Several Limit Theorems on Fuzzy Quantum Space

: 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.


Introduction
Selected limit theorems, which we shall deal with in the article, are well known from Kolmogorov's classical probability theory. Kolmogorov's work [1] introduced the theoretical axiomatic model in which events connected with the experiment form the Boolean σ-algebra of subsets S of the set Ω. Thus, the probability is that for the σ-additive, nonnegative final function P on S, with values in the interval [0, 1], if {A n } is a sequence of mutually exclusive events from S, then P(∪ n A n ) = ∑ n P(A n ) and P(Ω) = 1. 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 [2]. 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 [3] 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 [4] and Schrödinger [5] 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 [6] 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,[7]). 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 M) as system M, and such a definition is compared with that of the P-measure on fuzzy sets (according to Piasecki [8]), it follows that both objects have a similar algebraic structure. Piasecki submitted a model called soft σ-algebra in the fuzzy set theory in 1985. His model demonstrated several characteristics identical to quantum 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 [50]. A GEV can be identified by the real parameter γ and the extreme value index, and as a stable distribution it is a characteristic exponent α ∈ [0, 2]. Subsequently, several researchers have provided useful applications of extreme value distributions. They may be found in several works [51][52][53][54].

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 (Ω, S, P) is replaced by the couple (A, M) where A is a nonempty set, M ⊂ [0, 1] A is fuzzy σ-algebra of fuzzy subsets of A, such that the following conditions are satisfied: Elements of the set M are called fuzzy subsets of the universe A. In particular, if f is the characteristic function, we call it a crisp set. The symbols ∨ ∞ n=1 f n sup n f n and ∧ ∞ n=1 f n inf n f n indicate a fuzzy union and a fuzzy intersection of the sequence of fuzzy sets is the so-called fuzzy complement. By Piasecki [8], the system M is called a soft σ-algebra.
To define and prove the law of large numbers and the central limit theorem, we need the following basic notions: According to Piasecki [8], a fuzzy state is called the σ-measure. The triplet (A, M, m) where m is a σ-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 [55].
x(E n ) where B R 1 denotes the Borel σ-algebra of the real line R 1 and E c denotes the complement of a set E in R 1 .

Definition 3. Let f ∈ M.
The mapping x f : B R 1 → M is defined by for every E ∈ B R 1 there is a fuzzy observable of fuzzy quantum space (A, M) called the indicator of fuzzy set f ∈ M.
In particular, the null fuzzy observable of fuzzy quantum space (A, M) maps to If τ : R 1 → R 1 is a Borel measurable function and x is a fuzzy observable, then τ • x : E → x τ −1 (E) , E ∈ B R 1 is a fuzzy observable, too. In this way, we define the functional calculus of fuzzy observables. For example, if τ(t) = t 2 , t ∈ R 1 , we write τ • x = x 2 and the like. In particular, if a ∈ R 1 , then ax : E → x t ∈ R 1 : at ∈ E for any E ∈ B R 1 .
Let x be a fuzzy observable of fuzzy quantum space (A, M) and let B x (t) = x((−∞, t)), t ∈ R 1 . Dvurečenskij and Tirpáková [57] proved that the system B x (t) : t ∈ R 1 of fuzzy sets of fuzzy quantum space (A, M) is a one-to-one correspondence to fuzzy observable x. Due to this result, the sum of any pair x and y of fuzzy observables of (A, M) can be introduced as follows: Definition 4. Let x and y be two fuzzy observables of fuzzy quantum space (A, M). If the system B x+y (t) : t ∈ R 1 , B x+y (t) = ∨ r∈Q B x (r) ∧ B y (t − r) , t ∈ R 1 , where Q is the set of all rational numbers, then we determine fuzzy observable z of (A, M). We call it the sum of x and y and write z = x + y.
In the text [57], it was proved that the sum of two observables always exists, and it coincides with the pointwisely-defined sum of observables for σ-algebra of crisp subsets. Moreover, x + y = y + x, (x + y) + z = x + (y + z) for fuzzy observables x, y, and z. The subtraction of fuzzy observables x and y is defined as follows: The mean value of a fuzzy observable on fuzzy quantum space (A, M) was defined by Riečan [58] as follows: Let x be a fuzzy observable, and let m be a fuzzy state. If the integral m(x) sense that if one side exists, then the second side exists too, and they are equal. Specially, if

