- Jingfeng Tian
^{1}Email author, - Zhiming Zhang
^{2}and - Dazeng Tian
^{3}

**2010**:290124

https://doi.org/10.1155/2010/290124

© Jingfeng Tian et al. 2010

**Received: **8 October 2009

**Accepted: **12 December 2009

**Published: **4 January 2010

## Abstract

## Keywords

## 1. Introduction

In 1974, the Japanese scholar Sugeno [1] presented a kind of typical nonadditive measure, Sugeno measure, which is an important generalization of probability measure [2–6]. As we all know, the definitions and properties of moment of random variable play an important role in probability theory [7–9]. Likewise, they are also very important for Sugeno measure. In this paper we present the analogous definitions and properties based on random variable on Sugeno measure space. Then some important moment estimation inequalities based on random variable are presented and proven.

## 2. Preliminaries

Let us recall some definitions and facts from [5].

Definition 2.1.

for any disjoint sequence of sets in whose union is also in .

Definition 2.2.

Let be a -algebra of subsets of . And is called Sugeno measure on if and only if it satisfies the - rule and . Usually, Sugeno measure on is denoted by .

We call the triple a Sugeno measure space, denoted by space, where . In the following, our discussion will be restricted to this space.

Theorem 2.3.

For all imply that (monotonicity).

Theorem 2.4.

In order to present the analogous definitions and properties based on random variable on Sugeno measure space, we recall some definitions and facts from [10].

Definition 2.5.

Let be a function mapping from to real line . Then is called a random variable.

Definition 2.6.

Definition 2.7.

is valid. The function is called a density function of .

In the following, our discussion will be restricted to the continuous random variable.

Definition 2.8.

Let be the distribution function of random variable . If , then we call an expected value of random variable , denoted by .

Theorem 2.9.

Definition 2.10.

Let be a random variable. If exists, then is called the variance of , denoted by .

## 3. Moment Estimation Inequalities Based on Random Variable

We begin this section with a short lemma (see [11]), which will be useful in the sequel.

Lemma 3.1.

Theorem 3.2.

holds true.

Definition 3.3.

Let be a random variable and a positive number. Then the expected value is called the th moment, the expected value is called the th absolute moment, the expected value is called the th central moment, and the expected value is called the th absolute central moment.

Theorem 3.4.

Proof.

Similar to the case of credibility theory [12], we have the following: Theorems 3.5, 3.6, and 3.7.

Theorem 3.5.

Theorem 3.6.

Theorem 3.7.

Theorem 3.8.

Let be a random variable and Then if and only if .

Proof.

From and Theorem 3.2, the conclusion is valid.

Theorem 3.9.

Let be a random variable and . If then Conversely, if there exists one positive number such that , then for any where

## Declarations

### Acknowledgment

This work was supported by the NNSF of China (no. 60773062), the NSF of Hebei Province of China (no. 2008000633), the foundation of North China Electric Power University (no. 200911033), the KSRP of Department of Education of Hebei Province of China (no. 2005001D), and the KSTRP of Ministry of Education of China (no. 206012).

## Authors’ Affiliations

## References

- Sugeno M:
*Theory of fuzzy integrals and its applications, Ph.D. dissertation*. Tokyo Institute of Technology; 1974.Google Scholar - Basile A: Sequential compactness for sets of Sugeno fuzzy measures.
*Fuzzy Sets and Systems*1987, 21(2):243–247. 10.1016/0165-0114(87)90168-0MathSciNetView ArticleMATHGoogle Scholar - Berres M:
-additive measures on measure spaces.
*Fuzzy Sets and Systems*1988, 27(2):159–169. 10.1016/0165-0114(88)90146-7MathSciNetView ArticleMATHGoogle Scholar - Chen T-Y, Wang J-C, Tzeng G-H: Identification of general fuzzy measures by genetic algorithms based on partial information.
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*2000, 30(4):517–528.View ArticleGoogle Scholar - Wang ZY, Klir GJ:
*Fuzzy Measure Theory*. Plenum Press, New York, NY, USA; 1992:x+354.View ArticleMATHGoogle Scholar - Wierzchon ST: An algorithm for identification of fuzzy measure.
*Fuzzy Sets and Systems*1983, 9(1):69–78. 10.1016/S0165-0114(83)80005-0MathSciNetView ArticleMATHGoogle Scholar - Graversen SE, Peškir G: Maximal inequalities for Bessel processes.
*Journal of Inequalities and Applications*1998, 2(2):99–119. 10.1155/S102558349800006XMathSciNetMATHGoogle Scholar - Sung SH: Moment inequalities and complete moment convergence.
*Journal of Inequalities and Applications*2009, 2009:-14.Google Scholar - Zang Q-P: A limit theorem for the moment of self-normalized sums.
*Journal of Inequalities and Applications*2009, 2009:-10.Google Scholar - Ha M, Li Y, Li J, Tian D: The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space.
*Science in China. Series F*2006, 49(3):372–385. 10.1007/s11432-006-0372-8MathSciNetView ArticleMATHGoogle Scholar - Ha M, Zhang H, Pedrycz W, Xing H: The expected value models on Sugeno measure space.
*International Journal of Approximate Reasoning*2009, 50(7):1022–1035. 10.1016/j.ijar.2009.03.008MathSciNetView ArticleMATHGoogle Scholar - Liu B:
*Uncertainty Theory. An Introduction to Its Axiomatic Foundation, Studies in Fuzziness and Soft Computing*.*Volume 154*. Springer; Plenum, Berlin, Germany; 2004:xii+411.Google Scholar

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