Open Access

On Inverse Moments for a Class of Nonnegative Random Variables

Journal of Inequalities and Applications20102010:823767

DOI: 10.1155/2010/823767

Received: 1 April 2010

Accepted: 20 May 2010

Published: 15 June 2010

Abstract

Using exponential inequalities, Wu et al. (2009) and Wang et al. (2010) obtained asymptotic approximations of inverse moments for nonnegative independent random variables and nonnegative negatively orthant dependent random variables, respectively. In this paper, we improve and extend their results to nonnegative random variables satisfying a Rosenthal-type inequality.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq1_HTML.gif be a sequence of nonnegative random variables with finite second moments. Let us denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ1_HTML.gif
(1.1)
We will establish that, under suitable conditions, the inverse moment can be approximated by the inverse of the moment. More precisely, we will prove that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq3_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq4_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq5_HTML.gif The left-hand side of (1.2) is the inverse moment and the right-hand side is the inverse of the moment. Generally, it is not easy to compute the inverse moment, but it is much easier to compute the inverse of the moment.

The inverse moments can be applied in many practical applications. For example, they appear in Stein estimation and Bayesian poststratification (see Wooff [1] and Pittenger [2]), evaluating risks of estimators and powers of test statistics (see Marciniak and Weso https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq6_HTML.gif owski [3] and Fujioka [4]), expected relaxation times of complex systems (see Jurlewicz and Weron [5]), and insurance and financial mathematics (see Ramsay [6]).

For nonnegative asymptotically normal random variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq7_HTML.gif , (1.2) was established in Theorem 2.1 of Garcia and Palacios [7]. Unfortunately, that theorem is not true under the suggested assumptions, as pointed out by Kaluszka and Okolewski [8]. Kaluszka and Okolewski [8] also proved (1.2) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq8_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq9_HTML.gif in the i.i.d. case) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq10_HTML.gif is a sequence of nonnegative independent random variables satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq12_HTML.gif (Lyapunov's condition of order 3), that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq13_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq14_HTML.gif Hu et al. [9] generalized the result of Kaluszka and Okolewski [8] by considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq15_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq16_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq17_HTML.gif

Recently, Wu et al. [10] obtained the following result by using the truncation method and Bernstein's inequality.

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq18_HTML.gif be a sequence of nonnegative independent random variables such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq20_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq21_HTML.gif is defined by (1.1). Furthermore, assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ4_HTML.gif
(1.4)

Then (1.2) holds for all real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq23_HTML.gif

For a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq24_HTML.gif of nonnegative independent random variables with only https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq25_HTML.gif th moments for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq26_HTML.gif Wu et al. [10] also obtained the following asymptotic approximation of the inverse moment:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ5_HTML.gif
(1.5)
for all real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq28_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq29_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ6_HTML.gif
(1.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq30_HTML.gif is a sequence of positive constants satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ7_HTML.gif
(1.7)

Specifically, Wu et al. [10] proved the following result.

Theorem 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq31_HTML.gif be a sequence of nonnegative independent random variables. Suppose that, for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq32_HTML.gif ,

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq33_HTML.gif is uniformly integrable,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq34_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq35_HTML.gif for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq36_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq37_HTML.gif for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq38_HTML.gif

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq39_HTML.gif is the same as in (1.6) for some positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq40_HTML.gif satisfying (1.7). Then (1.5) holds for all real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq42_HTML.gif .

Wang et al. [11] obtained some exponential inequalities for negatively orthant dependent (NOD) random variables. By using the exponential inequalities, they extended Theorem 1.1 for independent random variables to NOD random variables without condition (1.3).

The purpose of this work is to obtain asymptotic approximations of inverse moments for nonnegative random variables satisfying a Rosenthal-type inequality. For a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq43_HTML.gif of independent random variables with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq45_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq46_HTML.gif Rosenthal [12] proved that there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq47_HTML.gif depending only on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq48_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ8_HTML.gif
(1.8)

Note that the Rosenthal inequality holds for NOD random variables (see Asadian et al. [13]).

In this paper, we improve and extend Theorem 1.2 for independent random variables to random variables satisfying a Rosenthal type inequality. We also extend Wang et al. [11] result for NOD random variables to the more general case.

