A Note on Strong Laws of Large Numbers for Dependent Random Sets and Fuzzy Random Sets
© Ke-Ang Fu 2010
Received: 16 August 2009
Accepted: 7 February 2010
Published: 23 March 2010
Recently, great progress has been made towards the theories and applications of random sets and fuzzy random sets in the areas of information science, probability, and statistics. It is well known that Robbins [1, 2] first proposed the concept of random sets and investigated the relationships between random sets and geometric probabilities in his early work. After that, Kendall  and Matheron  provided a comprehensive mathematical theory of random sets which was greatly influenced by the geometric probability prospective. Their proposed framework exerted a strong influence on the limit theorems developed in the recent decades. Notice that strong laws of large numbers (SLLNs) play an important role in probability limit theorems, and several variants of SLLNs were built by Artstein and Vitale , Puri and Ralescu , Hiai , Inoue , Taylor et al. [9–11], Uemura , and so forth. Among them, Artstein and Vitale  proved limit theorems concerning random sets in and Puri and Ralescu  were the first to obtain the SLLNs for independent identically distributed (i.i.d.) Banach space-valued compact convex random sets. Among others, SLLNs were obtained under more relaxed conditions, and a detailed survey of these results is available in Taylor and Inoue .
The theory of fuzzy sets was introduced by Zadeh  (for an outline recently, one can refer to [14, 15]), and the concept of fuzzy random variables was promoted by Kwakernaak , where useful basic properties were developed. Puri and Ralescu  used the concept of fuzzy random variables in generalizing results for random sets to fuzzy random sets. With respect to laws of large numbers, Kruse  proved an SLLN for i.i.d. fuzzy random variables. Klement et al.  considered fuzzy versions of random sets in Euclidean spaces and obtained an i.i.d. SLLN. Inoue  derived SLLNs for independent, tight fuzzy random sets, and i.i.d. fuzzy random sets in a separable Banach space. Recently, SLLNs have been established under various conditions, and one can refer to the following papers [8–11, 21–26]. Also for more detailed results about limit theorems of random sets and fuzzy random sets, we refer the readers to Li et al.  and references therein.
However, to the best of our knowledge, many limit theorems, especially the laws of large numbers, were obtained for independent random sets or fuzzy random sets in the past decades, and little is known of dependent random sets or fuzzy random sets except the exchangeable dependence involved in Inoue [8, 28], Taylor et al. , and Terán . In this paper, we aim to propose a new kind of dependence for random sets and fuzzy random sets, and then establish several strong laws of large numbers in Kuratowski-Mosco convergence without the restriction of compactness, where random sets take values of closed subsets in separable Banach spaces.
The layout of this paper is as follows. In Section 2, we give some basic definitions and properties, and the new dependence is proposed in Section 3. In the last section we show several SLLNs for a sequence of dependent random sets and fuzzy random sets, and their proofs.
2. Definitions and Preliminaries
Clearly, the Hausdorff convergence is generally stronger than Kuratowski-Mosco convergence, since the former implies the latter when is infinite dimensional, and in finite dimensional spaces they coincide with bounded sets (cf. ).
where is the usual Bochner integral in Define for . This definition was introduced by Aumann in 1965 as a natural generalization of the integral of real-valued random variables in . If a Bochner integral can be defined as and (cf. ). The random set is said to be integrably bounded if the real-valued random variable is integrable (cf. [27, 32]). Hiai and Umegaki  showed that a random set is integrably bounded if and only if is bounded in Thus an integrably bounded random set may take unbounded sets.
3. Mixing Dependence
Many statistical results are concerned with independent and identically distributed (i.i.d.) random sets or fuzzy random sets. While it is not always possible to assume that random sets or fuzzy random sets are independent, the sequence can be often dependent. However, for dependent case, it seems that only the exchangeability is involved. In what follows, we propose a new kind of dependence for random sets which is popular with random variables and random elements. Similarly, it can be defined for fuzzy random sets.
If (resp. tends to zero as then we say that the sequence is -mixing (resp., -mixing). Obviously, a -mixing sequence is a -mixing sequence. Also it is well known that many limit results were derived for real-valued mixing random sequences and random fields in the past thirty years (cf. [34, 35] and references therein). Zhang [36, 37] extended them to the Banach space-valued mixing random fields and established some moment inequalities. As far as we know, there is little concerning the dependent random sets or fuzzy random sets except the exchangeability dependence. The main purpose of this paper is to establish limit theorems for mixing dependent random sets or fuzzy random sets which extend the results of independent case.
