Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables

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  • ABSTRACT

    In this paper, we present some results on weak laws of large numbers for weighted sums of fuzzy random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in a separable real Banach space. First, we give weak laws of large numbers for weighted sums of strong-compactly uniformly integrable fuzzy random variables. Then, we consider the case that the weighted averages of expectations of fuzzy random variables converge. Finally, weak laws of large numbers for weighted sums of strongly tight or identically distributed fuzzy random variables are obtained as corollaries.


  • KEYWORD

    Fuzzy sets , Random sets , Fuzzy random variables , Weak law of large numbers , Compactly uniform integrability , Tightness , Weighted sum.

  • 1. Introduction

    In recent years, the theory of fuzzy sets introduced by Zadeh [1] has been extensively studied and applied the fields of statistics and probability. Statistical inference for fuzzy probability models led to the requirement for laws of large numbers to ensure consistency in estimation problems.

    Since Puri and Ralescu [2] introduced the concept of fuzzy random variables as a natural generalization of random sets, several authors have studied laws of large numbers for fuzzy random variables. Among others, several variants of strong law of large numbers (SLLN) for independent fuzzy random variables were built on the basis of SLLN for independent random sets. A rich variety of SLLN for fuzzy random variables can be found in the literature, e.g., Colub et al. [3,4], Feng [5], Fu and Zhang [6], Inoue [7], Klement et al. [8], Li and Ogura [9], Molchanov [10], Proske and Puri [11].

    However, weak laws of large numbers (WLLN) for fuzzy random variables are not as popular as SLLN. Taylor et al. [12] obtained WLLN for fuzzy random variables in a separable Banach space under varying hypotheses of independence, exchangeability, and tightness. Joo [13] established WLLN for convex-compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in a finite-dimensional Euclidean space.

    Generalizing the above results for sums of fuzzy random variables to the case of weighted sums is a significant problem. In this regard, Guan and Li [14] obtained some results on WLLN for weighted sums of fuzzy random variables under a restrictive condition, and Joo et al. [15] established some results on strong convergence for weighted sums of fuzzy random variables different from those of Guan and Li [14]. Moreover, Kim [16] studied WLLN for weighted sums of level-continuous fuzzy random variables.

    The purpose of this paper is to present some results on WLLN for the weighted sum of fuzzy random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in a real separable Banach space. First, we give WLLN for the weighted sum of strong-compactly uniformly integrable fuzzy random variables. Then, we give WLLN for the weighted sum of fuzzy random variables such that the weighted averages of its expectations are convergent.

    2. Preliminaries

    Let Y be a real separable Banach space with norm |?| and let K(Y) denote the family of all non-empty compact subsets of Y. Then the space K(Y) is metrizable by the Hausdorff metric h defined by

    image

    A norm of AK(Y) is defined by

    image

    It is well-known that K(Y) is complete and separable with respect to the Hausdorff metric h (See Debreu [17]).

    The addition and scalar multiplication on K(Y) are defined as usual:

    A ? B = {a + b : a ∈ A, b ∈ B}, λA = {λa : a ∈ A}

    for A,BK(Y) and λ ∈ R.

    The convex hull and closed convex hull of AY are denoted by co(A) and

    image

    , respectively. If dim(Y) < ∞ and AK(Y), then co(A) ∈ K(Y). But if dim(Y)= ∞, it is well-known that co(A) may not be an element of K(Y) even though AK(Y), but

    image

    K(Y) if AK(Y).

    Let F(Y) denote the family of all fuzzy sets u : Y → [0,1] with the following properties;

    (i) u is normal, i.e., there exists x ∈ Y such that u(x)= 1;

    (ii) u is upper-semicontinuous;

    (iii) supp u = cl{x ∈ Y : u(x) > 0} is compact, where cl(A) denotes the closure of A in Y.

    For a fuzzy subset u of Y, the α-level set of u is defined by

    image

    Then it follows immediately that uF(Y) if and only if LαuK(Y) for each a ∈ [0,1]: If we denote cl{xY : u(x) > α} by Lα+u, then

    image

    The linear structure on F(Y) is also defined as usual;

    image

    for u,vF(Y) and λ ∈ R, where

    image

    denotes the indicator function of {0}.

    Then it is known that for each α ∈ [0,1], Lα (uv)= LαuLαv and Lαu)= λLαu.

    Recall that a fuzzy subset u of Y is said to be convex if

    u(λx+(1?λ)y) ≥ min(u(x),u(y)) for x,y ∈ Y and λ ∈ [0,1].

    The convex hull of u is defined by

    co(u) = inf{v : v is convex and v ≥ u}.

    Then it is known that for each α ∈ [0,1], Lαco(u)= co(Lαu).

