On generalized versions of central limit theorems for IF-events

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On generalized versions of central limit theorems for IF-events Piotr Nowak∗, Olgierd Hryniewicz

Q1

Systems Research Institute Polish Academy of Sciences ul, Newelska 6, 01-447 Warszawa, Poland

a r t i c l e

i n f o

Article history: Received 6 October 2015 Revised 18 March 2016 Accepted 30 March 2016 Available online xxx Keywords: Central limit theorem IF-events IF-probability M-probability

a b s t r a c t IF-sets were introduced by Atanassov to model imprecision, which can be often met in real-world applications. IF-sets were also used for definition of IF-events, for which probability theories were considered, including IF-probability theory and M-probability theory. Within these theories IF-observables and M-observables were introduced and their properties have been studied. In this paper we consider limit behavior of the row sums of triangular arrays of independent IF-observables and M-observables. We prove generalized versions of the central limit theorem (CLT for short) for both types of observables, i.e., the Lindeberg CLT and the Lyapounov CLT. Furthermore, we prove the Feller theorem for null arrays of IF-observables and M-observables. Finally, we illustrate our theoretical results by examples. © 2016 Published by Elsevier Inc.

1

1. Introduction

2

The classical Kolmogorov’s theory of probability is generally accepted as the mathematical model of randomness. It has been described in hundreds of textbooks, and is well known both for scientists and practitioners dealing with random phenomena. However, there exist other approaches to model such phenomena, and one of them, the boolean algebraic probability theory, was proposed for the case of quantum systems. Its fundamentals were introduced by Carathéodory and von Neumann (see [5] and [7]), who considered states and observables of a quantum system, which are counterparts of probability and random variables in the Kolmogorov theory. The concepts of Carathéodory and von Neumann were developed in [16,28,39,43], where quantum logics were considered as orthomodular posets. In many practical applications randomness is not the only source of uncertainty. The second such source is imprecision which can be modeled by Zadeh’s fuzzy sets. When uncertain phenomena of interest are both random and imprecise the concept of a fuzzy random variable can be applied. There exist several definitions of the fuzzy random variable which have different interpretations (see [17,18,21,29] for various definitions). According to one of them, introduced by Kwakernaak [18], the fuzzy random variable can be interpreted as fuzzy perception of an original crisp random variable. According to this interpretation the fuzzy random variable is a (disjunctive) fuzzy set of classical random variables, and is described by a fuzzy set of classical probability distributions. This interpretation of the fuzzy random variable is nowadays called epistemic. Another definition was proposed by Puri and Ralescu [29]. According to their definition, to describe the fuzzy random variable σ -algebras of fuzzy sets are used. Thus, it is a classical random variable with values belonging to a set of

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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∗

Corresponding author. Tel.: +48 223810393. E-mail addresses: [email protected], [email protected] (P. Nowak), [email protected] (O. Hryniewicz).

http://dx.doi.org/10.1016/j.ins.2016.03.052 0020-0255/© 2016 Published by Elsevier Inc.

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functions. The interpretation of the fuzzy random variable understood in the sense of Puri and Ralescu is nowadays called ontic. For more information about different interpretations of fuzzy random variables see [11,12]. Non-random imprecision may have different interpretations. Attempts to describe sometimes subtle differences between these interpretations have resulted in several generalizations of fuzzy sets. One of such generalizations, the theory of IF-sets, was introduced by Atanassov (see [1,2] and references therein) in 1980s and has been essentially developed since that time. IF-sets can be regarded as generalizations of fuzzy sets, since they are described by two functions defined on a universe of discourse: one expressing the degree of membership and the other one expressing the degree of nonmembership, which do not need to sum up to one. The majority of results about the fuzzy generalizations of the probability have been related to the classical Kolmogorov’s definition of probability. However, fuzzy models of quantum mechanics have also been studied recently. One should mention the theories of F-quantum spaces and fuzzy quantum logics (for details see [39] and references therein). The fuzzy quantum logic of all measurable functions with values in the interval [0, 1] is an example of MV-algebra, introduced by Chang in [8]. Fundamentals and the most important theorems of MV-algebraic probability theory, including the central limit theorem, can be found in [27,31,37–39]. The MV-algebraic probability theory was also applied in the Atanassov’s IF-sets setting, which gave the possibility for the development of probability theory for IF-events. Riecˇ an in [36] considered two probability theories, involving two types of connectives between IF-sets. The first is the IF-probability theory corresponding to the Łukasiewicz connectives  and . The second, M-probability theory involves the Zadeh connectives ∨ and ∧. In [36] Riecˇ an proved the CLT for independent, identically distributed IF-observables and M-observables. Other results of probability theories concerning IF-events can be found e.g. in [9,19,20,30,32–35]. An appropriate theory of imprecisely- or ill-perceived random observations is not sufficient for many applications. Practitioners also need a well-developed theory of statistical inference for such data. Such a theory exists only for random variables defined in the epistemic sense of Kwakernaak. It was proposed in the seminal book by Kruse and Meyer [17] (for this reason the epistemic approach to fuzzy random variables is sometimes coined as “Kwakernaak–Kruse approach”). References to important results related to this approach can be found in, e.g., [14]. Unfortunately, for the ontic approach to fuzzy random variables such a well-developed theory does not exist. Statistical reasoning in this case can be done using bootstrap simulation techniques and appropriate central limit theorems. The review of recent results related to this approach can be found in [6]. The same problems seem to be evident when more complex models of imprecision, such as, e.g. IF-sets, are considered. Therefore, the introduction of central limit theorems seems to be a prerequisite for the development of workable statistical techniques when observed data are described by complex models. As it has been noted above, some central limits theorems have been already proposed for random IF-sets. They can be viewed upon as generalizations of classical central limit theorems for samples consisted of independent random variables having the same probability distributions. However, for more complex statistical data, coming from observation of , e.g., stochastic processes, we need more general central limit theorems, such as CLT for triangular arrays of random variables. For example, central limit theorems for triangular arrays of independent random variables have applications in time series. In [13] the Lindeberg CLT is used to obtain limiting distribution of a parameter estimator of a periodic first-order autoregressive (PAR(1)) model. In [42] it is shown that the Lindeberg CLT and the Lindeberg condition are closely related to central limit theorems for martingale difference arrays, which are used in asymptotic theory of statistical inference for time series. Another application is described in [26]. Therefore, future development of statistical methods for IF-set based data definitely requires the introduction of such general CLT for random IF-sets. In this paper we consider the limit behavior of the row sums of triangular arrays of independent IF-observables and Mobservables. We prove generalized versions of the central limit theorem for both types of observables, i.e., the Lindeberg CLT and the Lyapounov CLT. In addition, we prove the Feller theorem, which is a converse of the Lindeberg CLT for null arrays of IF-observables and M-observables. In proofs of limit theorems for IF-observables we use their MV-algebraic counterparts from [27]. In this paper we illustrate our theoretical results, applying them to examples of IF-observables and M-observables with convergent scaled row sums. The first example concerns a sequence of independent IF-observables with different distributions, satisfying the Lindeberg condition. In the second one, the distributions of independent M-observables are the same in each row of their array, but they differ between rows. Since they satisfy the Lyapounov condition, similarly as in the previous example, their scaled row sums converge to normal distribution. As IF-probability and M-probability we used the probability of IF-events considered in [40,41] and defined in general form in [15]. A more general notion of monotone measure of IF-sets was studied in [23–25]. The paper is organized as follows. Section 2 contains preliminaries from the classical probability theory and the theory of MV-algebras. Main results are included in Section 3 and Section 4, where the Lindeberg CLT, Lyapounov’s CLT, and the Feller theorem for IF-observables and M-observables are formulated and proved. Examples of applications of the limit theorems are presented in Section 5. Section 6 contains conclusions.

