Journal of Inequalities and Applications
VolumeΒ 2010Β (2010), Article IDΒ 290124, 10 pages
doi:10.1155/2010/290124
Research Article

Moment Estimation Inequalities Based on 𝑔 πœ† Random Variable on Sugeno Measure Space

Jingfeng Tian,1Β Zhiming Zhang,2Β and Dazeng Tian3

1College of Science and Technology, North China Electric Power University, Baoding, Hebei Province 071051, China
2College of Mathematics and Computer Sciences, Hebei University, Baoding, Hebei Province 071002, China
3College of Physics Science and Technology, Hebei University, Baoding, Hebei Province 071002, China

Received 8 October 2009; Accepted 12 December 2009

Academic Editor: AndreiΒ Volodin

Copyright Β© 2010 Jingfeng Tian et al. 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.

Abstract

The definitions and properties of moment of 𝑔 πœ† random variable are provided on Sugeno measure space. Then some important moment estimation inequalities based on 𝑔 πœ† random variable are presented and proven.

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 𝑔 πœ† random variable on Sugeno measure space. Then some important moment estimation inequalities based on 𝑔 πœ† random variable are presented and proven.

2. Preliminaries

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

Definition 2.1. Let 𝑋 be a nonempty set, let 𝜁 be a nonempty class of subsets of 𝑋 , and let πœ‡ be a nonnegative real valued set function defined on 𝜁 . Therefore πœ‡ satisfies the 𝜎 - πœ† rule (on 𝜁 ) if and only if there exists ξ‚΅ βˆ’ 1 πœ† ∈ ξ‚Ά s u p πœ‡ , ∞ βˆͺ { 0 } ( 2 . 1 ) such that πœ‡  ∞  𝑖 = 1 𝐸 𝑖 ξƒͺ = ⎧ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎩ 1 πœ† ξƒ― ∞  𝑖 = 1 ξ€Ί ξ€· 𝐸 1 + πœ† β‹… πœ‡ 𝑖 ξƒ° , ξ€Έ ξ€» βˆ’ 1 a s πœ† β‰  0 , ∞  𝑖 = 1 πœ‡ ξ€· 𝐸 𝑖 ξ€Έ , a s πœ† = 0 , ( 2 . 2 ) for any disjoint sequence { 𝐸 𝑖 } of sets in 𝜁 whose union is also in 𝜁 .

Definition 2.2. Let β„± be a 𝜎 -algebra of subsets of 𝑋 . And πœ‡ is called Sugeno measure on β„± if and only if it satisfies the 𝜎 - πœ† rule and πœ‡ ( 𝑋 ) = 1 . Usually, Sugeno measure on β„± is denoted by 𝑔 πœ† .

We call the triple ( 𝑋 , β„± , 𝑔 πœ† ) a Sugeno measure space, denoted by 𝑔 πœ† space, where πœ† ∈ ( βˆ’ 1 , ∞ ) . In the following, our discussion will be restricted to this space.

Theorem 2.3. For all 𝐸 , 𝐹 ∈ β„± , 𝐸 βŠ‚ 𝐹 imply that 𝑔 πœ† ( 𝐸 ) ≀ 𝑔 πœ† ( 𝐹 )   (monotonicity).

Theorem 2.4. Let 𝑔 πœ† be a Sugeno measure on β„± . Then, for any 𝐸 ∈ β„± and 𝐹 ∈ β„± , 𝑔 πœ† 𝑔 ( 𝐸 βˆͺ 𝐹 ) = πœ† ( 𝐸 ) + 𝑔 πœ† ( 𝐹 ) βˆ’ 𝑔 πœ† ( 𝐸 ∩ 𝐹 ) + πœ† 𝑔 πœ† ( 𝐸 ) 𝑔 πœ† ( 𝐹 ) 1 + πœ† 𝑔 πœ† ( , 𝑔 𝐸 ∩ 𝐹 ) πœ† ( 𝑔 𝐸 βˆ’ 𝐹 ) = πœ† ( 𝐸 ) βˆ’ 𝑔 πœ† ( 𝐸 ∩ 𝐹 ) 1 + πœ† 𝑔 πœ† , 𝑔 ( 𝐸 ∩ 𝐹 ) πœ† ( 𝐸 𝑐 ) = 1 βˆ’ 𝑔 πœ† ( 𝐸 ) 1 + πœ† 𝑔 πœ† . ( 𝐸 ) ( 2 . 3 )

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

Definition 2.5. Let πœ‰ be a function mapping from ( 𝑋 , β„± , 𝑔 πœ† ) to real line ℝ . Then πœ‰ is called a 𝑔 πœ† random variable.