    If Y is finite dimensional space and uF(Y), then co(u) ∈ F(Y). But if Y is infinite dimensional space, it may not be true. So we need the notion of the closed convex hull of u. The closed convex hull

    image

    of u is defined by

    image

    Then it is well-known that

    image

    for each α ∈ [0,1] and

    image

    The uniform metric d and norm ||?|| on F(Y) as usual;

    image

    It is well-known that (F(Y),d) is complete but is not separable (see Klement et al. [8]).

    3. Main Results

    Throughout this paper, let (Ω,A,P) be a probability space. A set-valued function X : Ω → (K(Y), h) is called a random set if it is measurable. A random set X is said to be integrably bounded if E||X|| < ∞. The expectation of integrably bounded random set X is defined by

    E(X)= {E(ξ) : ξ ∈ L(Ω,Y) and ξ(ω) ∈ X(ω)a.s.},

    where L(Ω,Y) denotes the class of all Y-valued random variables ξ such that E|ξ| < ∞.

    A fuzzy set valued function

    image

    is called a fuzzy random variable (or fuzzy random set) if for each

    image

    is a random set. It is well-known that if

    image

    is measurable, then

    image

    is a fuzzy random variable. But the converse is not true (For details, see Colubi et al. [18], Kim [19]).

    A fuzzy random set

    image

    is said to be integrably bounded if

    image

    The expectation of integrably bounded fuzzy random variable

    image

    is a fuzzy subset

    image

    of Y defined by

    image

    For more details for expectations of random sets and fuzzy random variables, the readers may refer to Li et al. [20].

    Let

    image

    be a sequence of integrably bounded fuzzy random variables and {λni} be a double array of real numbers that not necessarily Toeplitz but satisfying

    image

    where C > 0 is a constant not depending on n.

    The problem that we will consider is to establish sufficient conditions for

    image

    where

    image

    denotes the closed convex hull of

    image

    To this end, we need the concepts of tightness and compact uniform integrability for a sequence of fuzzy random variables.

    Definition 3.1. Let {Xn} be a sequence of random sets.

    (i) {Xn} is said to be tight if for each ε > 0, there exists a compact subset K of (K(Y), h) such that

    P(Xn ? K) < ε for all n.

    (ii) {Xn} is said to be compactly uniformly integrable(CUI) if for each ε > 0, there exists a compact subset K of (K(Y),h) such that

    image

    Definition 3.2. Let

    image

    be a sequence of fuzzy random variables.

    (i)

    image

    is said to be level-wise independent if for each α ∈ [0,1], the sequence

    image

    of random sets is independent.

    (ii)

    image

    is said to be independent if the sequence

    image

    of σ-fields is independent, where

    image

    is the smallest σ-field which

    image

    is measurable for all α ∈ [0, 1].

    (iii)

    image

    is said to be tight if for each ε > 0, there exists a compact subset K of (K(Y), h) such that

    image

    (iv)

    image

    is said to be strongly tight if for each ε > 0, there exists a compact subset K of (F(Y),d∞) such that

    image

    (v)

    image

    is said to be compactly uniformly integrable (CUI) if for each ε > 0 there exists a compact subset K of (K(Y),h) such that

    image

    (vi)

    image

    is said to be strong-compactly uniformly integrable (SCUI) if for each ε > 0 there exists a compact subset K of (F(Y),d∞) such that

    image

    It is trivial that strong-compactly uniform integrability (resp. strong tightness) implies compactly uniform integrability (resp. tightness). But, the converse is not true even though Y is finite dimensional.

    First, we establish weak law of large numbers for weighted sums of strong-compactly uniformly integrable fuzzy random variables.

    Theorem 3.3. Let

    image

    be a sequence of integrably bounded fuzzy random variables and let {λni} be a double array of real numbers satisfying

    image

    Then

    image

    if and only if for each α ∈ [0,1],

    image

    To prove the above theorem, we need some lemmas obtained by Kim (submitted) which is based on the characterization of relatively compact subsets of (F(Y),d) established by Greco and Moschen [21]. For easy references, we list them without proof.

    Lemma 3.4. Let K be a relatively compact subset of (F(Y),d). Then

    image

    is also relatively compact in (F(Y),d).

    Recall that we can define the concept of convexity on F(Y) as in the case of a vector space even though F(Y) is not a vector space. That is, KF(Y) is said to be convex if λu?(1?λ)vK whenever u,vK and 0 ≤ λ ≤ 1. Also, the convex hull co(K) of K is defined to be the intersection of all convex sets that contains K. Then we can easily show that co(K) is equal to the family of consisting of all fuzzy sets in the form λ1u1 ? … ? λkuk, where u1,...,uk are any elements of K, λ1,...,λk are nonnegative real numbers satisfying

    image

    Lemma 3.5. Let K be a relatively compact subset of (F(Y),d). Then co(K) is also relatively compact in (F(Y),d).

