R E S E A R C H
Open Access
Some probability inequalities for a class of
random variables and their applications
Aiting Shen
*and Ranchao Wu
*Correspondence:
[email protected] School of Mathematical Science, Anhui University, Hefei, 230601, China
Abstract
Some probability inequalities for a class of random variables are presented. As applications, we study the complete convergence for it. Our main results generalize the corresponding ones for negatively associated random variables and negatively orthant dependent random variables.
MSC: 60E15; 60F15
Keywords: acceptable random variables; probability inequality; complete convergence
1 Introduction
Let{Xn,n≥}be a sequence of random variables defined on a fixed probability space
(,F,P). The exponential inequality for the partial sumsni=(Xi–EXi) plays an
impor-tant role in various proofs of limit theorems. In particular, it provides a measure of con-vergence rate for the strong law of large numbers. The main purpose of the paper is to present some probability inequalities for a class of random variables. As applications, we will give some complete convergence for a class of random variables.
Firstly, we will recall the definitions of negatively orthant dependent random variables and acceptable random variables.
Definition . A finite collection of random variablesX,X, . . . ,Xnis said to be negatively
orthant dependent (NOD, in short) if the following two inequalities:
P(X>x,X>x, . . . ,Xn>xn)≤ n
i=
P(Xi>xi) (.)
and
P(X≤x,X≤x, . . . ,Xn≤xn)≤ n
i=
P(Xi≤xi) (.)
hold for all real numbersx,x, . . . ,xn. An infinite sequence{Xn,n≥}is said to be NOD
if every finite subcollection is NOD.
The notion of NOD random variables was introduced by Lehmann [] and developed in Joag-Dev and Proschan []. Obviously, independent random variables are NOD. Joag-Dev
and Proschan [] pointed out that negatively associated (NA, in short) random variables are NOD, but neither NUOD nor NLOD implies NA. They also presented an example in whichX= (X,X,X,X) possesses NOD, but does not possess NA. So, we can see that
NOD is weaker than NA.
Recently, Giuliano Antoniniet al.[] introduced the following notion of acceptability.
Definition . We say that a finite collection of random variablesX,X, . . . ,Xnis
accept-able if for any realλ,
Eexp
λ
n
i=
Xi
≤ n
i=
Eexp(λXi). (.)
An infinite sequence of random variables{Xn,n≥}is acceptable if every finite
subcol-lection is acceptable.
As is mentioned in Giuliano Antoniniet al.[], a sequence of NOD random variables with a finite Laplace transform or finite moment generating function near zero (and hence a sequence of negatively associated random variables with finite Laplace transform, too) provides us an example of acceptable random variables. For example, Xinget al.[] con-sider a strictly stationary negatively associated sequence of random variables. According to the sentence above, the sequence of strictly stationary and negatively associated ran-dom variables is acceptable. Hence, the model of acceptable ranran-dom variables is more general than models considered in the previous literature. Studying the limiting behavior of acceptable random variables is of interest.
The main purpose of the paper is to present some exponential probability inequalities for a sequence of acceptable random variables and give some applications by using these exponential probability inequalities. For more details about the exponential probability inequality, one can refer to Wanget al.[–], Sung [], Sunget al.[] and Xinget al.[, ], and so forth.
The paper is organized as follows. The exponential probability inequalities for a se-quence of acceptable random variables are presented in Section , and the complete con-vergence for it is obtained in Section . Our results are based on some moment conditions, while the main results of Sunget al.[] are under the condition of moment and identical distribution.
Throughout the paper, let{Xn,n≥}be a sequence of acceptable random variables and
denoteSn=
n
i=Xifor eachn≥.
2 Probability inequalities for acceptable random variables
Theorem . Let{Xn,n≥}be a sequence of acceptable random variables and{gn,n≥}
be a sequence of positive numbers with Gn=
n
i=gifor each n≥.For fixed n≥,if there
exists a positive number T such that
EetXk≤egkt, ≤t≤T,k= , , . . . ,n, (.)
then
P(Sn≥x)≤ ⎧ ⎨ ⎩
e– x
Gn, ≤x≤GnT,
e–Tx, x≥GnT.
Proof For eachx, by Markov’s inequality, Definition . and (.), we can see that
P(Sn≥x)≤e–txEetSn≤e–tx n
i=
EetXi≤eGnt
–tx, ≤t≤T, (.)
which implies that
P(Sn≥x)≤ inf
<t≤Te Gnt
–tx=einf<t≤T(Gnt
–tx). (.)