Throughout this paper, the symbol https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq49_HTML.gif denotes a positive constant which is not necessarily the same one in each appearance, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq50_HTML.gif denotes the indicator function of the event https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq51_HTML.gif

2. Main Results

Throughout this section, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq52_HTML.gif is a sequence of nonnegative random variables satisfying a Rosenthal type inequality (see (2.1)).

The following theorem gives sufficient conditions under which the inverse moment is asymptotically approximated by the inverse of the moment.

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq53_HTML.gif be a sequence of nonnegative random variables. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq55_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq57_HTML.gif are defined by (1.6), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq58_HTML.gif is a sequence of positive real numbers. Suppose that the following conditions hold:

(i)for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq59_HTML.gif there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq60_HTML.gif depending only on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq61_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ9_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq62_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq63_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq64_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq65_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq66_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq67_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq68_HTML.gif

Then (1.5) holds for all real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq70_HTML.gif .

Proof.

Let us decompose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq71_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ10_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq73_HTML.gif Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq74_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq75_HTML.gif we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq76_HTML.gif It follows by (ii) and (iii) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ11_HTML.gif
(2.3)
Now, applying Jensen's inequality to the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq77_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq78_HTML.gif Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ12_HTML.gif
(2.4)
Hence it is enough to show that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ13_HTML.gif
(2.5)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq79_HTML.gif we can take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq80_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq82_HTML.gif Namely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq83_HTML.gif Let us write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ14_HTML.gif
(2.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq85_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq86_HTML.gif we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ15_HTML.gif
(2.7)
which implies by (ii) and (2.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ16_HTML.gif
(2.8)
It remains to show that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ17_HTML.gif
(2.9)
Observe by Markov's inequality and (i) that, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq87_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ18_HTML.gif
(2.10)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq88_HTML.gif we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ19_HTML.gif
(2.11)
Substituting (2.11) into (2.10), we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ20_HTML.gif
(2.12)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq89_HTML.gif we have by (iv) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ21_HTML.gif
(2.13)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq90_HTML.gif we first note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ22_HTML.gif
(2.14)
which entails by (iii) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ23_HTML.gif
(2.15)
It follows by (iv) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ24_HTML.gif
(2.16)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq91_HTML.gif we have by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq92_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ25_HTML.gif
(2.17)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq93_HTML.gif we have by (2.15) and (iv) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ26_HTML.gif
(2.18)
Substituting (2.13) and (2.16)–(2.18) into (2.12), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ27_HTML.gif
(2.19)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq94_HTML.gif we can take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq95_HTML.gif large enough such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq96_HTML.gif Then we have by (2.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ28_HTML.gif
(2.20)
Hence all the terms in the second brace of (2.19) converge to 0 as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq97_HTML.gif Moreover, we have by (ii) and (iii) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ29_HTML.gif
(2.21)

Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq98_HTML.gif and so (2.9) is proved.

Remark 2.2.

In (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq99_HTML.gif are monotone transformations of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq100_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq101_HTML.gif is a sequence of independent random variables, then (2.1) is clearly satisfied from the Rosenthal inequality (1.8). There are many sequences of dependent random variables satisfying (2.1) for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq102_HTML.gif Examples include sequences of NOD random variables (see Asadian et al. [13]), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq103_HTML.gif -mixing identically distributed random variables satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq104_HTML.gif (see Shao [14]), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq105_HTML.gif -mixing identically distributed random variables satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq106_HTML.gif (see Shao [15]), negatively associated random variables (see Shao [16]), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq107_HTML.gif -mixing random variables (see Utev and Peligrad [17]).

We can extend Theorem 1.1 for independent random variables to the more general random variables by using Theorem 2.1. To do this, the following lemma is needed.

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq108_HTML.gif be a sequence of nonnegative random variables with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq109_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq110_HTML.gif be a sequence of positive real numbers satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq111_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq112_HTML.gif Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ30_HTML.gif
(2.22)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq113_HTML.gif

Proof.

Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq114_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq115_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq116_HTML.gif there exists a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq117_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq118_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq119_HTML.gif We have by (2.22) that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq120_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ31_HTML.gif
(2.23)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ32_HTML.gif
(2.24)
Similar to the above case, we get that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq121_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ33_HTML.gif
(2.25)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ34_HTML.gif
(2.26)

Hence the result is proved by (2.24) and (2.26).