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 (A, M), we used the method of F-σ-ideals, which enabled us to reformulate and prove many of the known limit theorems of the classical probability theory for the fuzzy quantum space (A, M). The basic idea of the F-σ-ideals method is described in [54], and we can shortly describe it as follows: Let m be a fuzzy state on fuzzy  [16] proved that I m is a σ-algebra of fuzzy quantum space (A, M). The relation "∼ m " defined on fuzzy quantum space is a probability measure on the Boolean σ-algebra M/ ∼ m . According to the Loomis-Sikorski theorem in [60], there is a measurable space (Ω, S) and σ-homomorphism ϕ from S onto M/ ∼ m , and due to Varadarajan [7] there are functions u, u 1 , u 2 , . . . : where h • x is an observable of a Boolean σ-algebra M/ ∼ m . Moreover, mapping µ ϕ : S → [0, 1] , defined as µ ϕ (Λ) = µ(ϕ(Λ)), Λ ∈ S, is a probability measure on S. Gudder and Mullikin [51] introduced many types of convergences for observables in quantum logics. Inspired by their definition, Dvurečenskij and Tirpaková introduced [57] the following definition: (viii) almost uniformly in fuzzy state m, if for every ε > 0 there is an element f ∈ M, such that m( f ) ≤ ε and a sequence {x n } ∞ n=1 converges uniformly to x on f .
To prove the law of large numbers and the central limit theorem on fuzzy quantum space (A, M), we also need the next theorem, which was proved in Dvurečenskij and Tirpaková [57]. In the following, we will continue to introduce the notion of the independence of fuzzy observables {x n } ∞ n=1 in fuzzy state m. Now, we define the joint fuzzy observable of fuzzy observables. Definition 6. Let x 1 , x 2 , . . . , x n , n ≥ 2, be a finite system of fuzzy observables on fuzzy quantum space (A, M). A joint fuzzy observable of fuzzy observables x 1 , x 2 , . . . , x n is a σ-homomorphism . . , n}, E ∈ B R 1 where π i : R n → R 1 is the projection into the i-th coordinate.

Definition 7.
Fuzzy observables x 1 , x 2 , . . . , x n on fuzzy quantum space (A, M) are independent in fuzzy state m if for every n ≥ 2 there exists joint fuzzy observable T n and for any E i ∈ B R 1 , i = 1, 2, . . . , n.
According to the assumption of independence of the sequence of fuzzy observables {x n } ∞ n=1 , for every n ≥ 2 there exists joint fuzzy observable T n . To each fuzzy observable x i : B R 1 → M , i = 1, 2, . . . , n exists the observable x i = h • x i : B R 1 → M/ ∼ m and a real function u i : Ω → R 1 , such that x i (E) = ϕ u −1 i (E) . We define function Φ : Ω → R 1 , such that Φ n (ω) = (u 1 (ω), u 2 (ω), . . . , u n (ω)), ω ∈ Ω. If n . The main idea of the proof can be illustrated by Figure 1.

Limit Theorems for Fuzzy Quantum Space
Theorem 4.1. (Central limit theorem). Let be a sequence of independent fuzzy observables, identically distributed in fuzzy state , with mean value and variance ∈ (0, ∞). Then, for any ∈ ℝ , the following equality holds:

Limit Theorems for Fuzzy Quantum Space
Theorem 2 (Central limit theorem). Let {x n } ∞ n=1 be a sequence of independent fuzzy observables, identically distributed in fuzzy state m, with mean value a and variance σ 2 ∈ (0, ∞). Then, for any s ∈ R 1 , the following equality holds: Proof. We define the real function k n : R n → R 1 as follows k n (r 1 , r 2 , . . . , Calculate: where k n (Φ n (ω)) = 1 Regarding Theorem 1.7.5 [63], the validity of the following argument is obvious: Theorem 3 (Weak law of large numbers). Let {x n } ∞ n=1 be a sequence of independent fuzzy observables, identically distributed in fuzzy state m, with the mean value a. Then, converges to null fuzzy observable o in fuzzy state m.
Proof. We define real function k n : R n → R 1 as follows If T n is the joint fuzzy observable of fuzzy observables x 1 , x 2 , . . . , x n , then we define fuzzy observable y n = T n • k −1 n . According to Definition 5 (i) a sequence of fuzzy observables {y n } ∞ n=1 converges to null fuzzy observable o in fuzzy state m, if for every ε > 0 it holds that Calculate: , ε)) .
If k n (Φ n (ω)) = 1 n n ∑ i=1 u i (ω) − a, then according to Theorem 1 A) (i) the sequence of fuzzy observables {y n } ∞ n=1 converges to null fuzzy observable o in fuzzy state m if and only if the sequence of the functions {k n (Φ n )} ∞ n=1 converges to null in measure µ ϕ . According to Theorem 1.10.3 [61], the validity of the arguments is obvious.