For fixedx≥, ifT≥Gx
n ≥, then
einf<t≤T(Gnt–tx)=e– x
Gn; (.)
ifT≤Gx
n, then
einf<t≤T(Gnt–tx)=eGnT–Tx≤e–Tx . (.)
The desired result (.) follows from (.)-(.) immediately.
Corollary . Let{Xn,n≥}be a sequence of acceptable random variables and{gn,n≥}
be a sequence of positive numbers with Gn=
n
i=gifor each n≥.For fixed n≥,if there
exists a positive number T such that
EetXk≤egkt, |t| ≤T, k= , , . . . ,n, (.)
then
P(Sn≤–x)≤ ⎧ ⎨ ⎩
e– x
Gn, ≤x≤GnT,
e–Tx , x≥GnT,
(.)
and
P|Sn| ≥x≤
⎧ ⎨ ⎩
e– x
Gn, ≤x≤GnT,
e–Tx, x≥G
nT.
(.)
Proof It is easily seen that{–Xn,n≥}is still a sequence of acceptable random variables.
By Theorem ., we can see that
P(–Sn≥x)≤ ⎧ ⎨ ⎩
e– x
Gn, ≤x≤GnT,
e–Tx , x≥GnT,
(.)
which implies that (.) is valid. Finally, (.) follows (.) and (.) immediately.
Corollary . Let{Xn,n≥}be a sequence of acceptable random variables with EXk=
and EXk=σk<∞for each k≥.Denote B
n=
n
there exists a positive number H such that
EXkm≤m!
σ
kHm–, k= , , . . . ,n (.)
for any positive integer m≥,then
P|Sn| ≥x≤
⎧ ⎨ ⎩e
–x
Bn, ≤x≤B n H,
e–Hx , x≥B n H.
(.)
Proof By (.), we can see that
EetXk= +t
σ
k +
t
EX
k+· · · ≤ +
t
σ
k
+H|t|+Ht+· · ·
fork= , , . . . ,n. When|t| ≤H, it follows that
EetXk≤ +t σ
k
·
–H|t|≤ +t
σ
k ≤et σ
k =. egkt, k= , , . . . ,n, (.)
where gk= σk. TakeGn=.
n
k=gk= BnandT =H. Hence, the conditions of
Corol-lary . are satisfied. Therefore, (.) follows from CorolCorol-lary . immediately.
3 Complete convergence for acceptable random variables
In this section, we will present some complete convergence for a sequence of acceptable random variables by using the probability inequality.
Theorem . Let{Xn,n≥}be a sequence of acceptable random variables with EXk=
and EXk=. σk<∞for each k≥.Denote B
n=
n
i=σi,n≥.For fixed n≥,suppose that
there exists a positive number H such that(.)holds true.If for anyε> , ∞
n=
exp
–b
nε
B
n
<∞, (.)
and
∞
n=
exp
–bnε H
<∞, (.)
where{bn,n≥}is a sequence of positive numbers, then bn
n
i=Xi→completely as
n→ ∞.
Proof By Corollary ., we have for anyx≥,
P
n
i=
Xi
≥x
≤exp
– x
B
n
+ exp
– x H
which implies that ∞ n= P bn n i= Xi ≥ε ≤ ∞ n= exp –b
nε
B
n + ∞ n= exp
–bnε H
<∞.
This completes the proof of the theorem.
It is easily seen that (.) holds ifbn=n. So, we have the following corollary.
Corollary . Let{Xn,n≥}be a sequence of acceptable random variables with EXi=
and EXi=. σi<∞for each i≥.Denote Bn=ni=σi,n≥.Suppose that conditions(.)
and(.)hold with bn=n.Then n
n
i=Xi→completely as n→ ∞.
Theorem . Let{Xn,n≥}be a sequence of acceptable random variables with EXi=
and EXi=. σi<∞for each i≥.Denote B
n=
n
i=σiand
cn=max
ess sup|Xi|
B
n
, ≤i≤n
, n≥.
If for anyε> , ∞
n=
exp
– nε
c
nBn
<∞, (.)
thennni=Xi→completely as n→ ∞.