Definition 2.6. Let πœ‰ be a 𝑔 πœ† random variable. Then the distribution function of πœ‰ is defined by 𝐹 𝑔 πœ† ( π‘₯ ) = 𝑔 πœ† { πœ‰ ≀ π‘₯ } , βˆ€ π‘₯ ∈ ℝ . ( 2 . 4 )

Definition 2.7. Let 𝐹 𝑔 πœ† ( π‘₯ ) be the distribution function of 𝑔 πœ† random variable πœ‰ . Then πœ‰ is called continuous 𝑔 πœ† random variable if there exists a nonnegative real valued function 𝑓 𝑔 πœ† ( π‘₯ ) such that 𝐹 𝑔 πœ† ξ€œ ( π‘₯ ) = π‘₯ βˆ’ ∞ 𝑓 𝑔 πœ† ( 𝑑 ) 𝑑 𝑑 , βˆ€ π‘₯ ∈ ℝ ( 2 . 5 ) is valid. The function 𝑓 𝑔 πœ† ( π‘₯ ) is called a density function of πœ‰ .

In the following, our discussion will be restricted to the continuous 𝑔 πœ† random variable.

Definition 2.8. Let 𝐹 𝑔 πœ† ( π‘₯ ) be the distribution function of 𝑔 πœ† random variable πœ‰ . If ∫ ∞ βˆ’ ∞ | π‘₯ | 𝑑 𝐹 𝑔 πœ† ( π‘₯ ) < ∞ , then we call ∫ ∞ βˆ’ ∞ π‘₯ 𝑑 𝐹 𝑔 πœ† ( π‘₯ ) an expected value of 𝑔 πœ† random variable πœ‰ , denoted by 𝐸 𝑔 πœ† ( πœ‰ ) .

Theorem 2.9. Let πœ‰ , πœ‚ be 𝑔 πœ† random variables; let C and D be constants. Then 𝐸 𝑔 πœ† ( 𝐢 πœ‰ + 𝐷 πœ‚ ) = 𝐢 𝐸 𝑔 πœ† ( πœ‰ ) + 𝐷 𝐸 𝑔 πœ† ( πœ‚ ) . ( 2 . 6 )

Definition 2.10. Let πœ‰ be a 𝑔 πœ† random variable. If 𝐸 𝑔 πœ† { [ πœ‰ βˆ’ 𝐸 𝑔 πœ† ( πœ‰ ) ] 2 } exists, then 𝐸 𝑔 πœ† { [ πœ‰ βˆ’ 𝐸 𝑔 πœ† ( πœ‰ ) ] 2 } is called the variance of πœ‰ , denoted by 𝐷 𝑔 πœ† ( πœ‰ ) .

3. Moment Estimation Inequalities Based on 𝑔 πœ† Random Variable

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

Lemma 3.1. Let πœ‰ be a 𝑔 πœ† random variable whose Sugeno density function 𝑓 𝑔 πœ† exists. If the Lebesgue integral ξ€œ 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ βˆ’ 0 βˆ’ ∞ 𝑔 πœ† ξ€œ { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ ( 3 . 1 ) is finite, then 𝐸 𝑔 πœ† ξ€œ ( πœ‰ ) = 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ βˆ’ 0 βˆ’ ∞ 𝑔 πœ† ξ€œ { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ . ( 3 . 2 )

Theorem 3.2. Let πœ‰ be a nonnegative 𝑔 πœ† random variable. When πœ† β‰₯ 0 , the inequality ∞  𝑖 = 1 𝑔 πœ† { πœ‰ β‰₯ 𝑖 } ≀ 𝐸 𝑔 πœ†  ( πœ‰ ) ≀ ( 1 + πœ† ) 1 + ∞  𝑖 = 1 𝑔 πœ† ξƒͺ { πœ‰ β‰₯ 𝑖 } ( 3 . 3 ) is valid; when πœ† < 0 , the inequality ( 1 + πœ† ) ∞  𝑖 = 1 𝑔 πœ† { πœ‰ β‰₯ 𝑖 } ≀ 𝐸 𝑔 πœ† ( πœ‰ ) ≀ 1 + ∞  𝑖 = 1 𝑔 πœ† { πœ‰ β‰₯ 𝑖 } ( 3 . 4 ) holds true.