73

2. Basic notions and facts

74

2.1. Selected facts from the classical probability theory

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

75

We recall necessary notions and facts from the classical probability theory. Please cite this article as: P. Nowak, O. Hryniewicz, On generalized versions of central limit theorems for IF-events, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.052

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Let N be the set of all positive integers. For n ∈ N we use the symbols N[n] and B (Rn ) to denote the set {1, 2, . . . , n} and the σ -algebra of Borel subsets of Rn , respectively. The following theorem (see [4], Theorem 16.12) concerning the change of variable for integrals will be useful in the further part of the paper. Let (X, X ) and (X  , X  ) be measurable spaces. Let T: X → X be an X /X  measurable function, i.e. T −1 (A ) ∈ X for each A ∈ X  . For a measure μ on X we define a measure μT −1 on X  given by μT −1 (A ) = μ(T −1 (A )), A ∈ X  .

82

Theorem 1. Let f : X  → R be an X  -measurable function. If f is non-negative, then

76 77 78 79 80



X

83 84

f (T x )μ(dx ) =

86 87 88 89 90 91 92 93 94 95 96 97 98 99 100



( A )

103 104 105

108 109 110

A

  −1     μT dx , A ∈ X  ,

f x

(2)

d

We will write Yn −−−→ N (0, 1 ) if {Yn }∞ converge in distribution to a random variable Y0 and Y0 has the standard normal n=1 n→∞

distribution N(0, 1) (see [3] for further details). We will denote by  the cdf of the standard normal distribution. In this subsection we will consider real-valued random variables. Let {kn }n∈N be a fixed sequence of positive integers such that limn→∞ kn = ∞. Let {((n ) , S(n ) , P(n ) )}n∈N be a sequence of arbitrary probability spaces. For each n ∈ N and a random variable X on ((n ) , S(n ) , P(n ) ), we denote by E(n ) X the expected value of X and by D(2n ) X the variance of X with respect to P(n) . Definition 2. Let, for each n ∈ N, {Xn1 , Xn2 , ..., Xnkn } be a sequence of independent random variables on ((n ) , S(n ) , P(n ) ). Then {Xn j }n∈N, j∈N[kn ] is called a triangular array of independent random variables (TR for short). We present modified versions of the Lindeberg CLT, the Lyapounov CLT, and the Feller theorem in [3], assuming more generally that expected values of Xnj , n ∈ N, j ∈ N[kn ], are not necessarily equal to zero. The presented versions of theorems are straightforward consequences of their counterparts in [3].



Definition 3. Let Xn j



Then Xn j



be a TR such that for each n ∈ N

n∈N, j∈N[kn ]

kn  D(2n ) Xn j ∈ (0, ∞ ).

j ∈ N[kn ], Sn2 =

(3)

j=1

 n∈N, j∈N[kn ]

satisfies the Lindeberg condition if for each ε > 0

n 1 −→ E(n ) ((Xn j − E(n ) Xn j )2 I|Xn j −E(n) Xn j |>εSn )− n→∞ 0. 2 Sn j=1

k



(4)



Theorem 4. (Lindeberg CLT) Let Xn j be a TR satisfying (3) and the Lindeberg condition (4). 

n∈N, j∈N[kn ] kn d kn Then X − j=1 E(n ) Xn j /Sn −−−→ N (0, 1 ). j=1 n j n→∞

We also recall the Lyapounov condition and the Lyapounov CLT.



Definition 5. A TR Xn j

Sn2+δ

107



Let {Yn }∞ be a collection of random variables and let Fn denote the cumulative distribution functions (cdfs for short) of n=0 Yn , n ≥ 0. Then {Yn }∞ n=1 is said to converge in distribution to Y0 if limn→∞ Fn (t ) = F0 (t ) for every point t of continuity of F0 .

kn 1 

106

(1)

hold. Moreover, for any non-negative f, identity (2) always holds.

Ln ( ε ) = 102

  −1    μT dx .

f x

f (T x )μ(dx ) =

E(n ) Xn2j < ∞, 101

X

A function f (not necessarily non-negative) is integrable with respect to μT −1 if and only if fT is integrable with respect to μ, in which case (1) and T −1

85





n∈N, j∈N[kn ]

satisfying (3) is said to satisfy the Lyapounov condition if there exists δ > 0 such that

−→ E(n ) |Xn j − E(n ) Xn j |2+δ − n→∞ 0.

j=1



(5)



be a TR satisfying (3) and Lyapounov’s condition (5). Theorem 6. (Lyapounov CLT) Let Xn j 

n∈N, j∈N[kn ] kn d kn Then X − j=1 E(n ) Xn j /Sn −−−→ N (0, 1 ). j=1 n j n→∞

The following Feller theorem one can treat as converse of the Lindeberg CLT. Theorem 7. (Feller) Let {Xn j }n∈N, j∈N[kn ] be a TR satisfying (3) and such that for each ε > 0 max P (|Xn j | > ε Sn ) −−−→ 0. If (

kn j=1

Xn j −

k n j=1

1≤ j≤kn

n→∞

d

E(n ) Xn j )/Sn −−−→ N (0, 1 ), then (4) is satisfied. n→∞

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2.2. MV-algebras

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MV-algebras can be treated as non-commutative generalizations of boolean algebras. Basic MV-algebraic notions and preliminary results concerning MV-algebras one can find in [10] (see also [22]). In this subsection we recall only selected elements of the general theory of MV-algebras and the MV-algebraic probability theory from [38] and [27] with minor modifications. The possibilities of application of MV-algebras for description of quantum mechanical systems with infinitely many degrees of freedom were discussed in [38].

113 114 115 116 117 118

Definition 8. An MV-algebra (M, 0, 1, ¬, , ) is a system where M is a non-empty set, the operation  is associativecommutative with a neutral element 0, ¬0 = 1, ¬1 = 0, and additionally, for all x, y ∈ M x  1 = 1,

y  ¬(y  ¬x ) = x  ¬(x  ¬y ), x  y = ¬(¬x  ¬y ). 119

In each MV-algebra (M, 0, 1, ¬, , ) the relation ≤ given by

x ≤ y ⇔ x  ¬y = 0 121

is a partial order. Furthermore, the operations x ∨ y = ¬(¬x  y )  y and x ∧ y = ¬(¬x ∨ ¬y ) make M into a distributive lattice (called the underlying lattice of M) with least element 0 and greatest element 1.

122

Definition 9. An MV-algebra M is σ - complete if every non-empty countable subset of M has the supremum in M.