Proof By Markov’s inequality and Definition ., for anyε> andt> ,
P n i= Xi ≥ε =P n i=
Xi≥ε
+P
n
i=
(–Xi)≥ε
≤e
–√tε BnEexp
t B n n i= Xi +e
–√tε BnEexp
–√t
Bn n i= Xi ≤e
–√tε Bn
n i= Ee tXi √
Bn + n i= Ee –tXi √
Bn
≤exp
–tε
B
n
+nt
c
n
.
Takingt= ε
nc n
√ B
n
in the inequality above, we can get that
P n i= Xi ≥ε ≤exp
– ε
nc
nBn
.
It follows from the inequality above and (.) that
∞ n= P n n i= Xi ≥ε ≤ ∞ n= exp
– nε
c
nBn
<∞,
Hanson and Wright () obtained a bound on tail probabilities for quadratic forms in independent random variables using the following condition: for alln≥ and allx≥, there exist positive constantsMandγ such that
P|Xn| ≥x≤M
+∞
x
e–γtdt. (.)
Wright [] proved that the bound established by Hanson and Wright [] for indepen-dent symmetric random variables also holds when the random variables are not symmet-ric but condition (.) is valid. We will study the complete convergence for a sequence of acceptable random variables under condition (.). The main result is as follows.
Theorem . Let{Xn,n≥}be a sequence of acceptable random variables satisfying
con-dition(.)for all n≥and all x≥,where M andγ are positive constants.Suppose that there exists a positive constant C not depending on n such that
E
n
i=
Xi
≤C
n
i=
EXi. (.)
Then for allβ> ,nβ
n
i=Xi→completely as n→ ∞.
Proof By Markov’s inequality and assumption (.), we have that for anyε> , ∞
n=
P
nβ n
i=
Xi >ε
≤
∞
n=
nβεE
n
i=
Xi
≤C
∞
n=
nβε
n
i=
EXi.
In the following, we will estimateEXi. By (.), we can see that
EXi=
+∞
xP|Xi| ≥xdx≤
+∞
x
M
+∞
x
e–γtdt
dx
=M
+∞
e–γt t
x dx
dt=M
+∞
te–γtdt=M
√
π γ/.
Hence,
∞
n=
P
nβ n
i=
Xi >ε
≤CM
√
π γ/ε
∞
n=
nβ–<∞.
This completes the proof of the theorem.
Acknowledgements
The authors are most grateful to the editor and the anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001, 11126176 and 11226207), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, 1208085QA03), the 211 project of Anhui University and the Students Science Research Training Program of Anhui University (KYXL2012007).
References
1. Lehmann, E: Some concepts of dependence. Ann. Math. Stat.37, 1137-1153 (1996)
2. Joag-Dev, K, Proschan, F: Negative association of random variables with applications. Ann. Stat.11(1), 286-295 (1983) 3. Giuliano, AR, Kozachenko, Y, Volodin, A: Convergence of series of dependentϕ-sub-Gaussian random variables. J.
Math. Anal. Appl.338, 1188-1203 (2008)
4. Xing, GD, Yang, SC, Liu, AL, Wang, X: A remark on the exponential inequality for negatively associated random variables. J. Korean Stat. Soc.38, 53-57 (2009)
5. Wang, XJ, Hu, SH, Yang, WZ, Li, XQ: Exponential inequalities and complete convergence for a LNQD sequence. J. Korean Stat. Soc.39(4), 555-564 (2010)
6. Wang, XJ, Hu, SH, Yang, WZ, Ling, NX: Exponential inequalities and inverse moment for NOD sequence. Stat. Probab. Lett.80(5-6), 452-461 (2010)
7. Wang, XJ, Hu, SH, Shen, AT, Yang, WZ: An exponential inequality for a NOD sequence and a strong law of large numbers. Appl. Math. Lett.24, 219-223 (2011)
8. Sung, SH, Srisuradetchai, P, Volodin, A: A note on the exponential inequality for a class of dependent random variables. J. Korean Stat. Soc.40, 109-144 (2011)
9. Sung, SH: On the exponential inequalities for negatively dependent random variables. J. Math. Anal. Appl.381, 538-545 (2011)
10. Xing, GD, Yang, SC, Liu, AL: Exponential inequalities for positively associated random variables and applications. J. Inequal. Appl.2008, Article ID 385362 (2008)
11. Wright, FT: A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab.1(6), 1068-1070 (1973)
12. Hanson, DL, Wright, FT: A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat.42, 1079-1083 (1971)
doi:10.1186/1029-242X-2013-57