Theorem 4 (Strong law of large numbers).
Let {x n } ∞ n=1 be a sequence of fuzzy observables independent in fuzzy state m, such that converges to null fuzzy observable o almost everywhere in fuzzy state m.
Proof. We define real function q n : R n → R 1 as follows: q n (r 1 , r 2 , . . . , r n ) = 1 n (r 1 − E(r 1 ) + r 2 − E(r 2 ) + . . . + r n − E(r n )), where E(r i ) is the mean value r i defined as E(r i ) = R 1 tdµ ϕ (t), r i ∈ R, i = 1, 2, . . . n. Then, q n (Φ n (ω)) = 1 n (∑ n i=1 (u i (ω) − E(u i (ω)))). If T n is a joint fuzzy observable of fuzzy observables x 1 , x 2 , . . . , x n , then we define fuzzy observable z n = T n • q −1 n = 1 n (∑ n i=1 (x i − m(x i ))). According to Definition 5 (ii), a sequence of fuzzy observables {z n } ∞ n=1 converges to null fuzzy observable o almost everywhere in fuzzy state m if for any ε > 0 holds: Then, according to Theorem 1 A) (iii), a sequence of fuzzy observables {z n } ∞ n=1 converges to null fuzzy observable o almost everywhere in fuzzy state m if and only if the sequence of functions {q n (Φ n )} ∞ n=1 converges to null almost everywhere in measure µ ϕ . According to Theorem 1.10.5 [61], the validity of the arguments is obvious.

Extreme Value Theorems for Fuzzy Quantum Space
Let {x n } ∞ n=1 be a sequence of independent, identically-distributed fuzzy observables of fuzzy quantum space (A, M). For any n ≥ 1 we define the real function k n : R n → R 1 as follows: k n (r 1 , r 2 , . . . , r n ) = max{r 1 , r 2 , . . . , r n }.
Let T n be the joint fuzzy observable of fuzzy observables x 1 , x 2 , . . . , x n . We define the maximum fuzzy observables of x 1 , x 2 , . . . , x n as where M n is the fuzzy observable.
Theorem 6 (Fisher-Tippett-Gnedenko theorem). Let {x n } ∞ n=1 be a sequence of fuzzy independent observables identically distributed in fuzzy state m. Let there exist norming constants a n > 0, b n ∈ R and some non-degenerate distribution function H such that lim n→∞ m( 1 a n (M n − b n )(−∞, t)) = H(t) for any t ∈ R.
Then, H belongs to the type of one of the following three types of standard extreme value distributions: Gumbel, Fréchet, or Weibull.
Then q n (Φ n (ω)) = 1 a n (max{u 1 (ω), u 2 (ω), . . . , u n (ω)} − b n ) and We have 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 (A, M). Function F x is called the distribution function of an observable x on fuzzy quantum space (A, M). Proposition 1. If the function F x is the distribution function of an observable x on fuzzy quantum space (A, M), then it satisfies the following conditions: Proof.
We proved that lim n→∞ F x = 1.

Definition 9.
For w > 0 we define excess distribution function F w on fuzzy quantum space for every 0 < t < ω F = sup t; F(t) < 1 . Value ω F is called the right endpoint of distribution function F.

Theorem 7 (Balkema, de Haan-Pickands).
For a sufficiently large w, the excess distribution F w converges to the generalized pareto distribution. Parameter β = β(w) is dependent on threshold w, and for every α > 0 Proof. Let x i : B R 1 → M, i = 1, 2, . . . , n be fuzzy observables. Then, there exist observables x i = h • x i : B R 1 → M/I m and real functions u i : Ω → R 1 , such that is the distribution function of real random variable u. It is obvious that

Conclusions
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 [52]. 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 [64].