120

123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141

We introduce the following notations. Let {An }∞ n=1 be a sequence of subsets of a set X. Then An A iff A1 ⊆A2 ⊆... and ∞ n An = A. For a sequence {xn }∞ n=1 ⊆ R, xn x iff x1 ≤ x2 ≤ ... and x = supi xi . ∞ Finally, for {bn }n=1 included in an MV-algebra M, bn b iff b1 ≤ b2 ≤ ... and b = supi bi with respect to the underlying order of M. In the probability theory of MV-algebras the notions of state and observable were introduced, similarly to the boolean algebraic probability case, by abstracting the properties of probability measure and classical random variable. Definition 10. Let M be a σ - complete MV-algebra. A state on M is a map m: M → [0, 1] satisfying the following conditions for all a, b, c ∈ M and {an }∞ n=1 ⊂ M: (i) m(1 ) = 1; (ii) if b  c = 0, then m(b) + m(c ) = m(b  c ); (iii) if an a, then m(an ) m(a). We say that m is faithful if m(x) = 0 whenever x = 0 and x ∈ M. Definition 11. A probability MV-algebra is a pair (M, m), where M is a σ - complete MV-algebra and m is a faithful state on M. Definition 12. Let M be a σ - complete MV-algebra. An n -dimensional observable of M is a map x : B (Rn ) → M satisfying the following conditions: (i) x(Rn ) = 1; (ii) whenever A, B ∈ B (Rn ) and A ∩ B = ∅, then

x (A )  x (B ) = 0 142 143 144 145 146 147 148

x(A ∪ B ) = x(A )  x(B );

(iii) for all A, A1 , A2 , ... ∈ B (Rn ), if An A, then x(An ) x(A). Theorem 13. Let M be a σ - complete MV-algebra with an n-dimensional observable x : B (Rn ) → M and a state m. Then the map mx : B (Rn ) → [0, 1] given by mx (A ) = (m ◦ x )(A ) = m(x(A ) ), A ∈ B (Rn ), is a probability measure on B (Rn ). For the proof we refer the reader to [27]. Definition 14. Let (M, m) be a probability MV-algebra. Let x : B (R ) → M be an observable of M. We say that x is integrable in (M, m), and we write x ∈ L1m , if the expectationE(x ) = R t mx (dt ) exists. We say that x is square-integrable in (M, m), and we write x ∈ L2m , if R t 2 mx (dt ) exists. If x ∈ L2m , then its variance also exists and is described by the equality

D2 ( x ) = 149

and



R

t 2 mx (dt ) − (E(x ) )2 = p



R

(t − E(x ) )2 mx (dt ).

More generally, we write x ∈ Lm for p ≥ 1 if

R

|t | p mx (dt ) < ∞.

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Definition 15. Let (M, m) be a probability MV-algebra. Observables x1 , x2 , ..., xn are independent (with respect to m) if there exists an n-dimensional observable h : B (Rn ) → M such that for all C1 , C2 , ..., Cn ∈ B (R )

m(h(C1 × C2 × ... × Cn )) = m(x1 (C1 ) ) · m(x2 (C2 ) ) · ... · m(xn (Cn ) ) = mx1 (C1 ) · mx2 (C2 ) · ... · mxn (Cn ). 152 153 154 155 156 157

Remark 16. Let x1 , x2 , ..., xn : B (R ) → M be independent observables in a probability MV-algebra (M, m). Let g : Rn → R be a Borel measurable function and let h : B (Rn ) → M be the joint observable of x1 , x2 , ..., xn . Then g(x1 , x2 , ..., xn ) = h ◦ g−1 is an observable. Let {kn }n∈N be a fixed sequence of positive integers such that limn→∞ kn = ∞. Let {(M(n ) , m(n ) )}n∈N be a sequence of probability MV-algebras. For each n ∈ N and observable x : B (R ) → M(n ) , we denote by E(n ) (x ) the expected value of x and by D2(n ) (x ) the variance of x with respect to m(n) . For each s > 0 we also consider the R-valued function



2

lnx (ε , s ) = E(n ) ( x − E(n ) (x ) I|x−E(n) (x )|>εs ). 159

Definition 17. Let, for n ∈ N, {xn1 , xn2 , ..., xnkn } be a sequence of independent (with respect to m(n) ) observables of the MValgebra M(n) . Then {xn j }n∈N, j∈N[kn ] is called a triangular array of independent observables (TA for short).

160

Definition 18. Let {xn j }n∈N, j∈N[kn ] be a TA such that for each n ∈ N

158

xn j ∈ L2m(n) , j ∈ N[kn ], s2n =

kn    D2(n ) xn j ∈ (0, ∞ ).

(6)

j=1

161

Then {xn j }n∈N, j∈N[kn ] satisfies the Lindeberg condition if for each ε > 0

Ln ( ε ) =

n 1 x −→ lnn j (ε , sn )− n→∞ 0. s2n j=1

k



162

Definition 19. Let xn j

163

>0

 n∈N, j∈N[kn ]

be a TA satisfying (6). Then {xn j }n∈N, j∈N[kn ] satisfies the Lyapounov condition if for some δ

kn     1  −→ E(n ) |xn j − E(n ) xn j |2+δ − n→∞ 0. 2+δ

sn 164 165 166 167 168 169 170 171

(7)

(8)

j=1

Definition 20. Let {(M(n ) , m(n ) )}n∈N be a sequence of probability MV-algebras. Let {xn : B (R ) → M(n ) }n∈N be a sequence of observables. The sequence {xn }∞ is convergent in distribution to a function F : R → [0, 1] if limn→∞ m(n ) (xn ( (−∞, t )) ) = n=1 F (t ) for each t ∈ R. The following MV-algebraic versions of limit theorems were proved in [27]. Theorem 21. (Lindeberg CLT) Let {xn j }n∈N, j∈N[kn ] be a TA satisfying (6) and the Lindeberg condition (7).   

n kn −→ Then x − kj=1 E(n ) xn j /sn − n→∞ N (0, 1 ) in distribution. j=1 n j Theorem 22. (Lyapounov CLT) Let {xn j }n∈N, j∈N[kn ] be a TA satisfying (6) and Lyapounov’s condition (8).   

kn kn −→ Then x − E xn j /sn − n→∞ N (0, 1 ) in distribution. n n j ( ) j=1 j=1

174

Theorem 23. (Feller) Let {xn j }n∈N, j∈N[kn ] be a TA satisfying (6) and such that for each ε > 0 max (m(n ) )xn j ((−∞, −ε sn ) ∪ 1≤ j≤kn −→ (ε sn , ∞ ))− n→∞ 0. −→ If ( kn xn j − kn E n (xn j ))/sn − n→∞ N (0, 1 ) in distribution, then the Lindeberg condition (7) is satisfied.

175

3. IF-probability

176

3.1. Basic definitions and theorems

172 173

177 178 179

j=1

j=1

( )

Let (, S ) be a measurable space. By an IF-eventwe mean any pair A = (μA , νA ) of S-measurable functions such that μA ≥ 0, ν A ≥ 0 and μA + νA ≤ 1. The functions μA and ν A are called the membership function and nonmembership function, respectively. We denote by F (, S ) the set of all IF-events and we introduce the following operations on F (, S ). For Please cite this article as: P. Nowak, O. Hryniewicz, On generalized versions of central limit theorems for IF-events, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.052

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A = (μA , νA ), B = (μB , νB ) ∈ F (, S ), {An }n∈N = {(μAn , νAn )}n∈N ⊂ F (, S ):

A  B = (μA  μB , νA  νB ) = ( (μA + μB ) ∧ 1, (νA + νB − 1 ) ∨ 0 ); A  B = ( μA  μB , νA  ν B ) = ( ( μA + μB − 1 ) ∨ 0 , ( νA + νB ) ∧ 1 ) 181 182 183 184 185 186 187 188 189 190 191

and we write An A ⇔ μAn μA , νAn  νA . Moreover, we consider the product A.B = (μA μB , 1 − (1 − νA )(1 − νB ) ).F (, S ) is ordered as follows: A ≤ B⇔μA ≤ μB , ν A ≥ νB. It was proved in [36] that F (, S ) can be embedded to MV-algebra (M(, S ), 0 , 1 , ¬, ,  ), where 0 =(0 , 1 ), 1 =(1 , 0 ),M(, S ) = {(μA , νA ) : μA , νA :  → [0, 1] are S -measurable functions },¬(μA , νA ) = (1 − μA , 1 − νA ), (μA , νA )  (μB , νB ) = (μA  μB , νA  νB ),(μA , νA )  (μB , νB ) = (μA  μB , νA  νB ). Definition 24. An IF-state on F (, S ) is a map m : F (, S ) → [0, 1] satisfying the following conditions for all A, B ∈ F (, S ) and {An }∞ n=1 ⊂ F (, S ): (i) m(1 ) = 1, m(0 ) = 0; (ii) m(A ) + m(B ) = m(A  B ) + m(A  B ); (iii) if An A, then m(An ) m(A). Let J be the family of all compact intervals on R.

