10.3390/math9040438

Viliam Ďuriš, Renáta Bartková + Show 1 more

Open Access

https://doi.org/10.3390/math9040438

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Abstract

The probability theory using fuzzy random variables has applications in several scientific disciplines. These are mainly technical in scope, such as in the automotive industry and in consumer electronics, for example, in washing machines, televisions, and microwaves. The theory is gradually entering the domain of finance where people work with incomplete data. We often find that events in the financial markets cannot be described precisely, and this is where we can use fuzzy random variables. By proving the validity of the theorem on extreme values of fuzzy quantum space in our article, we see possible applications for estimating financial risks with incomplete data.

Highlights

Selected limit theorems, which we shall deal with in the article, are well known from Kolmogorov’s classical probability theory
As the central limit theorem refers to the limit distribution of the averages of independent, -distributed random variables, extreme value theory (EVT)
When using random samples to estimate distributional parameters, we would like to know that as the sample size gets larger, the estimates are probably close to the parameters that they are estimating

Summary

Introduction

Selected limit theorems, which we shall deal with in the article, are well known from Kolmogorov’s classical probability theory. Among the vital concepts of probability theory are the different kinds of convergence of random variables They are especially significant for parts dealing with the validity of various forms of the law, the central limit theorem, and big numbers. We formulated convergences and proved many familiar limit theorems [34,36,45] for fuzzy quantum space on the basis of the analogy of the probability theory notions. As the central limit theorem refers to the limit distribution of the averages of independent, -distributed random variables, extreme value theory (EVT). Mathematics 2021, 9, 438 single process, the behavior of the maxima can be described by the three extreme value distributions: Gumbel, Fréchet and reversed Weibull distribution as suggested by the Fisher–Tippett–Gnedenko theorem. They may be found in several works [51,52,53,54]

Fuzzy Quantum Space

Convergences on a Fuzzy Space

Extreme Value Theorems for Fuzzy Quantum Space

Conclusions