1. Introduction
The study of the fuzzy random sets, defined as measurable mappings on a probability space, was initiated by Kwakernaak [12] where useful basic properties were developed. Puri and Ralescu [9] used the concept of fuzzy random variables in generating results for random sets to fuzzy random sets. Kruse [8] proved a strong law of large numbers for independent identically distributed fuzzy random variables. Artstein and Vitale [1] proved a strong law of large numbers(SLLN) for Rp-valued random sets and Cressie [3] proved a SLLN for some paticular class of Rp-valued random sets. Using Rådstrom embedding(e.g. Rådstrom [14]), Puri and Ralescu [12] proved a SLLN for Banach space valued random sets and they also proved SLLN for fuzzy random sets, which generalized all of previous SLLN for random sets. In recent year, Joo, Kim and Kwon [6] proved Chung’s type law of large numbers for fuzzy random variables and Kwon and Shim [11] obtained a uniform strong law of large numbers for partial sum processes of fuzzy random sets. In this paper we obtain Marcinkiewicz’s type laws for fuzzy random sets in the Euclidean space under the assumption that WLLN holds. The proofs of the results are based heavily on isometrical embeddings of the fuzzy sample spaces, endowed with Lp-metrics, into Lp-spaces. Our results give the fuzzy version of Marcinkiewicz’s type law of large numbers in general Banach spaces.
2. Preliminaries
Let K(Rn) (Kc(Rn) ) be the collection of nonempty compact (and convex ) subsets of Euclidean space Rn. The set can be viewed as a linear structure induced by the scalar multiplication and the Minkowski addition, that is
for all A, B ∈ K(Rn) and λ ∈ R. If d is the Hausdoff metric on K(Rn) which, for A, B ∈ K(Rn) , is given by
where |·| denotes the Euclidean norm. Then (K(Rn) , d) is a complete separable metric space [4,10].
A fuzzy set of Rn is a mapping A : Rn → [0, 1]. We will denote by Aα the α-level set of A (that is Aα = {x ∈ Rn : A(x) ≥ α} ) for all α ∈ (0, 1] and by A0 the closure of the support of A(that is A0 = cl{x ∈ Rn : A(x) > 0} ).
Let Fc(Rn)(Fcoc(Rn)) be the class of the fuzzy sets A satisfying the following conditions
And is the subset of Fc(Rn)( Fcoc(Rn)) with bounded support.
Given a measurable space (Ω,A) and the metric space (K(Rn) , d), a random set (or as a random compact set) is associated with a Borel measurable mapping X : Ω → K(Rn) . If X : Ω → K(Rn) is a set-valued mapping, then X is a random set if and only if X−1(C) = {ω ∈ Ω : X(ω) ∩ C ≠ ∅} ∈ A for all C ∈ K(Rn) .
If X is a random set, the mapping denoted by ∥X∥d and defined by
for all ω ∈ Ω, is a random variable, where is the fuzzy set where and otherwise.
A support function of a non-void bounded subset K of Rn is defined by
where
Now we endow Fc(Rn) with the initial topology generated by the mappings
then the topology mentioned above enables us to introduce a measurability concept for defining fuzzy random variable. We call a mapping X : Ω → Fc(Rn) fuzzy random variable over (Ω, A, μ) if it is A-measurable over the initial topology.
For a real number p ≥ 1 and A, B ∈ Fc(Rn) , define
and
where denotes the unit Lebesgue measure on the unit sphere in Rn. Then dp(ρp) becomes a separable metric on with the relation ρp ≤ dp which induce the same topology
Now consider Lp([0, 1]×Sn−1), the Lp-space with respect to [0, 1]×Sn−1, the obvious product σ-algebra and the product measure . Then under the Lp-norm ∥∥p we obtain Lp([0, 1]×Sn−1) as a separable Banach space. Next we can embed isometrically isomorphic into Lp([0, 1]×Sn−1) as a positive cone (for details see [5,7]). Embedding into Lp([0, 1]×Sn−1), we draw a convergence theorem in Banach space. For 1 ≤ p < ∞, Lp([0, 1] × Sn−1) is so called separable Banach space of type min(p, 2). It is known that separable Banach spaces of type 2 is exactly those separable Banach space where the classical strong law of large numbers for independent non-identically distributed random variables holds.
3. Main Results
To prove the main theorem we will need the following lemmas. Lemma 1 connects two metric spaces and Lp([0, 1] × Sn−1) isometrically.
Lemma 3.1. Let 1 ≤ p < ∞ be fixed. Then j : → Lp([0, 1] × Sn−1) by A defines an injection mapping satisfying
The following is a generalization of a classical result [15, p. 127-128].