192 193 194 195 196 197

Definition 25. An IF-probability is a mapping P : F (, S ) → J satisfying the following conditions for all A, B ∈ F (, S ) and {An }∞ n=1 ⊂ F (, S ): (i) P(1 ) = [1, 1], P(0 ) = [0, 0]; (ii) P(A ) + P(B ) = P(A  B ) + P(A  B ); (iii) if An A, then P(An ) P(A ).

199

An IF-probability space is a pair (F (, S ), P ), where P is an IF-probability on F (, S ). We will use the notation P(A )=[P (A ), P (A )] for each A ∈ F (, S ).

200

Definition 26. An IF-observable is a map x : B (R ) → F (, S ) satisfying the following conditions:

198

(i) x(R ) = 1 , x(∅ ) = 0 ; (ii) whenever A, B ∈ B (R ) and A ∩ B = ∅, then

201 202

x ( A )  x ( B ) = 0 203 204 205

Definition 27. If x, y : B (R ) → F (, S ) are IF-observables, then their joint IF-observable is the map h : B (R2 ) → F (, S ) satisfying the following conditions: (i) h(R2 ) = 1 , h(∅ ) = 0 ; (ii) whenever A, B ∈ B (R2 ) and A ∩ B = ∅, then

207

h ( A )  h ( B ) = 0 209



R2



and

h(A ∪ B ) = h(A ) + h(B );

(iii) for all A, A1 , A2 , ... ∈ B , if An A, then h(An ) h(A); (iv) h(C × D ) = x(C ).y(D ) for all C, D ∈ B (R ). We present important properties of the defined above notions.

210 211

x(A ∪ B ) = x(A )  x(B );

(iii) for all A, A1 , A2 , ... ∈ B (R ), if An A, then x(An ) x(A).

206

208

and

Theorem 28. The following properties of IF-probability, IF-states and IF-observables hold:

216

If P : F (, S ) → J is an IF-probability, then the mappings P and P are IF-states on F (, S ). To any IF-state p : F (, S ) → [0, 1] there exists a state p¯ : M(, S ) → [0, 1] such that p¯ |F (,S ) = p. To any IF-observables x, y : B (R ) → F (, S ), there exists their joint IF-observable. Since F (, S ) ⊂ M(, S ), any IF-observablex : B (R ) → F (, S ) is an observable in the sense of MV-algebraic probability theory andPx = P ◦ x, P x = P ◦ x : B (R ) → [0, 1] are probability measures.

217

For the proof we refer the reader to [36]. The following remark is also a consequence of considerations in [36].

212 213 214 215

218 219 220 221

1. 2. 3. 4.

Remark 29. Let x1 , x2 , ..., xn : B (R ) → F (, S ) be IF-observables . Let g : Rn → R be a Borel measurable function and let h : B (Rn ) → F (, S ) be the joint observable of x1 , x2 , ..., xn . Then g(x1 , x2 , ..., xn ) = h ◦ g−1 is an IF-observable. Definition 30. IF-observables x1 , x2 , ..., xn : B (R ) → F (, S ) are independent (with respect to P) if there exists an ndimensional observable h : B (Rn ) → F (, S ) such that or all C1 , C2 , ..., Cn ∈ B (R )

P (h(C1 × C2 × ... × Cn )) = Px1 (C1 ) · Px2 (C2 ) · ... · Pxn (Cn ), P (h(C1 × C2 × ... × Cn )) = P x1 (C1 ) · P x2 (C2 ) · ... · P xn (Cn ). Please cite this article as: P. Nowak, O. Hryniewicz, On generalized versions of central limit theorems for IF-events, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.052

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222 223 224 225

Definition 31. Let P : F (, S ) → J be an IF-probability. Let x : B (R ) → F (, S ) be an IF-observable. Then x is said to be integrable, and we write x ∈ L1P,1 , if the expectationsE (x ) = R t Px (dt ), E (x ) = R t P x (dt ) exist. We say that x is square integrable, and we write x ∈ L2P,2 , if R t 2 Px (dt ) and R t 2 P x (dt ) exist. Then the variances of x also exist and are described by the equalities



D,2 (x ) = D 226

7

,2



R

(x ) =

t

2

R

P x

 R

2

t − E (x ) Px (dt ),



R

p,p LP

p ,p LP1 2

for p1 , p2 ≥ 1 if

R

|t | p1 Px (dt ) < ∞ and

R

|t | p2 P x (dt ) < ∞. We will also use notation

x∈

228

The following lemma, which is a consequence of Theorem 1, shows the form of the expected value of a Borel function of an IF-observable.

230 231 232 233

for p ≥ 1.



227

229

instead of x ∈

 =

2   2 t − E (x ) P x (dt ). (dt ) − E (x ) = 

More generally, we write x ∈ p LP

2

t 2 Px (dt ) − E (x )

Lemma 32. Let P : F (, S ) → J be an IF-probability, ϕ be an R-valued Borel function, which domain is the whole set of real numbers R, x : B (R ) → F (, S ) be an IF-observable and y = ϕ (x ) = x ◦ ϕ −1 . Then E (y ) exists if and only if R |ϕ (t )|Px (dt ) <   ∞ and then E (y ) = R ϕ (t )Px (dt ). Moreover, the analogous assertion holds for E (y ) and the corresponding probability measure P x .

236

Proof. We use Theorem 1 for (X, X ) = (X  , X  ) = (R, B (R ) ), T = ϕ , and f (t ) = t. As we mentioned above, μ = Px is a probability measure. Furthermore, by direct computations one can check the equality μT −1 = mϕ (x ) = my , which ends the proof. One can prove analogously the assertion for E (y ). 

237

3.2. Central limit theorems

234 235

238 239 240 241

Let {kn }n∈N be a fixed sequence of positive integers such that limn→∞ kn = ∞. To formulate general versions of central limit theorems we need to introduce some additional notations and definitions.   for IF-observables   Let { F (n ) , S(n ) , P(n ) }n∈N be a sequence of IF-probability spaces. For each n ∈ N and an observable x : B (R ) →

F ((n ) , S(n ) ), we denote by E(n ) (x ), E (n ) (x ) the expected values of x and by D2(n,) (x ), D2(n, ) (x ) the variances of x with respect

243

to P(n ) and P (n ) , respectively.     Let n ∈ N, x : B (R ) → F (n ) , S(n ) be an observable of F (n ) , S(n ) , and x ∈ L2P

244

following R-valued functions:

242

lxn, (ε , s ) = E(n ) 245

lxn, (ε , s ) = E (n ) 246





2



2



(n )

. Then for each s > 0, we consider the

x − E(n ) (x ) I|x−E (x )|>εs , (n )

x − E (n ) (x ) I|x−E (x )|>εs . (n )

From Lemma 32 it follows that lxn, and lxn, are well-defined.

248

Definition 33. Let, for each positive integer n, {xn1 , xn2 , ..., xnkn } be a sequence of independent (with respect to P(n ) ) IFobservables of F ((n ) , S(n ) ). Then {xn j }n∈N, j∈N[kn ] is called a triangular array of independent IF-observables (TI for short).