    For a fixed partition π :0 = α0 < α1 < … < αr = 1 of [0,1], we define

    image

    Then it follows that

    image

    From this fact, we can prove easily that

    gπ(u ? v)= gπ(u) ? gπ(v) and gπ(λu)= λgπ(u).

    Lemma 3.6. Let K be a relatively compact subset of (F(Y),d). Then for each natural number m, there exists a partition πm of [0,1] such that

    image

    We are now in a position to prove the main theorem.

    Proof of Theorem 3. The necessity is trivial. To prove the sufficiency, We can assume that C = 1 without loss of generality. Let ε > 0 and 0 < δ < 1 be given. By strong-compactly uniform integrability of

    image

    , we can choose a compact subset K of (F(Y),d) such that

    image

    Without loss of generality, we may assume that

    image

    K is convex and symmetric (i.e., (?1)uK if uK), and that K contains

    image

    for all uK by lemmas 4 and 5.

    By lemma 6, we choose a partition πm :0 = αm,0 < αm,1 < … < αm,rm of [0, 1] such that

    image

    Now we denote

    image

    Then by assumptions of K and λni, we have

    image

    Thus by (2),

    image

    Then we have

    image

    Hence we obtain

    image

    This implies that

    image

    For (I), we first note that

    image

    And so

    image

    Now for (II), since

    image

    we have

    image

    for sufficiently large n by our assumption. This completes the proof.

    Corollary 3.7. Let {Xn} be a sequence of strongly tight fuzzy random variables such that

    image

    Then

    image

    if and only if for each α ∈ [0,1],

    image

    By applying Theorem 3, we can obtain WLLN for level-wise independent case.

    Theorem 3.8. Let

    image

    be a sequence of level-wise independent and strong-compactly uniformly integrable fuzzy random variables. Then for any Toeplitz sequence {λni} satisfying

    image

    for some γ > 0,

    image

    Proof. Let ε > 0 and 0 < δ < 1 be given and K be a compact subset of (F(Y),d) such that

    image
    image

    Let us denote

    image

    Then since

    image

    we have that

    image

    For (I), we note that for each α ∈ [0,1], the sequence {Lαn}of random sets is independent and tight. Since (4) implies

    image

    we have that by Corollary 3.2 of Taylor and Inoue [22],

    image

    By Corollary 7, this implies that (I) → 0 as n → ∞.

    Now for (II), since

    image

    we have that

    image

    Thus for large n,

    image

    which completes the proof.

    Corollary 3.9. Let

    image

    be a sequence of level-wise independent and strongly tight fuzzy random variables such that

    image

    Then for any Toeplitz sequence {λni} satisfying

    image

    for some γ > 0,

    image

    Unfortunately, the following example shows that a sequence of identically distributed fuzzy random variables may not be strong-compactly uniformly integrable.

    Example. Let Y = R. For 0 < λ < 1, we define

    image

    Then

    image

    and so d(uλ,uδ)= 1 for λ ≠ δ.

    Now we let Ω =(0,1), A = the Lebesque σ-field and P be the Lebesgue measure. and let

    image

    be a sequence of identically distributed fuzzy random variables with

    image

    defined by

    image

    Suppose that 0 < ε < 1 and that there is a compact subset K of (F(R),d) such that

    image

    Then K necessarily contains a set of the form

    KJ = {uλ : λ ∈ J},

    where P(J) > 1 ? ε. But this is impossible because KJ contains a sequence {uλn : λnJ} which does not have any convergent subsequence.

    The above example implies that Theorem 3 cannot be applied for identically distributed fuzzy random variables. Guan and Li [14] gave an WLLN for weighted sums of level-wise independent fuzzy random variables under the assumption that

    image

    is convergent. The next theorem is slightly different from the result of Guan and Li [14].

    Theorem 3.10. Let

    image

    be a sequence of integrably bounded fuzzy random variables such that for some vF(Y),

    image

    Then

    image

    if and only if for each α ∈ [0,1],

    image

    and

    image

    Proof. To prove the sufficiency, it suffices to prove that

    image

    Let

    image

    and let ε > 0 be given. By Lemma 4 of Guan and Li [8], there exists a partition 0 = α0 < α1 < … < αr = 1 such that

    image

    Then by our assumption, we can find a natural number N such that

    image

    First we note that if A1AA2 and B1BB2, then

    h(A, B) ≤ max[h(A1,B2),h(A2,B1)].

    If 0 < α ≤ 1, then αk?1 < α ≤ αk for some k. Since

    image

    we have that for nN,

    image

    Thus for nN,

    image

    Therefore, by assumption we obtain

    image

    This completes the proof.

    Corollary 3.11. Let

    image

    be a sequence of identically distributed fuzzy random variables with

    image

    and {λni} be a double sequence of real numbers satisfying

    image

    Then

    image

    if and only if for each α ∈ [0,1]

    image

    and

    image

    Proof. The necessity is trivial. To prove the sufficiency, we note that

    image

    Since

    image

    the desired result follows immediately.

