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.
Selected limit theorems, which we shall deal with in the article, are well known from Kolmogorov’s classical probability theory. Kolmogorov’s work  introduced the theoretical axiomatic model in which events connected with the experiment form the Boolean -algebra of subsets of the set . Thus, the probability is that for the -additive, nonnegative final function on , with values in the interval , if is a sequence of mutually exclusive events from , then and . Limit theorems have a wide range of use in this theory. Their validity has already been proven for other structures (spaces), e.g., MV-algebras defined in . We want to extend their use; therefore, in this article we prove that they also apply to sets in which we are working with incomplete data. Specifically, they also apply to Fuzzy quantum space, and that is the most significant finding in this article.
After some time, it became apparent that Kolmogorov’s classical model of the probability theory was not sufficient for describing quantum mechanics situations. Birkhoff and von Neumann  referred to the fact that the set of experimentally verifiable statements about the quantum mechanical system does not have the same algebraic structure as Boolean algebra. Heisenberg  and Schrödinger  put forth the earliest attempts at the mathematical formulation of quantum mechanics. Schrödinger presented the formalism of wave mechanics, while Heisenberg proposed the formalism of matrix mechanics.
Zadeh  wrote about the theory of fuzzy sets in the 1960s. The current quantum theory, basic mathematical model is that of von Neumann, grounded in the geometry of Hilbert space (Varadarajan, ). If we define all closed subspaces of a given Hilbert space (where, according to Varadarajan, the notion “a state of system” means a measure of probability on ) as system , and such a definition is compared with that of the -measure on fuzzy sets (according to Piasecki ), it follows that both objects have a similar algebraic structure. Piasecki submitted a model called soft -algebra in the fuzzy set theory in 1985. His model demonstrated several characteristics identical to quantum logics. That comparison was first noted by Riečan  and then by Pykacz , and it led us to the idea to build a quantum theory based on fuzzy sets. If is a non-empty set called a universum and is a system of fuzzy subsets of universum , i.e., the system of functions on with values in the interval , then according to Riečan  we say that is an -quantum space, also referred to by Dvurečenskij and Chovanec  as a fuzzy quantum space, or by Dvurečenskij  as a fuzzy measurable space.
Many writers have attempted to prove some known assertions from the classical probability theory in the theory of fuzzy quantum spaces. For example, Dvurečenskij , Navara , and Navara and Pták [14,15] studied the existence of a fuzzy state on fuzzy quantum space while Dvurečenskij and Riečan [16,17] examined joint fuzzy observables and joint distributions of fuzzy observables. The representation theorem was proved by Dvurečenskij, Kôpka, and Riečan ; it also includes the case in fuzzy quantum space. Riečan [19,20] looked at the theory of an indefinite integral on fuzzy quantum space. Mesiar [21,22,23], Piasecki [24,25], and Piasecki and Svitalski  investigated the extension of the validity of the Bayes formula for fuzzy sets. Markechová [27,28,29] researched the entropy on fuzzy quantum space, and Tirpáková and Markechová  investigated the fuzzy analogies of some ergodic theorems and Birkhoff’s individual ergodic theorem and maximal ergodic theorem for fuzzy dynamical systems .
The existence of the sum of fuzzy observables is a key fact for the analysis of many assertions in the fuzzy sets theory. The existence of the sum of compatible fuzzy observables was proved by Harman and Riečan .
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. As a result, the problem of generalizing different types of convergence for fuzzy quantum space became topical. A few authors studied particular types of convergences on quantum logic. Here, we mention the writings which were the basic material for the study of various types of convergences of fuzzy observables on fuzzy quantum space : Dvurečenskij and Pulmanová , Jajte , Ochs [35,36], Cushen , Gudder , and Révesz . Some types of convergences of fuzzy observables on fuzzy quantum space were dealt with by Dvurečenskij , Riečan [41,42], Chovanec and Kôpka , Kôpka and Chovanec , and others.