249

Definition 34. Let {xn j }n∈N, j∈N[kn ] be a TI such that for each n ∈ N

247

xn j ∈ L2P(n) , j ∈ N[kn ],

(9)

250

s2n, =

kn 



j=1

251



D2(n,) xn j , s2n, =

kn    D2(n, ) xn j ∈ (0, ∞ ).

Then {xn j }n∈N, j∈N[kn ] satisfies the Lindeberg condition if for each ε > 0

−→ Ln (ε ) = Ln (ε ) + L n (ε )− n→∞ 0, 252

(10)

j=1

where Ln (ε ) =

1

2, sn

kn   xn j  ε, sn , L n (ε ) = ln, j=1

(11) 1

2, sn

kn

  xn j  ε, s n ,sn = s2n, and s n = s2n, . ln, j=1

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253

Definition 35. Let xn j

254

pounov condition if there exist δ 1 , δ 2 > 0 such that

be a TI fulfilling conditions (9) and (10). Then the array xn j

n∈N, j∈N[kn ]

kn      E(n ) |xn j − E(n ) xn j |2+δ1 +

1

δ1 , s2+ n j=1 255

2+δ1 ,

where sn

 2+δ1



 k P (n )

259

n

j=1



xn j −

j=1

 k

n

j=1

xn j −

j=1

s n

(12)

.



E(n ) xn j

E (n ) xn j

n∈N, j∈N[kn ]

be a TI satisfying (9) and (10) and the Lindeberg condition (11). Then for t ∈ R





−→ ((−∞, t )) − n→∞ (t ),

(13)





−→ ((−∞, t )) − n→∞ (t ).

(14)





¯  : M (n ) , S(n ) → [0, 1] such that P ¯  |F  ,S Proof. Theorem 28 implies that for each n ∈ N there exists a state P = (n ) (n ) ( (n ) (n ) )       ¯ P(n ) . We consider the sequence of MV-algebras M (n ) , S(n ) with states P(n ) , n = 1, 2, ... Since F (n ) , S(n ) ⊂





M (n ) , S(n ) , the array

263

each n ∈ N and j ∈ N[kn ]



265

satisfies the Lya-

kn      −→ E (n ) |xn j − E (n ) xn j |2+δ2 − n→∞ 0,

1

 2+δ2



kn

262

264

n∈N, j∈N[kn ]

δ2 , s2+ n j=1

= s n



kn sn

P (n )

261

and sn

Theorem 36. (Lindeberg CLT) Let xn j

258

260

2+δ2 ,

= sn



We formulate and prove IF-probabilistic versions of central limit theorems.

256 257

[m3Gsc;April 8, 2016;16:57]

¯ P (n )





xn j

= P(n )



 xn j

xn j



n∈N, j∈N[kn ]

is a TA of the MV-algebras





M  (n ) , S (n )



n∈N

(see Theorem 28). Moreover, for

.

(15)









The equality (15) implies that for n ∈ N expected values E(n ) xn j , j ∈ N[kn ], variances D2(n,) xn j , j ∈ N[kn ], and Ln (ε ), L n (ε ) ¯  . Therefore the Lindeberg condition (11) implies its counterpart coincide with their MV-algebraic counterparts for state P (n )



266

(7) for the TA xn j

267

Let ξn =



kn

kn

x − j=1 n j





j=1  sn

E n ( xn j ) ( ) 





¯  . Moreover, the array satisfies (6). of MV-algebras M (n ) , S(n ) , considered with states P (n )

j∈N[kn ],n∈N









 −→ ¯  ξn ( (−∞, t )) − , n ∈ N. Theorem 21 implies that for t ∈ RP n→∞ (t ). Since for each n ∈ N ξn (n )





269

¯  ξn ( (−∞, t ) ) = P ξn ( (−∞, t )) , holding for each t ∈ R, ends the is an F (n ) , S(n ) -valued observable, the equality P (n ) (n ) proof. Formula (14) can be obtained analogously. 

270

Theorem 37. (Lyapounov CLT) Let xn j

271

(13)-(14) hold.

272

Proof. We use notations from the proof of the previous theorem. The array

268

273 274







 n∈N, j∈N[kn ]

be a TI satisfying (9)-(10) and Lyapounov’s condition (12). Then for each t ∈ R





xn j

 n∈N, j∈N[kn ]

is a TA of the MV-algebras





¯  , n = 1, 2, ... As it was noted above, for each n ∈ N expected values E xn j , j ∈ N[kn ], with states P (n ) (n ) n∈N  ¯  . This implies that the TA and variances D2(n,) xn j , j ∈ N[kn ], coincide with their MV-algebraic counterparts for state P (n )



M  (n ) , S (n )







278

of MV-algebras M (n ) , S(n ) satisfies (6). Moreover, the Lyapounov condition (12) implies its counterpart   −→ ¯  . By Theorem 22, P ¯  ξn ( (−∞, t ) ) − (8) for this array, considered with states P n→∞ (t ) for t ∈ R. The same reasoning as (n ) (n ) in the last part of the previous proof shows that (13) holds. The convergence (14) we obtain analogously. This completes the proof. 

279

Theorem 38. (Feller) Let xn j

275 276 277

xn j

n∈N, j∈N[kn ]





lim max P(n ) n→∞ 1≤ j≤kn

  xn j

 n∈N, j∈N[kn ]

be a TI satisfying (9) and (10) and such that for each ε > 0

 



 



−∞, −ε sn ∪

ε sn , ∞

= 0,

(16)

= 0.

(17)

280



lim max P (n )

n→∞ 1≤ j≤kn

281

  xn j

−∞, −ε s n ∪

ε s n , ∞

If for t ∈ R (13) and (14) hold, then the Lindeberg condition (11) is satisfied. Please cite this article as: P. Nowak, O. Hryniewicz, On generalized versions of central limit theorems for IF-events, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.052

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282

Proof. The equalities (15)–(17) imply



¯ lim max P (n )

n→∞ 1≤ j≤kn

283



¯ lim max P (n )

n→∞ 1≤ j≤kn

284

  xn j

  xn j

 

−∞, −ε sn ∪

 

−∞, −ε s n ∪

ε sn , ∞ ε s n , ∞

 

9

= 0,

(18)

= 0.

(19)

As it was noted in the proofs of the previous theorems, the TA



xn j





n∈N, j∈N[kn ]



of MV-algebras M (n ) , S(n ) , considered

286

−→ −−→ ¯  and P ¯ , satisfies (6). Theorem 23 implies that for each ε > 0Ln (ε )− with states P n→∞ 0, Ln (ε )n→∞ 0. Since Ln (ε ) = (n ) (n )  Ln (ε ) + Ln (ε ), the Lindeberg condition is satisfied. 