Lemma 3.2. Let {Xn|n ≥ 1} be a sequence of fuzzy random sets stochastically dominated by X with for 0 < r < ∞, that is, for any t > 0, . Then
Proof. . Notice that is a sequence of random variables stochastically dominated by . Now apply Stout’s result.
Lemma 3.3 ([5]). Let {Xk|1 ≤ k ≤ n} be -valued independent random variables. Let Then
Lemma 3.4 ([5]). Let {Xk|1 ≤ k ≤ n} be independent -valued random variables with for k = 1, 2, · · · n and 1 ≤ r ≤ 2. Then we have
where Cr is a positive constant depending only on r ; if r = 2 then it is possible to take C2 = 4.
Theorem 3.5. Let {Xn|n ≥ 1} be a sequence of independent identically distributed -valued fuzzy random variables with for 1≤r≤2 and let .Then the following are equivalent:
Proof. Let j : → Lp([0, 1]×Sn−1) be an isometry. Then {j ◦Xn|n ≥ 1} be a sequence of independent identically distributed random element in a Banach space Lp([0, 1] × Sn−1). Since in what follow we use X and j ◦ X interchangeably.
First we show that (i) ⇔ (ii) ⇔ (iii) in L1. Since , we have
Hence it is enough to show that
Since by lemma 1, these equivalence hold by applying Theorem 5 in [5] to
Now it remains to show that (iii) ⇒ (iv). Assume that in L1.
Then now
Thus it is enough to show that
From Lemma 4
By a standard calculation, we have Thus the proof is completed.
Remark 3.1. (1) For i.i.d real valued random variables, Pyke and Root [13] showed that
(2) For i.i.d. B-valued random variables with E∥X1∥ < ∞ for 1 ≤ r < 2, Choi and Sung [2] showed that
References
- Artstein, Z and Vitale, D., A strong law of large numbers for random compact sets, Ann. Probab. 3 (1975), 879-882. https://doi.org/10.1214/aop/1176996275
- Choi, B.D., Sung, S.H., On Chung's strong law of large numbers in general Banach spaces, Bull. Austral. Math. Soc. 37 (1988), 93-100. https://doi.org/10.1017/S0004972700004184
- Cressie, N., A strong limit theorem for random sets, Suppl. Adv. Appl. Prob. 10 (1978), 36-46. https://doi.org/10.2307/1427005
- Debreu, G., Integration of correspondences, in: Proc. 5th Berkely Symp. on Mathematical Statistics and Probability, Vol. II, Part I, University of California Press, Berkely. pp. 351-372, 1967.
- Hong, D.H., Moon, Eunho and Kim, J.D., Convexity and semicontinuity of fuzzy mappings using the support function, J. Appl. Math. & Informatics 28 (2010), 1419-1430.
- Joo, S.Y., Kim, Y.K. and Kwon, J.S. On Chung's type law of large numbers for fuzzy random variables, Stat, and Probab Letters 74 (2005), 67-75 https://doi.org/10.1016/j.spl.2005.04.030
- Kratschmer, V., Limit theorems for fuzzy random variables, Fuzzy Sets and Systems 126 (2002), 253-263. https://doi.org/10.1016/S0165-0114(00)00100-7
- Kruse, R., The strong law of large numbers for fuzzy random variables, Inform. Sci. 28 (1982), 233-241. https://doi.org/10.1016/0020-0255(82)90049-4
- Kwakernaak, H., Fuzzy random variables I, Inform. Sci. 15 (1978), 1-29. https://doi.org/10.1016/0020-0255(78)90019-1
- Kwon, J.S. and Shim, H.T., Overviews on limit concrepts of a sequence of fuzzy numbers I, J. Appl. Math. & Informatics 29 (2011), 1017-1025
- Kwon, J.S. and Shim, H.T., A uniform strong law of large numbers for partial sum processes of fuzzy random sets J. Appl. Math. & Informatics 30 (2012), 647-653
- Puri, M.L. and Ralescu, D.A., Strong law of large numbers for Banach space valued random sets, Ann. Probab. 11 (1983), 222-224. https://doi.org/10.1214/aop/1176993671
- Pyke, R and Root, D., On convregence in r-mean of normalized partial sums, Ann. Math. Statistics 39 (1968), 379-381. https://doi.org/10.1214/aoms/1177698400
- Radstrom, H., An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169. https://doi.org/10.1090/S0002-9939-1952-0045938-2
- Stout, W.F., Almost sure convergence, Academic press, New York, 1974.