Proof. (I) When πœ† β‰₯ 0 , since 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } is a monotone decreasing function of π‘Ÿ , we have 𝐸 𝑔 πœ† ξ€œ ( πœ‰ ) = 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† β‰₯ ξ€œ { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ† = { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† β‰₯ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† = { πœ‰ β‰₯ 𝑖 } 𝑑 π‘Ÿ ∞  𝑖 = 1 𝑔 πœ† 𝐸 { πœ‰ β‰₯ 𝑖 } , 𝑔 πœ† ξ€œ ( πœ‰ ) = 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† ≀ ξ€œ { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ = ( 1 + πœ† ) 0 + ∞ 𝑔 πœ† = { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ ( 1 + πœ† ) ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† ≀ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ ( 1 + πœ† ) ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ†  { πœ‰ β‰₯ 𝑖 βˆ’ 1 } 𝑑 π‘Ÿ = ( 1 + πœ† ) 1 + ∞  𝑖 = 1 𝑔 πœ† ξƒͺ . { πœ‰ β‰₯ 𝑖 } ( 3 . 5 )
(II) When πœ† < 0 , owing to the monotonicity of 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } we also have
𝐸 𝑔 πœ† ξ€œ ( πœ‰ ) = 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† β‰₯ ξ€œ { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ = ( 1 + πœ† ) 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ = ( 1 + πœ† ) ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ β‰₯ ( 1 + πœ† ) ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† { πœ‰ β‰₯ 𝑖 } 𝑑 π‘Ÿ = ( 1 + πœ† ) ∞  𝑖 = 1 𝑔 πœ† 𝐸 { πœ‰ β‰₯ 𝑖 } , 𝑔 πœ† ξ€œ ( πœ‰ ) = 0 + ∞ 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† ≀ ξ€œ { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ† = { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† ≀ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ ∞  𝑖 = 1 ξ€œ 𝑖 𝑖 βˆ’ 1 𝑔 πœ† { πœ‰ β‰₯ 𝑖 βˆ’ 1 } 𝑑 π‘Ÿ = 1 + ∞  𝑖 = 1 𝑔 πœ† { πœ‰ β‰₯ 𝑖 } . ( 3 . 6 )

Definition 3.3. Let πœ‰ be a 𝑔 πœ† random variable and π‘˜ a positive number. Then ( 1 ) the expected value 𝐸 𝑔 πœ† ( πœ‰ π‘˜ ) is called the π‘˜ th moment, ( 2 ) the expected value 𝐸 𝑔 πœ† ( | πœ‰ | π‘˜ ) is called the π‘˜ th absolute moment, ( 3 ) the expected value 𝐸 𝑔 πœ† { [ πœ‰ βˆ’ 𝐸 𝑔 πœ† ( πœ‰ ) ] π‘˜ } is called the π‘˜ th central moment, and ( 4 ) the expected value 𝐸 𝑔 πœ† { [ | πœ‰ βˆ’ 𝐸 𝑔 πœ† ( πœ‰ ) | ] π‘˜ } is called the π‘˜ th absolute central moment.