4. Limit Theorems
It follows from (2) easily.
Hiai  proved a strong law of large numbers of i.i.d. random variables in in Kuratowski-Mosco convergence. Recently, Inoue and Taylor  replaced i.i.d. by exchangeability and obtain a strong law of large numbers. Here we replace the i.i.d. by -mixing dependence which is a more extensive dependence and derive strong laws of large numbers for random sets and fuzzy random sets, respectively.
Here we only consider the -mixing case, since the -mixing case can be proved similarly. By Lemma 4.1(2), for a sequence of -mixing random set in , there exists a -measurable function and the corresponding random elements such that for all
The author thanks the referees for pointing out some errors in a previous version, as well as for several comments that have led to improvements in this work. This project was supported by the National Natural Science Foundation of China (nos. 10671176, 10771192, and 10901138) and the Research Grant of Zhejiang Gongshang University (X10-26).
- Robbins HE: On the measure of a random set. Annals of Mathematical Statistics 1944, 15: 70–74. 10.1214/aoms/1177731315MathSciNetView ArticleMATHGoogle Scholar
- Robbins HE: On the measure of a random set. II. Annals of Mathematical Statistics 1945, 16: 342–347. 10.1214/aoms/1177731060MathSciNetView ArticleMATHGoogle Scholar
- Kendall DG: Foundations of a theory of random sets. In Stochastic Geometry (A Tribute to the Memory of Rollo Davidson). Edited by: Harding EF, Kendall DG. John Wiley & Sons, New York, NY, USA; 1974:322–376.Google Scholar
- Matheron G: Random Sets and Integral Geometry, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York, NY, USA; 1975:xxiii+261.Google Scholar
- Artstein Z, Vitale RA: A strong law of large numbers for random compact sets. Annals of Probability 1975, 3(5):879–882. 10.1214/aop/1176996275MathSciNetView ArticleMATHGoogle Scholar
- Puri ML, Ralescu DA: Strong law of large numbers for Banach space valued random sets. The Annals of Probability 1983, 11(1):222–224. 10.1214/aop/1176993671MathSciNetView ArticleMATHGoogle Scholar
- Hiai F: Convergence of conditional expectations and strong laws of large numbers for multivalued random variables. Transactions of the American Mathematical Society 1985, 291(2):613–627. 10.1090/S0002-9947-1985-0800254-XMathSciNetView ArticleMATHGoogle Scholar
- Inoue H: Exchangeability and convergence for random sets. Information Sciences 2001, 133(1–2):23–37. 10.1016/S0020-0255(01)00075-5MathSciNetView ArticleMATHGoogle Scholar
- Taylor RL, Inoue H: Convergence of weighted sums of random sets. Stochastic Analysis and Applications 1985, 3(3):379–396. 10.1080/07362998508809069MathSciNetView ArticleMATHGoogle Scholar
- Taylor RL, Inoue H: Laws of large numbers for random sets. In Random Sets (Minneapolis, MN, 1996). Volume 97. Springer, New York, NY, USA; 1997:347–360. 10.1007/978-1-4612-1942-2_15View ArticleGoogle Scholar
- Taylor RL, Vidyashankar AN, Chen Y: Limit theorems for exchangeable random elements and exchangeable random sets. Asymptotics in Statistics and Probability 2000, 367–378.Google Scholar
- Uemura T: A law of large numbers for random sets. Fuzzy Sets and Systems 1993, 59(2):181–188. 10.1016/0165-0114(93)90197-PMathSciNetView ArticleMATHGoogle Scholar
- Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.MathSciNetMATHGoogle Scholar
- Zadeh LA: Toward a generalized theory of uncertainty (GTU)—an outline. Information Sciences 2005, 172(1–2):1–40. 10.1016/j.ins.2005.01.017MathSciNetView ArticleMATHGoogle Scholar
- Zadeh LA: Is there a need for fuzzy logic? Information Sciences 2008, 178(13):2751–2779. 10.1016/j.ins.2008.02.012MathSciNetView ArticleMATHGoogle Scholar
- Kwakernaak H: Fuzzy random variables. I. Definitions and theorems. Information Sciences 1978, 15(1):1–29. 10.1016/0020-0255(78)90019-1MathSciNetView ArticleMATHGoogle Scholar
- Puri ML, Ralescu DA: Fuzzy random variables. Journal of Mathematical Analysis and Applications 1986, 114(2):409–422. 10.1016/0022-247X(86)90093-4MathSciNetView ArticleMATHGoogle Scholar