    4. Conclusions

    In this paper, we obtained two types of necessary and sufficient conditions under which weak laws of large numbers for weighted sums of fuzzy random variables hold. One is the case of strong-compactly uniformly integrable fuzzy random variables. The other is the case that the weighted averages of its expectations converge. The former includes a strongly tight case and the latter contains the identically distributed case. We also provided WLLN for weighted sums of level-wise independent and strong-compactly uniformly integrable (or strongly tight) fuzzy random variables.

    It remains an open problem whether we can obtain a generalization for the above WLLN to the case of compactly uniform integrability.

  • 1. Zadeh L.A. 1965 “Fuzzy sets” [Information and Control] Vol.8 P.338-353 google doi
  • 2. Puri M. L., Ralescu D. A. 1986 “Fuzzy random variables” [Journal of Mathematical Analysis and Applications] Vol.114 P.409-422 google doi
  • 3. Colubi A., Lopez-Diiaz M., Domiinguez-Menchero J. S., Gil M. A. 1999 “A generalized strong law of large numbers” [Probability Theory and Related Fields] Vol.114 P.401-417 google doi
  • 4. Colubi A., Dominguez-Menchero J. S., Lopez-Diaz M., Korner R. 2001 “A method to derive strong laws of large numbers for random upper semicontinuous functions” [Statistics & Probability Letters] Vol.53 P.269-275 google doi
  • 5. Feng Y. 2002 “An approach to generalize laws of large numbers for fuzzy random variables” [Fuzzy Sets and Systems] Vol.128 P.237-245 google doi
  • 6. Fu K. A., Zhang L. X. 2008 “Strong laws of large numbers for arrays of rowwise independent random compact sets and fuzzy random sets” [Fuzzy Sets and Systems] Vol.159 P.3360-3368 google doi
  • 7. Inoue H. 1991 “A strong law of large numbers for fuzzy random sets” [Fuzzy Sets and Systems] Vol.41 P.285-291 google doi
  • 8. Klement E. P., Puri M. L., Ralescu D. A. 1986 “Limit theorems for fuzzy random variables” [Proceedings of the Royal Society A] Vol.407 P.171-182 google doi
  • 9. Li S., Ogura Y. 2006 “Strong laws of large numbers for independent fuzzy set-valued random variables” [Fuzzy Sets and Systems] Vol.157 P.2569-2578 google doi
  • 10. Molchanov I. S. 1999 “On strong laws of large numbers for random upper semicontinuous functions” [Journal of Mathematical Analysis and Applications] Vol.235 P.349-355 google doi
  • 11. Proske F. N., Puri M. L. 2002 “Strong law of large numbers for banach space valued fuzzy random variables” [Journal of Theoretical Probability] Vol.15 P.543-551 google doi
  • 12. Taylor R. L., Seymour L., Chen Y. 2001 “Weak laws of large numbers for fuzzy random sets” [Nonlinear Analysis: Theory, Methods & Applications] Vol.47 P.1245-1256 google doi
  • 13. Joo S. Y. 2004 “Weak laws of large numbers for fuzzy random variables” [Fuzzy Sets and Systems] Vol.147 P.453-464 google doi
  • 14. Guan L., Li S. 2004 “Laws of large numbers for weighted sums of fuzzy set-valued random variables” [International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems] Vol.12 P.811-825 google doi
  • 15. Joo S. Y., Kim Y. K., Kwon J. S. 2006 “Strong convergence for weighted sums of fuzzy random sets” [Information Sciences] Vol.176 P.1086-1099 google doi
  • 16. Kim Y. K. 2004 “Weak convergence for weighted sums of levelcontinuous fuzzy random variables” [Journal of Fuzzy Logic and Intelligent Systems] Vol.14 P.852-856 google
  • 17. Debreu G 1967 “Integration of correspondences” [in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability] P.351-372 google
  • 18. Colubi A., Domıinguez-Menchero J. S., Lopez-Diaz M., Ralescu D. A. 2001 “On the formalization of fuzzy random variables” [Information Sciences] Vol.133 P.3-6 google doi
  • 19. Kim Y. K. 2002 “Measurability for fuzzy valued functions” [Fuzzy Sets and Systems] Vol.129 P.105-109 google doi
  • 20. Li S., Ogura Y., Kreinovich V. 2002 Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables google
  • 21. Greco G. H., Moschen M. P. 2006 “Supremum metric and relatively compact sets of fuzzy sets” [Nonlinear Analysis: Theory, Methods & Applications] Vol.64 P.1325-1335 google doi
  • 22. Taylor R. L., Inoue H. 1985 “Convergence of weighted sums of random sets” [Stochastic Analysis and Applications] Vol.3 P.379-396 google doi