Theorem 3.4. Let πœ‰ be a nonnegative 𝑔 πœ† random variable and π‘˜ a positive number. Then 𝐸 𝑔 πœ† ξ€· πœ‰ π‘˜ ξ€Έ ξ€œ = π‘˜ 0 + ∞ π‘Ÿ π‘˜ βˆ’ 1 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + π‘˜ πœ† 0 + ∞ π‘Ÿ π‘˜ βˆ’ 1 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ . ( 3 . 7 )

Proof. From Lemma 3.1, we infer 𝐸 𝑔 πœ† ξ€· πœ‰ π‘˜ ξ€Έ = ξ€œ 0 + ∞ 𝑔 πœ† ξ€½ πœ‰ π‘˜ ξ€Ύ ξ€œ β‰₯ π‘₯ 𝑑 π‘₯ + πœ† 0 + ∞ 𝑔 πœ† ξ€½ πœ‰ π‘˜ ξ€Ύ β‰₯ π‘₯ β‹… 𝑔 πœ† ξ€½ πœ‰ π‘˜ ξ€Ύ = ξ€œ ≀ π‘₯ 𝑑 π‘₯ 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ π‘˜ ξ€œ + πœ† 0 + ∞ 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ π‘˜ ξ€œ = π‘˜ 0 + ∞ π‘Ÿ π‘˜ βˆ’ 1 𝑔 πœ† ξ€œ { πœ‰ β‰₯ π‘Ÿ } 𝑑 π‘Ÿ + π‘˜ πœ† 0 + ∞ π‘Ÿ π‘˜ βˆ’ 1 𝑔 πœ† { πœ‰ β‰₯ π‘Ÿ } β‹… 𝑔 πœ† { πœ‰ ≀ π‘Ÿ } 𝑑 π‘Ÿ . ( 3 . 8 )

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

Theorem 3.5. Let πœ‰ be a 𝑔 πœ† random variable that takes values in [ π‘š , 𝑛 ] and has expected value 𝐸 𝑔 πœ† ( πœ‰ ) , and let 𝑓 ( π‘₯ ) be a convex function on [ π‘š , 𝑛 ] . Then 𝐸 𝑔 πœ† [ 𝑓 ] ≀ ( πœ‰ ) 𝑛 βˆ’ 𝐸 𝑔 πœ† ( πœ‰ ) 𝑓 𝐸 𝑛 βˆ’ π‘š ( π‘š ) + 𝑔 πœ† ( πœ‰ ) βˆ’ π‘š 𝑓 𝑛 βˆ’ π‘š ( 𝑛 ) . ( 3 . 9 )

Theorem 3.6. Let πœ‰ be a 𝑔 πœ† random variable that takes values in [ π‘š , 𝑛 ] and has expected value 𝐸 𝑔 πœ† ( πœ‰ ) . Then 𝐷 𝑔 πœ† ξ€Ί 𝐸 ( πœ‰ ) ≀ 𝑔 πœ† ( πœ‰ ) βˆ’ π‘š ξ€» ξ€Ί 𝑛 βˆ’ 𝐸 𝑔 πœ† ξ€» . ( πœ‰ ) ( 3 . 1 0 )

Theorem 3.7. Let πœ‰ be a 𝑔 πœ† random variable that takes values in [ π‘š , 𝑛 ] and has expected value πœ‡ . Then, for any positive integer π‘˜ , 𝐸 𝑔 πœ† ξ‚€ | | πœ‰ | | π‘˜  ≀ 𝑛 βˆ’ πœ‡ 𝑛 βˆ’ π‘š | π‘š | π‘˜ + πœ‡ βˆ’ π‘š 𝑛 βˆ’ π‘š | 𝑛 | π‘˜ , 𝐸 𝑔 πœ† ξ‚€ | | | | πœ‰ βˆ’ πœ‡ π‘˜  ≀ 𝑛 βˆ’ πœ‡ | | | | 𝑛 βˆ’ π‘š πœ‡ βˆ’ π‘š π‘˜ + πœ‡ βˆ’ π‘š | | | | 𝑛 βˆ’ π‘š 𝑛 βˆ’ πœ‡ π‘˜ . ( 3 . 1 1 )

Theorem 3.8. Let πœ‰ be a 𝑔 πœ† random variable and 𝑑 > 0 . Then 𝐸 𝑔 πœ† ( | πœ‰ | 𝑑 ) < ∞ if and only if βˆ‘ ∞ 𝑖 = 1 𝑔 πœ† { | πœ‰ | > 𝑖 1 / 𝑑 } < ∞ .

Proof. From 𝑔 πœ† { | πœ‰ | 𝑑 β‰₯ 𝑖 } = 𝑔 πœ† { | πœ‰ | β‰₯ 𝑖 1 / 𝑑 } and Theorem 3.2, the conclusion is valid.