- Kruse R: The strong law of large numbers for fuzzy random variables. Information Sciences 1982, 28(3):233–241. 10.1016/0020-0255(82)90049-4MathSciNetView ArticleMATHGoogle Scholar
- Klement EP, Puri ML, Ralescu DA: Limit theorems for fuzzy random variables. Proceedings of the Royal Society A 1986, 407(1832):171–182. 10.1098/rspa.1986.0091MathSciNetView ArticleMATHGoogle Scholar
- Inoue H: A strong law of large numbers for fuzzy random sets. Fuzzy Sets and Systems 1991, 41(3):285–291. 10.1016/0165-0114(91)90132-AMathSciNetView ArticleMATHGoogle Scholar
- Colubi A, López-Díaz M, Domínguez-Menchero JS, Gil MA: A generalized strong law of large numbers. Probability Theory and Related Fields 1999, 114(3):401–417. 10.1007/s004400050229MathSciNetView ArticleMATHGoogle Scholar
- Fu K, Zhang L: Strong limit theorems for random sets and fuzzy random sets with slowly varying weights. Information Sciences 2008, 178(12):2648–2660. 10.1016/j.ins.2008.01.005MathSciNetView ArticleMATHGoogle Scholar
- Guan L, Li S: Laws of large numbers for weighted sums of fuzzy set-valued random variables. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2004, 12(6):811–825. 10.1142/S0218488504003223MathSciNetView ArticleMATHGoogle Scholar
- Joo SY, Kim YK, Kwon JS: Strong convergence for weighted sums of fuzzy random sets. Information Sciences 2006, 176(8):1086–1099. 10.1016/j.ins.2005.02.002MathSciNetView ArticleMATHGoogle Scholar
- Li S, Ogura Y: Strong laws of large numbers for independent fuzzy set-valued random variables. Fuzzy Sets and Systems 2006, 157(19):2569–2578. 10.1016/j.fss.2003.06.011MathSciNetView ArticleMATHGoogle Scholar
- Terán P: A strong law of large numbers for random upper semicontinuous functions under exchangeability conditions. Statistics & Probability Letters 2003, 65(3):251–258. 10.1016/j.spl.2003.08.004MathSciNetView ArticleMATHGoogle Scholar
- Li S, Ogura Y, Kreinovich V: Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Theory and Decision Library. Series B: Mathematical and Statistical Methods. Volume 43. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xiv+391.View ArticleGoogle Scholar
- Inoue H, Taylor RL: Laws of large numbers for exchangeable random sets in Kuratowski-Mosco sense. Stochastic Analysis and Applications 2006, 24(2):263–275. 10.1080/07362990500522114MathSciNetView ArticleMATHGoogle Scholar
- Salinetti G, Wets RJ-B: On the convergence of sequences of convex sets in finite dimensions. SIAM Review 1979, 21(1):18–33. 10.1137/1021002MathSciNetView ArticleMATHGoogle Scholar
- Aumann RJ: Integrals of set-valued functions. Journal of Mathematical Analysis and Applications 1965, 12: 1–12. 10.1016/0022-247X(65)90049-1MathSciNetView ArticleMATHGoogle Scholar
- Debreu G: Integration of correspondences. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1. University of California Press, Berkeley, Calif, USA; 1967:351–372.Google Scholar
- Hiai F, Umegaki H: Integrals, conditional expectations, and martingales of multivalued functions. Journal of Multivariate Analysis 1977, 7(1):149–182. 10.1016/0047-259X(77)90037-9MathSciNetView ArticleMATHGoogle Scholar
- Puri ML, Ralescu DA: Différentielle d'une fonction floue. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique 1981, 293(4):237–239.MathSciNetMATHGoogle Scholar
- Bradley RC: On the spectral density and asymptotic normality of weakly dependent random fields. Journal of Theoretical Probability 1992, 5(2):355–373. 10.1007/BF01046741MathSciNetView ArticleMATHGoogle Scholar
- Lin ZY, Lu CR: Limit Theory on Mixing Random Variables. Kluwer Academic Publishers and Science Press, Dordrecht, The Netherlands; 1997.Google Scholar
- Zhang L: Rosenthal type inequalities for -valued strong mixing random fields and their applications. Science in China 1998, 41(7):736–745. 10.1007/BF02901956MathSciNetView ArticleMATHGoogle Scholar
- Zhang LX, Wen JW: A strong law of large numbers for -valued random fields. Chinese Annals of Mathematics 2001, 22(2):205–216.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.