We formulated convergences and consequently 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, equally-distributed random variables, extreme value theory (EVT) addresses the limit distribution of the maximums of the independent, equally-distributed random variables [45,46]. EVT’s principal objective is to know or predict the statistical probabilities of events that have never or rarely been observed. Kotz and Nadarajah  indicated that the extreme value distributions could be traced back to Bernoulli’s 1709 work . The theory of max-stable distribution functions, the counterpart of Feller stable distributions , formed the basis of the probability background. First, the statistical analysis of extreme values was performed in order to study flood levels. These days, the areas of application include finance, meteorological events, insurance, industry, or the environmental sciences . Allow , , …, to be a sequence of , independently and identically distributed random variables with distribution function . The corresponding ordered sequence in non-decreasing order is indicated by , , …, , where , represents the -th order statistic. and stand for the sample minimum and the sample maximum, respectively. Then, examine the sequence of maxima , , for , obtained from the above sequence. All the sequence minimum results can be obtained from those of the sequence maximum since . ’ s exact distribution can be obtained from the distribution function . In fact, for all : For a 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. One can combine these three distributions into a single family of continuous cumulative distribution functions, known as the generalized extreme value (GEV) distributions . A GEV can be identified by the real parameter and the extreme value index, and as a stable distribution it is a characteristic exponent . Subsequently, several researchers have provided useful applications of extreme value distributions. They may be found in several works [51,52,53,54].
2. Fuzzy Quantum Space
First, we recall the definitions of basic notions and some facts that will be used in the following text. In the quantum space approach to the fuzzy quantum theory, the triple is replaced by the couple where is a nonempty set, is fuzzy -algebra of fuzzy subsets of , such that the following conditions are satisfied:
if for any then
if , then
, for any
if for any then
Elements of the set are called fuzzy subsets of the universe . In particular, if is the characteristic function, we call it a crisp set. The symbols and indicate a fuzzy union and a fuzzy intersection of the sequence of fuzzy sets . The event is the so-called fuzzy complement. By Piasecki , the system is called a soft -algebra.
To define and prove the law of large numbers and the central limit theorem, we need the following basic notions:
A fuzzy state on fuzzy quantum spaceis a mapping, such that
ifis a sequence of pairwise orthogonal fuzzy subsets from, i.e.,, , whenever, then
According to Piasecki , a fuzzy state is called the -measure. The triplet where is a-measure is called a fuzzy probability space. This structure was studied in [55,56].
For illustration, we give the following example of a nontrivial fuzzy quantum space .
Considerwhere,,,for every. It is evident that. We define the mappingby the equalities,, and. Then the tripletis a fuzzy probability space.
A fuzzy observable on fuzzy quantum spacemaps to, satisfying the following properties:
if , then
wheredenotes the Borel-algebra of the real lineanddenotes the complement of a setin.
Let. The mappingis defined by
for everythere is a fuzzy observable of fuzzy quantum spacecalled the indicator of fuzzy set.
In particular, the null fuzzy observable of fuzzy quantum space maps to defined by
If is a Borel measurable function and is a fuzzy observable, then is a fuzzy observable, too. In this way, we define the functional calculus of fuzzy observables. For example, if , we write and the like. In particular, if then for any .
Let x be a fuzzy observable of fuzzy quantum space and let , . Dvurečenskij and Tirpáková  proved that the system of fuzzy sets of fuzzy quantum space is a one-to-one correspondence to fuzzy observable . Due to this result, the sum of any pair and of fuzzy observables of can be introduced as follows:
Letandbe two fuzzy observables of fuzzy quantum space. If the systemwhereis the set of all rational numbers, then we determine fuzzy observableof. We call it the sum ofandand write.
In the text , it was proved that the sum of two observables always exists, and it coincides with the pointwisely-defined sum of observables for -algebra of crisp subsets. Moreover, , for fuzzy observables , and . The subtraction of fuzzy observables and is defined as follows: , where . The mean value of a fuzzy observable on fuzzy quantum space was defined by Riečan  as follows: Let be a fuzzy observable, and let be a fuzzy state. If the integral exists, then is called the mean value of in , where , is a probability measure on . In addition, if is a Borel measurable function, then in the sense that if one side exists, then the second side exists too, and they are equal. Specially, if , then is called the dispersion of fuzzy observable x in fuzzy state .