Theorem 3.9. Let πœ‰ be a 𝑔 πœ† random variable and 𝑑 > 0 . If 𝐸 𝑔 πœ† ( | πœ‰ | 𝑑 ) < ∞ , then l i m π‘₯ β†’ ∞ π‘₯ 𝑑 𝑔 πœ† { | πœ‰ | β‰₯ π‘₯ } = 0 . Conversely, if there exists one positive number 𝑑 such that l i m π‘₯ β†’ ∞ π‘₯ 𝑑 𝑔 πœ† { | πœ‰ | β‰₯ π‘₯ } = 0 , then 𝐸 𝑔 πœ† ( | πœ‰ | 𝑠 ) < ∞ for any 𝑠 , where 0 ≀ 𝑠 < 𝑑 .

Proof. (1) When πœ† β‰₯ 0 , we have 𝐸 𝑔 πœ† ξ‚€ | | πœ‰ | | 𝑑  = ξ€œ 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ β‹… 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ ξ€œ ≀ π‘Ÿ 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ 𝑑 π‘Ÿ . ( 3 . 1 2 ) Since 𝐸 𝑔 πœ† ( | πœ‰ | 𝑑 ) < ∞ , we obtain ∫ 0 + ∞ 𝑔 πœ† { | πœ‰ | 𝑑 β‰₯ π‘Ÿ } 𝑑 π‘Ÿ < ∞ . Consequently, l i m π‘₯ β†’ ∞ ξ€œ ∞ π‘₯ 𝑑 / 2 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ 𝑑 π‘Ÿ = 0 . ( 3 . 1 3 ) Since ξ€œ ∞ π‘₯ 𝑑 / 2 𝑔 πœ†  | | πœ‰ | | 𝑑  ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ β‰₯ π‘₯ 𝑑 π‘₯ 𝑑 / 2 𝑔 πœ†  | | πœ‰ | | 𝑑  1 β‰₯ π‘Ÿ 𝑑 π‘Ÿ β‰₯ 2 π‘₯ 𝑑 𝑔 πœ† ξ€½ | | πœ‰ | | ξ€Ύ , β‰₯ π‘₯ ( 3 . 1 4 ) we have l i m π‘₯ β†’ ∞ π‘₯ 𝑑 𝑔 πœ† ξ€½ | | πœ‰ | | ξ€Ύ β‰₯ π‘₯ = 0 . ( 3 . 1 5 )
(2) When πœ† < 0 , we have
𝐸 𝑔 πœ† ξ‚€ | | πœ‰ | | 𝑑  = ξ€œ 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ β‹… 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ ξ€œ ≀ π‘Ÿ 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ = ( 1 + πœ† ) 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ 𝑑 π‘Ÿ . ( 3 . 1 6 ) Since 𝐸 𝑔 πœ† ξ‚€ | | πœ‰ | | 𝑑  < ∞ , ( 3 . 1 7 ) we obtain ξ€œ ( 1 + πœ† ) 0 + ∞ 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ 𝑑 π‘Ÿ < ∞ . ( 3 . 1 8 ) Consequently, l i m π‘₯ β†’ ∞ ξ€œ ( 1 + πœ† ) ∞ π‘₯ 𝑑 / 2 𝑔 πœ†  | | πœ‰ | | 𝑑  β‰₯ π‘Ÿ 𝑑 π‘Ÿ = 0 . ( 3 . 1 9 ) Since ξ€œ ( 1 + πœ† ) ∞ π‘₯ 𝑑 / 2 𝑔 πœ†  | | πœ‰ | | 𝑑  ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ β‰₯ ( 1 + πœ† ) π‘₯ 𝑑 π‘₯ 𝑑 / 2 𝑔 πœ†  | | πœ‰ | | 𝑑  1 β‰₯ π‘Ÿ 𝑑 π‘Ÿ β‰₯ 2 ( 1 + πœ† ) π‘₯ 𝑑 𝑔 πœ† ξ€½ | | πœ‰ | | ξ€Ύ , β‰₯ π‘₯ ( 3 . 2 0 ) we have l i m π‘₯ β†’ ∞ π‘₯ 𝑑 𝑔 πœ† ξ€½ | | πœ‰ | | ξ€Ύ β‰₯ π‘₯ = 0 . ( 3 . 2 1 ) Conversely, if l i m π‘₯ β†’ ∞ π‘₯ 𝑑 𝑔 πœ† { | πœ‰ | β‰₯ π‘₯ } = 0 , then there exists one number 𝑙 such that π‘₯ 𝑑 𝑔 πœ† { | πœ‰ | β‰₯ π‘₯ } ≀ 1 , for all π‘₯ β‰₯ 𝑙 .
(3) When πœ† β‰₯ 0 , for any 𝑠 , where 0 ≀ 𝑠 < 𝑑 , we have