287

4. M-probability

288

4.1. Basic notions

285

289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324

In this section we consider the set of all IF-events F (, S ) with the Zadeh connectives, defined for each A = (μA , νA ), B = (μB , νB ) ∈ F (, S ) by A ∨ B = (μA ∨ μB , νA ∧ νB ), A ∧ B = (μA ∧ μB , νA ∨ νB ). According to the terminology introduced in [36], the probability theory corresponding to IF-events with such defined connectives we will call M-probability theory. At the beginning we define some basic notions of M-probability theory. For further details we refer the reader to [36]. Definition 39. An M-state on F (, S ) is a map s : F (, S ) → [0, 1] satisfying the following conditions for all A, B ∈ F (, S ) ∞ and {An }∞ n=1 , {Bn }n=1 ⊂ F (, S ): (i) s(1 ) = 1, s(0 ) = 0; (ii) s(A ) + s(B ) = s(A ∨ B ) + s(A ∧ B ); (iii) if An A, Bn B, then s(An ) s(A), s(Bn )s(B). Definition 40. An M-probability is a mapping P =[P  ,P ] : F (, S ) → J satisfying the following conditions for all A, B ∈ ∞ F (, S ) and {An }∞ n=1 ,{Bn }n=1 ⊂ F (, S ): (i) P (1 ) = [1, 1], P (0 ) = [0, 0]; (ii) P (A ) + P (B ) = P (A ∨ B ) + P (A ∧ B ); (iii) if An A, Bn B, then P(An ) P(A), P(Bn )P(B). An M-probability  space is a pair (F (, S ), P ), where P is an M-probability on F (, S ). We will use notation P (A )= P  (A ),P (A ) for each A ∈ F (, S ). Definition 41. An M-observable is a map x : B (R ) → F (, S ) satisfying the following conditions for A, B ∈ B (R ) and ∞ {An }∞ n=1 , {Bn }n=1 ⊂ B (R ): (i) x(R ) = 1 , x(∅ ) = 0 ; (ii) x(A ∪ B ) = x(A ) ∨ x(B ) and x(A ∩ B ) = x(A ) ∧ x(B ); (iii) if An A and Bn B, then x(An ) x(A) and x(Bn )x(B). Proposition 42. For any M-observable x : B (R ) → F (, S ) and M-states : F (, S ) → [0, 1] sx = s ◦ x : B (R ) → [0, 1] is a probability measure. For the proof we refer the reader to [36], Proposition 6.





Definition 43. If x, y : B (R ) → F (, S ) are M-observables, then their joint M-observable is a map h : B R2 → F (, S ) sat-



 2



isfying the following conditions for A, B ∈ B R , {An }∞ , {Bn }∞ ⊂B R n=1 n=1 (i) (ii) (iii) (iv)



 2

 2

and C, D ∈ B (R ):

h R = 1 , h ( ∅ ) = 0 ; h(A ∪ B ) = h(A ) ∨ h(B ) and h(A ∧ B ) = h(A ) ∧ h(B ); if An A and Bn B, then h(An ) h(A) and h(Bn )h(B); h(C × D ) = x(C ) ∧ y(D ).

In the case of M-observables the following theorem and remark hold (see [36] for proof). Theorem 44. To any M-observables x, y : B (R ) → F (, S ), there exists their joint M-observable. Remark 45. Let x1 , x2 , ..., xn : B (R ) → F (, S ) be M-observables . Let g : Rn → R be a Borel measurable function and let h : B (Rn ) → F (, S ) be the joint M-observable of x1 , x2 , ..., xn . Then g(x1 , x2 , ..., xn ) = h ◦ g−1 is an M-observable. Please cite this article as: P. Nowak, O. Hryniewicz, On generalized versions of central limit theorems for IF-events, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2016.03.052

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Definition 46. M-observables x1 , x2 , ..., xn : B (R ) → F (, S ) are independent (with respect to P) if there exists an ndimensional M-observable h : B (Rn ) → F (, S ) such that for all C1 , C2 , ..., Cn ∈ B (R )

P  (h(C1 × C2 × ... × Cn )) = P x1 (C1 ) · P x2 (C2 ) · ... · P xn (Cn ), P (h(C1 × C2 × ... × Cn )) = P x1 (C1 ) · P x2 (C2 ) · ... · P xn (Cn ). 327 328 329 330

Definition 47. Let P : F (, S ) → J be an M-probability. Let x : B (R ) → F (, S ) be an M-observable. Then x is said to be integrable, and we write x ∈ L1P,1 , if the expectationsE (x ) = R t P x (dt ), E (x ) = R t P x (dt ) exist. We say that x is square  integrable, and we write x ∈ L2P,2 , if R t 2 P x (dt ) and R t 2 P x (dt ) exist. Then the variances of x also exist and are described by the equalities

D2, (x ) = D 331 332 333 334 335 336 337 338 339 340

341 342 343 344 345 346

2,

(x ) =





R R

tx2 P x



=

 R

2

t − E (x ) P x (dt ),

2   2 t − E (x ) P x (dt ). (dt ) − E (x ) = x 



R

p ,p

More generally, we write x ∈ LP1 2 for p1 , p2 ≥ 1 if p p,p use the symbol LP instead of LP for each p ≥ 1.

R

|t | p1 P x (dt ) < ∞ and

R

|t | p2 P x (dt ) < ∞. To shorten notation we will

The following lemma, which is a consequence of Theorem 1, shows the form of expected values of a Borel function of an M-observable. Lemma 48. Let P : F (, S ) → J be an M-probability, ϕ be an R-valued Borel function, which domain is the whole set of real numbers R, x : B (R ) → F (, S ) be an M-observable and y = ϕ (x ) = x ◦ ϕ −1 . Then E (y) exists if and only if R |ϕ (t )|P x (dt ) < ∞ and then E (y ) = R ϕ (t )P x (dt ). Moreover, the analogous assertion holds for E (y) and the corresponding probability measure P x .





Proof. We apply Theorem 1, assuming that (X, X )= X  , X  = (R, B (R ) ), T = ϕ , and f (t ) = t. Since P is an M-state, by Proposition 42, μ = P x is a probability measure. Furthermore, by direct computations one can verify the equality μT −1 = P ϕ (x ) = P y , which ends the proof. The assertion for E (y) can be proved analogously.  4.2. Central limit theorems

    F  (n ) , S (n ) , P (n ) be a sequence of Mn∈N    probability spaces. For each n ∈ N and M-observable x : B (R ) → F (n ) , S(n ) , we denote by E(n ) (x ), E (n ) (x ) the expected values of x and by D2(n,) (x ), D2(n, ) (x ) the variances of x with respect to P (n ) and P (n ) , respectively.   For n ∈ N, an M-observable x : B (R ) → F (n ) , S(n ) belonging to L2P and s > 0 we consider the following R-valued n Let {kn }n∈N ⊂ N be a fixed sequence such that limn→∞ kn = ∞. Let

( )

functions:

lxn, (ε , s ) = E(n ) 347

lxn, (ε , s ) = E (n ) 348

2

t 2 P x (dt ) − E (x )

 

2



2



x − E(n ) (x ) I|x−E (x )|>εs , (n )

x − E (n ) (x ) I|x−E (x )|>εs . (n )

Lemma 48 implies that lxn, and lxn, exist.

350

Definition 49. Let, for each positive integer n, {xn1 , xn2 , ..., xnkn } be a sequence of independent (with respect to P) M  observables of F (n ) , S(n ) . Then {xn j }n∈N, j∈N[kn ] is called a triangular array of independent M-observables (TM for short).

351

Definition 50. Let xn j

349





n∈N, j∈N[kn ]

be a TM such that for each n ∈ N

xn j ∈ L2P (n) , j ∈ N[kn ], 352

s2n, =

kn 



(20)



D2(n,) xn j , s2n, =

j=1

353

kn 





D2(n, ) xn j ∈ (0, ∞ ).

Then {xn j }n∈N, j∈N[kn ] satisfies the Lindeberg condition if for each ε > 0

−→ Ln (ε ) = Ln (ε ) + L n (ε )− n→∞ 0, 354

(21)

j=1

where Ln (ε ) =

1

2, sn

kn  j=1

x





nj ε, sn , L n (ε ) = ln,

(22) 1

2, sn

kn

  xn j  ε, s n ,sn = s2n, and s n = s2n, . ln, j=1

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kn

C , where C j ∈ B (R ), j ∈ N[kn ], the probability measures P(n ) , P( n ) : B (Rkn ) → j=1 j     n n P(n ) (C ) = kj=1 (P (n) )xn j C j ,P( n) (C ) = kj=1 (P (n) )xn j C j . Then {(Rkn , B (Rkn ), P(n) )}n∈N

355

Definition 51. Let, for each n ∈ N and C =

356

[0, 1] be defined by the equalities:

357

and {(Rkn , B (Rkn ), P( n ) )}n∈N are sequences of probability spaces.