3. Convergences on a Fuzzy Space
Various types of convergences of random variables belong among important concepts of the probability theory. Therefore, the notion of a fuzzy observable is an analogy to the notion of a random variable. When defining different types of convergence and for the proof of limit theorems on fuzzy quantum space we used the method of --ideals, which enabled us to reformulate and prove many of the known limit theorems of the classical probability theory for the fuzzy quantum space . The basic idea of the --ideals method is described in , and we can shortly describe it as follows: Let be a fuzzy state on fuzzy quantum space . Denote . Dvurečenskij and Riečan  proved that is a -algebra of fuzzy quantum space . The relation “” defined on fuzzy quantum space via if and only if is the congruence, and, moreover, is the Boolean -algebra (in the sense of Sikorski ), where complementation “ ” in is defined with properties , and , , . Then, according to these properties, the mapping defined by is a -homomorphism from onto . The mapping from a Boolean -algebra into the interval , defined by for every , is a probability measure on the Boolean -algebra . According to the Loomis–Sikorski theorem in , there is a measurable space and -homomorphism from onto , and due to Varadarajan  there are functions such that
where is an observable of a Boolean -algebra . Moreover, mapping , defined as , is a probability measure on .
Gudder and Mullikin  introduced many types of convergences for observables in quantum logics. Inspired by their definition, Dvurečenskij and Tirpaková introduced  the following definition:
We say that sequenceof fuzzy observables on fuzzy quantum spaceconverges to fuzzy observable
in fuzzy state, if for every, we have
almost everywhere in the fuzzy state, if for every , we have
in a mean, where, if
everywhere on, if
uniformly on, if for every, there is an integer, such that
uniformly, if for every, there is an integer, such that
almost uniformly in fuzzy state, if for everythere is an elementsuch thatand a sequenceconverges uniformly to on .
To prove the law of large numbers and the central limit theorem on fuzzy quantum space , we also need the next theorem, which was proved in Dvurečenskij and Tirpaková .
Letbe a fuzzy state of fuzzy quantum space,be fuzzy observables ofandbe functions with properties (1) and (2). Then,
The sequence of fuzzy observablesconverges to fuzzy observable
in fuzzy stateif and only if the sequence of functionsconverges toin measure,
almost uniformly in fuzzy stateif and only if the sequence of functionsconverges almost uniformly toin measure,
almost everywhere in fuzzy stateif and only if the sequence of functionsconverges almost everywhere toin measure,
in meanif and only ifconverges toin meanin measure.
If the sequence of fuzzy observablesconverges to fuzzy observable
everywhere, then there issuch thatand the sequenceconverges toeverywhere on,
uniformly, then there issuch thatand the sequenceconverges touniformly on,
uniformly on, then there issuch thatand the sequenceconverges touniformly on.
Conversely, if the sequence of functionsdefined by (1), (2) converges to
everywhere, thenconverges to fuzzy observableeverywhere on, where
uniformly, thenconverges to fuzzy observableuniformly on, where
uniformly on, , then fuzzy observablesconverges to fuzzy observableuniformly on, where.
In the following, we will continue to introduce the notion of the independence of fuzzy observables in fuzzy state . Now, we define the joint fuzzy observable of fuzzy observables.
Let,, be a finite system of fuzzy observables on fuzzy quantum space. A joint fuzzy observable of fuzzy observablesis a-homomorphism, such that
whereis the projection into the-th coordinate.
In accordance with Riečan  and Riečan and Neubrunn , a sufficient condition for the existence of the joint fuzzy observable of fuzzy observables , , meets the condition for every .
Fuzzy observableson fuzzy quantum spaceare independent in fuzzy stateif for everythere exists joint fuzzy observableand
According to the assumption of independence of the sequence of fuzzy observables for every there exists joint fuzzy observable . To each fuzzy observable , exists the observable and a real function , such that . We define function such that . If , then . The main idea of the proof can be illustrated by Figure 1.
4. Limit Theorems for Fuzzy Quantum Space
Theorem2(Central limit theorem).
Letbe a sequence of independent fuzzy observables, identically distributed in fuzzy state, with mean valueand variance. Then, for anythe following equality holds:
We define the real function as follows
Regarding Theorem 1.7.5 , the validity of the following argument is obvious:
Theorem3(Weak law of large numbers).
Letbe a sequence of independent fuzzy observables, identically distributed in fuzzy state, with the mean value. Then,
converges to null fuzzy observablein fuzzy state.
We define real function as follows
If is the joint fuzzy observable of fuzzy observables , then we define fuzzy observable . According to Definition 5 (i) a sequence of fuzzy observables converges to null fuzzy observable in fuzzy state , if for every it holds that
If , then according to Theorem 1 A) (i) the sequence of fuzzy observables converges to null fuzzy observable in fuzzy state if and only if the sequence of the functions converges to null in measure . According to Theorem 1.10.3 , the validity of the arguments is obvious. □
Theorem4 (Strong law of large numbers).