By using Theorem 2.1, we can obtain the following theorem which improves and extends Theorem 1.1 for independent random variables to the more general random variables satisfying the Rosenthal-type inequality (2.1).

Theorem 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq122_HTML.gif be a sequence of nonnegative random variables with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq123_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq125_HTML.gif be defined by (1.1). Assume that the Rosenthal-type inequality (2.1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq126_HTML.gif holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq127_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq128_HTML.gif is the same as in (ii). Furthermore, assume that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq129_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq130_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq131_HTML.gif

Then (1.2) holds for all real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq133_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq135_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq137_HTML.gif are defined by (1.6). Note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ35_HTML.gif
(2.27)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ36_HTML.gif
(2.28)
But, we have by (ii) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ37_HTML.gif
(2.29)
Substituting (2.29) into (2.28), we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ38_HTML.gif
(2.30)
Now we will apply Theorem 2.1 to the random variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq138_HTML.gif By (2.30) and (i), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ39_HTML.gif
(2.31)
We also get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ40_HTML.gif
(2.32)
since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq139_HTML.gif by (i) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq140_HTML.gif by (ii). From (2.31) and (2.32), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq141_HTML.gif and so we have by (2.30) that, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq142_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ41_HTML.gif
(2.33)
Hence all conditions of Theorem 2.1 are satisfied. By Theorem 2.1,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ42_HTML.gif
(2.34)

Note that the norming constants in (2.34) are different from those in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq143_HTML.gif

To complete the proof, we will use Lemma 2.3. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq144_HTML.gif we have by Lemma 2.3 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ43_HTML.gif
(2.35)
Namely,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ44_HTML.gif
(2.36)
By (i) and (2.30),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ45_HTML.gif
(2.37)

Combining (2.36) with (2.37) gives the desired result.

Remark 2.5.

Wang et al. [11] extended Wu et al. [10] result (see Theorem 1.1) to NOD random variables without condition (1.3). As observed in Remark 2.2, (2.1) holds for not only independent random variables but also NOD random variables. Hence Theorem 2.4 improves and extends the results of Wu et al. [10] and Wang et al. [11] to the more general random variables.

Theorem 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq145_HTML.gif be a sequence of nonnegative random variables. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq147_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq149_HTML.gif are defined by (1.6), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq150_HTML.gif is a sequence of positive real numbers satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ46_HTML.gif
(2.38)

Assume that the Rosenthal-type inequality (2.1) holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq151_HTML.gif Furthermore, assume that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq152_HTML.gif is uniformly integrable,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq153_HTML.gif for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq154_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq155_HTML.gif for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq156_HTML.gif

Then (1.5) holds for all real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq158_HTML.gif .

Proof.

We first note by (i) and (ii) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ47_HTML.gif
(2.39)
We next estimate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq159_HTML.gif By (2.39),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ48_HTML.gif
(2.40)
Combining (2.40) with (iii) gives
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ49_HTML.gif
(2.41)
Now we will apply Theorem 2.1 to the random variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq160_HTML.gif By (ii), (2.41), and (2.38), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ50_HTML.gif
(2.42)
We also get by (ii) and (2.39) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ51_HTML.gif
(2.43)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq161_HTML.gif we can take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq162_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq163_HTML.gif Then we have by (ii), (iii), (2.38), and (2.43) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_Equ52_HTML.gif
(2.44)

since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq165_HTML.gif Hence all conditions of Theorem 2.1 are satisfied. The result follows from Theorem 2.1.

Remark 2.7.

The conditions of Theorem 2.6 are much weaker than those of Theorem 1.2 in the following three directions.

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq166_HTML.gif is a sequence of independent random variables, then (2.1) is satisfied from the Rosenthal inequality. Hence (2.1) is weaker than independence condition.

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq167_HTML.gif satisfies (1.7), then it also satisfies (2.38) by the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq168_HTML.gif Hence (2.38) is weaker than (1.7).

(iii)The condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F823767/MediaObjects/13660_2010_Article_2264_IEq169_HTML.gif in Theorem 1.2 is not needed in Theorem 2.6. Therefore Theorem 2.6 improves and extends Wu et al. [10] result (see Theorem 1.2) to the more general random variables.

Declarations

Acknowledgments

The author is grateful to the editor Andrei I. Volodin and the referees for the helpful comments. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pai Chai University

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Copyright

© Soo Hak Sung. 2010

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.