For each n ∈ N and j ∈ N[kn ] we define random variables ιnj : Rkn → R and vectors ¯ιnj : Rkn → R j on the probability space

358 359 360 361

( R kn , B ( R kn ), P

363

365 366 367

n n (n ) ) by ι j (u1 , u2 , ..., ukn ) = u j ,¯ι j (u1 , u2 , ..., ukn ) = (u1 , u2 , ..., u j ).

We denote by E respectively.

362

364

R kn



P (n )

369 370 371

fn (t1 , t2 , ..., tkn ) =

the expected values and by D



2,P ( n )

and D



2,P ( n )

the variances with respect to P(n ) and P( n ) ,



kn t j=1 j

−



kn

E (n ) xn j

j=1

2+δ1 ,

where sn



   |xn j − E(n) xn j |2+δ1 +

 2+δ1

= sn

2+δ2 ,

and sn

fn , fn :



/s n .



condition if for some δ 1 , δ 2 > 0

kn 1  E (n ) 2+δ2 , sn j=1

 2+δ2

= s n



   −→ |xn j − E (n) xn j |2+δ2 − n→∞ 0,

 n∈N, j∈N[kn ]

satisfies the Lyapounov

(23)

.

We formulate and prove the M-probabilistic version of the Lindeberg CLT.



Theorem 53. (Lindeberg CLT) Let t∈R

 k

xn j −

k n

j=1

sn

 k

372

n

j=1

P (n )

n

j=1



P (n )

xn j −

j=1

s n

xn j



E(n ) xn j



k n

E (n ) xn j



be a TM satisfying (20) and (21) and the Lindeberg condition (22). Then for

n∈N, j∈N[kn ]





−→ ( (−∞, t )) − n→∞ (t ),

(24)





−→ ( (−∞, t )) − n→∞ (t ).

(25)

Proof. We consider two sequences of spaces of IF-events with M-states:

 



F (n ) , S(n ) , P (n )

374



P (n )

Definition 52. Let {xn j }n∈N, j∈N[kn ] be a TM, fulfilling conditions (20) and (21). Then xn j



373

and E

Let, for each n ∈ N, hn : B (Rkn ) → F ((n ) , S(n ) ) be the joint observable of the sequence xn1 , xn2 , ..., xnkn and → R be the functions: 

n    kn fn (t1 , t2 , ..., tkn ) = t − kj=1 E(n ) xn j /sn , j=1 j

kn 1  E(n ) 2+δ1 , sn j=1

368

11



and

n∈N

The following equalities hold:



P (n ) ◦ hn = P (n )





xn1

× P (n )

 



F (n ) , S(n ) , P (n )





xn2

× ... × P (n )

 n −1    = P (n ) for j ∈ N[kn ]; ιj xn j  −1 = P (n ) ◦ hn . P(n ) ◦ ¯ιnkn

 xnkn



n∈N

.

;

P(n ) ◦

375

Let t ∈ R. Clearly, P (n )

 kn

k n  x − j=1 E ( n ) ( xn j ) j=1 n j 

sn

 



(26)



( (−∞, t ))





 −1

= P (n ) fn xn1 , xn2 , ..., xnkn ( (−∞, t ) ) = P (n ) hn ◦ fn 376

=



P (n )





xn1

× P (n )



377

=

P(n )

=

P(n )

u∈R

kn

:



378

u∈R

kn

:





xn2

× ... × P (n )



 −1 

( (−∞, t ))

k n



fn

xnkn

u1 + u2 + ... + ukn −

((−∞, t ))

j=1

sn

E(n ) xn j

ιn1 (u ) + ιn2 (u ) + ... + ιnkn (u ) − sn





k n j=1




E(n ) xn j




.

(27)

379

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From (26) it follows that, for n ∈ N and j ∈ N[kn ], the random variables ιnj are independent, have distribution (P (n ) )xn j , and kn     P 2,P 2,P E(n ) (xn j ) = E (n) ιnj , D2(n,) (xn j ) = D (n) ιnj , s2n, = D (n) ιnj . j=1

383 384

Moreover,

xn j ln, (ε , sn )

=E



P (n )

((ιnj



P − E (n) ιnj )2 I ).  P |ιnj −E (n ) ιnj |>εsn

−→ Since 0 ≤ Ln (ε ) ≤ Ln (ε )− n→∞ 0, Theorem 4 implies convergence

of (27) to (t) as n → ∞ and therefore (24) holds. Analogously, we obtain (25).



386

Theorem 54. (Lyapounov CLT) Let {xn j }n∈N, j∈N[kn ] be a TM satisfying (20) and (21) and Lyapounov’s condition (23). Then for t ∈ R (24) and (25) hold.

387

Proof. We apply Theorem 53. For each M-observable y : B (R ) → F (, S ), y ∈ L2P , of an M-probability space {(F (, S ), P )}

385

388 389

2+δ1 ,2+δ2

and θ > 0, we define an M-observable ϕ θ (y) by the formula: ϕθ (y ) = y2 I|y|>θ . Moreover, if additionally y ∈ LP δ 2 > 0, then, by Lemma 48,

E ( ϕθ ( y ) ) =



≤ 390

E ( ϕθ ( y ) ) =

t 2 I| t |>1 Py (dt ) θ

R



t |t |δ1 2

θ δ1

R

 

≤

I|t |>θ Py (dt ) ≤

t |t |δ2

θ δ2

I|t |>θ P y (dt ) ≤

We will show that the TM

392

E(n ) ϕεs xn j − E(n ) xn j n



1

 2,

sn







(28)

x

 

xn j

 

1

θ δ2

E |y|2+δ2 .

(29) x

n∈N, j∈N[kn ]









, applying the inequalities (28) and (29), we obtain

kn kn    1     1     xn j − E E ϕ x + E (n ) ϕεs n xn j − E (n ) xn j n j ε s (n ) (n ) 2, 2, n sn j=1 sn j=1

kn      E(n ) |xn j − E(n ) xn j |2+δ1 +  δ 1 

ε sn



nj satisfies the Lindeberg condition. Since, for each ε > 0ln, ε, sn =

nj ε, s n = E (n) ϕεs xn j − E (n) xn j andln, n

Ln (ε ) = Ln (ε ) + L n (ε ) =

≤



E |y|2+δ1 ,

θ

2

R



1

θ δ1

t 2 I| t |>1 P y (dt )

R

391



for δ 1 ,

j=1

 2,

1

sn

 δ2

ε sn

kn 





E (n ) |xn j − E (n ) xn j

 2+δ  | 2 .

j=1

394

−→ Since, by the Lyapounov condition, the right side of the above inequality converges to 0 as n tends to infinity, Ln (ε )− n→∞ 0, which ends the proof. 

395

Theorem 55. (Feller) Let xn j

393





lim max P(n )

 

n→∞ 1≤ j≤kn

396



lim max P (n )

xn j

 

n→∞ 1≤ j≤kn

397 398 399 400

n∈N, j∈N[kn ]

be a TM satisfying (20) and (21) and such that for each ε > 0

  



 



ε sn , ∞

−∞, −ε sn ∪ −∞, −ε s n ∪

ε s n , ∞

= 0,

(30)

= 0.