Letbe a sequence of fuzzy observables independent in fuzzy statesuch that
converges to null fuzzy observablealmost everywhere in fuzzy state.
We define real function as follows:
where is the mean value defined as . Then, . If is a joint fuzzy observable of fuzzy observables , then we define fuzzy observable . According to Definition 5 (ii), a sequence of fuzzy observables converges to null fuzzy observable almost everywhere in fuzzy state if for any holds:
Then, according to Theorem 1 A) (iii), a sequence of fuzzy observables converges to null fuzzy observable almost everywhere in fuzzy state if and only if the sequence of functions converges to null almost everywhere in measure . According to Theorem 1.10.5 , the validity of the arguments is obvious. □
5. Extreme Value Theorems for Fuzzy Quantum Space
Let be a sequence of independent, identically-distributed fuzzy observables of fuzzy quantum space . For any we define the real function as follows:
Let be the joint fuzzy observable of fuzzy observables . We define the maximum fuzzy observables of as
where is the fuzzy observable.
Letbe a sequence of independent random variables with the same distribution functionin fuzzy state. Put. Let there exist,such that
whereis a continuous distribution function, increasing on an interval. Then,has one of three distributions with parameters(Figure 2):
Letbe a sequence of fuzzy independent observables identically distributed in fuzzy state. Let there exist norming constants,and some non-degenerate distribution functionsuch that
Then,belongs to the type of one of the following three types of standard extreme value distributions: Gumbel, Fréchet, or Weibull.
We define the real function as follows:
Then, according to Theorem 5, the validity of the arguments is obvious. □
Now, we define the distribution function and excess distribution function on fuzzy quantum space .
Letbe a fuzzy state andbe a fuzzy observable on fuzzy quantum space. For anywe define functionas
Functionis called the distribution function of an observable x on fuzzy quantum space.
If the functionis the distribution function of an observableon fuzzy quantum space, then it satisfies the following conditions:
is left continuous
Let then it follows that
We proved that the function is non-decreasing.
Let then it follows that
We proved that the function is left continuous.
Let then , it follows that
We proved that
Let then , it follows that
We proved that □
Now we define the excess distribution function on fuzzy quantum space .
Forwe define excess distribution functionon fuzzy quantum spaceas
for everyValueis called the right endpoint of distribution function.
Theorem7 (Balkema, de Haan–Pickands).
For a sufficiently large, the excess distributionconverges to the generalized pareto distribution. Parameteris dependent on threshold, and for every
Let M, be fuzzy observables. Then, there exist observables and real functions such that . Then,
is the distribution function of real random variable . It is obvious that
The seminal theoretical results in probability theory are limit theorems. 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. In statistical inference, the central limit theorem is the dominant and most useful theorem. It allows us to make the assumption that, for a population, a normal distribution will occur regardless of what the initial distribution looks like for a sufficiently large sample size. When the distribution shape is not known or the population is not normally distributed, the theorem is used to make assumptions. The law of large numbers is an invaluable tool that is expected to state definite things about the real-world results of unexpected events. The law of large numbers is the postulate of statistics and probability theory that states that the greater the number of samples are used from an event, the closer the monitoring results will be to the average population. Thus, the law of large numbers describes the stability of big random variables. Both the strong and weak laws refer to the convergence of the sample mean to the population mean as the sample size gets bigger.
This paper generalizes the central limit theorem, the law of large numbers, and extreme value theorems of classical probability theory to fuzzy quantum spaces. Extreme value theory models rare events outside the range of allowable observations with high impact. This method has become a widely-used tool for risk assessment in recent years. It is used in the areas of insurance, banking, operational risk, market risk, and credit risk . By applying these limit theorems to the Atanassov set, it gives us the space to work with incomplete data, which we can use in the area of finance. The basic advantage of fuzzy logic is the ability to mathematically express information expressed verbally. Thanks to this, fuzzy logic proves to be a very good tool for working with behavioral data. Behavioral finance takes into account the human factor when making financial decisions. For this reason, behavioral finance often uses linguistic data, and therefore it is appropriate to use methods based on fuzzy logic to describe them. Behavioral finance is a financial field examining the effect of social, cognitive, and emotional factors on the economic decisions of individuals and institutions as well as the consequences of these decisions on market prices .
All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Our thanks belong to Beloslav Riečan, who contributed the results used in our work. Rest in peace, our beloved friend, co-worker, and teacher
Conflicts of Interest
The authors declare that they have no conflicts of interests.
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