𝐸 𝑔 πœ† ξ€· | | πœ‰ | | 𝑠 ξ€Έ = ξ€œ 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ β‰₯ π‘Ÿ β‹… 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ≀ ξ€œ ≀ π‘Ÿ 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ = ( 1 + πœ† ) 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ‚΅ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ = ( 1 + πœ† ) 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑙 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ‚Ά ξ‚΅ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ = ( 1 + πœ† ) 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑙 + ∞ 𝑠 π‘Ÿ 𝑠 βˆ’ 1 𝑔 πœ† ξ€½ | | πœ‰ | | ξ€Ύ ξ‚Ά ξ‚΅ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ ≀ ( 1 + πœ† ) 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑠 𝑙 + ∞ π‘Ÿ 𝑠 βˆ’ 𝑑 βˆ’ 1 ξ‚Ά ξ‚΅ ξ€œ 𝑑 π‘Ÿ ≀ ( 1 + πœ† ) 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑠 0 + ∞ π‘Ÿ 𝑠 βˆ’ 𝑑 βˆ’ 1 ξ‚Ά . 𝑑 π‘Ÿ ( 3 . 2 2 ) Since ∫ 0 + ∞ π‘Ÿ 𝑝 𝑑 π‘Ÿ < ∞ for any 𝑝 < βˆ’ 1 , we have 𝐸 𝑔 πœ† ξ€· | | πœ‰ | | 𝑠 ξ€Έ ξ‚΅ ξ€œ ≀ ( 1 + πœ† ) 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑠 0 + ∞ π‘Ÿ 𝑠 βˆ’ 𝑑 βˆ’ 1 ξ‚Ά 𝑑 π‘Ÿ < ∞ . ( 3 . 2 3 ) (4) When πœ† < 0 , for any 𝑠 , where 0 ≀ 𝑠 < 𝑑 , we have 𝐸 𝑔 πœ† ξ€· | | πœ‰ | | 𝑠 ξ€Έ = ξ€œ 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + πœ† 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ β‰₯ π‘Ÿ β‹… 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ≀ ξ€œ ≀ π‘Ÿ 𝑑 π‘Ÿ 0 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ = ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑙 + ∞ 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ = ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑙 + ∞ 𝑠 π‘Ÿ 𝑠 βˆ’ 1 𝑔 πœ† ξ€½ | | πœ‰ | | ξ€Ύ ≀ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑠 𝑙 + ∞ π‘Ÿ 𝑠 βˆ’ 𝑑 βˆ’ 1 ≀ ξ€œ 𝑑 π‘Ÿ 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑠 0 + ∞ π‘Ÿ 𝑠 βˆ’ 𝑑 βˆ’ 1 𝑑 π‘Ÿ . ( 3 . 2 4 ) Since ∫ 0 + ∞ π‘Ÿ 𝑝 𝑑 π‘Ÿ < ∞ for any 𝑝 < βˆ’ 1 , we have 𝐸 𝑔 πœ† ξ€· | | πœ‰ | | 𝑠 ξ€Έ ≀ ξ€œ 𝑙 0 𝑔 πœ† ξ€½ | | πœ‰ | | 𝑠 ξ€Ύ ξ€œ β‰₯ π‘Ÿ 𝑑 π‘Ÿ + 𝑠 0 + ∞ π‘Ÿ 𝑠 βˆ’ 𝑑 βˆ’ 1 𝑑 π‘Ÿ < ∞ . ( 3 . 2 5 )

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).

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