(31)

If for each t ∈ R (24) and (25) hold, then the Lindeberg condition (22) is satisfied. Proof. We use the notation from the proof of Theorem 53. For each n ∈ N, the random variables ιn1 , ιn2 , ..., ιnk are indepenn   n 2,P n n P 2,P dent, for each j ∈ N[kn ] ιnj has distribution (P(n ) )xn j , E(n ) (xn j ) = E (n) ιnj , D2(n,) (xn j ) = D (n) ιnj . Moreover, s2n, = kj=1 D ( )ιj and

P(n ) 401

xn j





       |ιnj | > ε sn = P(n) x −∞, −ε sn ∪ ε sn , ∞ . nj

For each t ∈ R



P(n )

u∈R



402

= P (n )

kn

:

ιn1 (u ) + ιn2 (u ) + ... + ιnkn (u ) − sn

xn1 + xn2 + ... + xnkn − sn

k n j=1



E(n ) xn j



k n j=1



E(n ) xn j




 −→ ( (−∞, t )) − n→∞ (t ).

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P. Nowak, O. Hryniewicz / Information Sciences xxx (2016) xxx–xxx

403

Therefore, by the classical Feller theorem,

Ln (ε ) =

1



E P (n ) 2,



2  ιnj − E P(n) ιnj I

|ι

sn

13

  n −E P (n ) n j j

ι |>εsn

− −→ n→∞ 0.

404

−→ Analogously, we obtain L n (ε )− n→∞ 0, which ends the proof.

405

5. Examples of applications

406

414

In this section we present and analyse two examples of arrays of IF-observables and M-observables with convergent scaled row sums. The considered observables are not identically distributed and therefore the versions of CLT proved in [36] cannot be applied for them. To prove the convergence in distribution of the considered scaled row sums to standard normal distribution we use Theorem 36 and Theorem 54. In both examples observables take values in F (, S ), where  = {ω1 , ω2 }, S =2 , considered with connectives ,  in the case of IF-observables and ∨, ∧ in the case of scaled row sums of M-observables. In both examples four particular elements 0 , π 10 = (μ1 , ν1 ), π01 = (μ2 , ν2 ), 1 ∈ F (, S ) are used, where the membership functions μ1 , μ2 and nonmembership functions ν 1 , ν 2 are defined by μ1 ( ω 1 ) = 1 − μ 2 ( ω 1 ) = 1 − ν 1 ( ω 1 ) = ν 2 ( ω 1 ) = 1 , 1 − μ 1 ( ω 2 ) = μ2 ( ω 2 ) = ν 1 ( ω 2 ) = 1 − ν 2 ( ω 2 ) = 1 .

415

5.1. Array of IF-observables

407 408 409 410 411 412 413

416 417

Let P0 be probability defined on (, S ) by the equalities: P0 ({ω1 } ) = P0 ({ω2 } ) = each n ∈ N has the form considered in [15]:

P ( ( μ, ν ) ) = 418 419





μdP0 , 1 −

⎧ ⎪ ⎪ 0 ⎪ ⎨π 10 x j (A ) = ⎪π01 ⎪ ⎪ ⎩



j





   j f ,  j j   1j  A ∩ f1 , f2 = f2 ,  j j A ∩ f1j , f2j =

if if

f1 , f2 ⊂ A,

if j

 

where f1 = −1 − 1j , f2 = 1 + 1j . We also assume that x j

 

 

s2n, = s2n, =

n  

 

1 j

1+

j=1

 Since the supports of x

nj ln,



P(n )



j∈N



are independent. Then for j ∈ N

xj

P (n )

xj

2

,

− −→ n→∞ ∞.

are bounded, for a fixed ε > 0 and sufficiently large n

( )

kn   xn j  ε, sn = L n (ε ) = ln,

and Ln (ε ) =

423

for t ∈ R (13) and (14) hold.

424

5.2. Array of M-observables

427

 and

2

1 j

     2 ε , sn = lxn, n j ε , s n = E(n) ( x j − E(n) x j I|x j −En (x j )|>εsn ) = 0,

422

426

(32)

A ∩ f1j , f2j = ∅,

if

 

425

IF-probability P : F (, S ) → J for

 ν dP0 , (μ, ν ) ∈ F (, S ).



E(n ) x j = E (n ) x j = 0 and D2(n,) x j = D2(n, ) x j = 1 +

421

1 2.

We assume that xn j = x j and kn = n for each n ∈ N and for arbitrary A ∈ B (R ) observable x j : B (R ) → F (, S ), j = 1, 2, 3, ..., has the form

1

420





1

2, sn

j=1



1

2, sn

kn   xn j  −→ ε, s n = 0. Therefore Ln (ε ) = 2Ln (ε )− ln, n→∞ 0 and Theorem 36 implies that j=1

Let γ ∈ 0, 12 , γ˜ = γ (1 − γ ) and {ξn }n∈N ⊂ (γ , 1 − γ ). For each n ∈ N we consider the sequence {P0n }n∈N of probabilities defined on (, S ) by equalities: 1 − P0n ({ω1 } ) = P0n ({ω2 } ) = ξn and the sequence of M-probabilites P(n ) : F (, S ) → J of the form analogous to (32):

P ( n ) ( ( μ, ν ) ) =





μdP0n , 1 −

 

 ν dP0n , (μ, ν ) ∈ F (, S ).

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For each n ∈ N we assume that kn = n and for arbitrary A ∈ B (R ) observable xn j : B (R ) → F (, S ), j = 1, 2, 3, ..., is given by the formula

⎧ 0 ⎪ ⎪ ⎨ π10 xn j ( A ) = ⎪ π01 ⎪ ⎩ 1

430 431

if if if if





A ∩ gn1 , gn2 = ∅,     A ∩ gn1 , gn2 = gn1 ,  n n  n A ∩ g1 , g2 = g2 ,  n n g1 , g2 ⊂ A,

where gn1 = −ξn , gn2 = 1 − ξn . We also assume that {xnj }j ≤ n are independent for each positive integer n.









Let n ∈ N and δ > 0. For each j ∈ N[n]E(n ) xn j = E (n ) xn j = 0,







E(n ) |xn j |2+δ = E (n ) |xn j |2+δ



2+δ

= ξn2+δ (1 − ξn ) + (1 − ξn ) 432









D2(n,) xn j = D2(n,) xn j = E(n ) 433

[m3Gsc;April 8, 2016;16:57]

Therefore sn = s n >

√ nγ˜ and

1



2+δ, sn

xn j

kn

2

ξn ≤ ξn ( 1 − ξn ) < 1 ,

= E (n )



xn j

2

= ξn (1 − ξn ) > γ˜ 2 .

E(n ) (|xn j − E(n ) (xn j )|2+δ )+ 2+1δ, sn

j=1

kn





E (n ) (|xn j − E (n ) xn j |2+δ ) <

j=1

2 −−→ n→∞ 0. δ n 2 γ˜ 2+δ

Applying

434

Theorem 54 we obtain the convergence (24) and (25) for t ∈ R.

435

6. Conclusions

436

445

The paper was devoted to formulation and proof of the Lindeberg CLT, the Lyapounov CLT, and the Feller theorem for IF-events. The obtained theoretical results generalize versions of central limit theorems known from the literature, involving the case of not necessarily identically distributed observables. We took into account two pairs of connectives for IF-sets, i.e. the Łukasiewicz and the Zadeh one, considering IF-probability theory and M-probability theory. In spite of differences in both theories, the assumptions of the obtained limit theorems are similar. This unifies our approach and shows that there are no essential differences between limit behavior of observables defined for IF-sets with the two most popular pairs of connectives. We presented examples of applications of our results for scaled sums of IF-observables and M-observables. Our future work will concern further development of the probability theory for IF-events. It is worth to note that this direction seems interesting from statistical point of view. The presented approach can be also applied to observables with values in the space of interval fuzzy sets.

446

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