Springer - Publisher Connector

Gender differences in beliefs about health: a comparative qualitative study with Ghanaian and Indian migrants living in the United Kingdom

Im Grenzbereich des Räumlichen

The effect of unilateral disruption of the centrifugal visual system on normal eye development in chicks raised under constant light conditions

Co-occurrence of  and  pollen seasons against the background of the synoptic situations in Poland

The accumulation of vitrified oocytes is a strategy to increase the number of euploid available blastocysts for transfer after preimplantation genetic testing

Cristina Kraemer Zimpel

Juliana Sperotto Brum

Antônio Francisco de Souza Filho

Alexander Welker Biondo

João Henrique Perotta

Cristina Corsi Dib

Marcelo Bonat

José Soares Ferreira Neto

Paulo Eduardo Brandão

Marcos Bryan Heinemann

Ana Marcia Sa Guimaraes


            Mycobacterium bovis in a European bison (Bison bonasus) raises concerns about tuberculosis in Brazilian captive wildlife populations: a case report

Yong Zhang

Xue Ding

Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 
DOI 10.1186/s13660-016-1287-6
RESEARCH Open Access
Limit properties for ratios of order
statistics from exponentials
Yong Zhang* and Xue Ding
*Correspondence:
zyong2661@jlu.edu.cn
College of Mathematics, Jilin
University, Changchun, 130012,
P.R. China
Abstract
In this paper, we study the limit properties of the ratio for order statistics based on
samples from an exponential distribution and obtain the expression of the density
functions, the existence of the moments, the strong law of large numbers for Rnij with
1≤ i < j <mn =m. We also discuss other limit theorems such as the central limit
theorem, the law of iterated logarithm, the moderate deviation principle, the almost
sure central limit theorem for self-normalized sums of Rnij with 2≤ i < j <mn =m.
Keywords: exponential distribution; order statistics; strong law of large numbers;
central limit theorem; law of iterated logarithm
1 Introduction andmain results
Throughout this note, let {Xni, ≤ i≤mn} be a sequence of independent exponential ran-
dom variable with mean λn, let {Xn,n ≥ } =: {(Xni,  ≤ i ≤ mn),n ≥ } be an independent
random sequence, where {mn ≥ } denotes the sample size. Denote the order statistics be
Xn() ≤ Xn() ≤ · · · ≤ Xn(mn), and the ratios of those order statistics
Rnij =
Xn(j)
Xn(i)
, ≤ i < j ≤mn.
As we know, the exponential distribution can describe the lifetimes of the equipment, and
the ratios Rnij can measure the stability of equipment, it shows whether or not our system
is stable. Adler [] established the strong law of the ratio Rnj for j ≥  with ﬁxed sample
size mn =m, and the strong law of Rn formn → ∞ as follows.
Theorem A For ﬁxed sample size mn =m and all α > –, ≤ j ≤m, we know
lim
N→∞

(logN)α+
N∑
n=
(logn)α
n Rnj =
m!
(j – )!(m – j)!(α + )
j–∑
l=
Clj–
(–)j–l–
(m – l – ) a.s.
For mn → ∞ and all α > –,
lim
N→∞

(logN)α+
N∑
n=
(logn)α
n Rn =

α +  a.s.
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-
vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 2 of 8
Later on, Miao et al. [] proved the central limit theorem and the almost sure central
limit for Rn with ﬁxed sample size, we state their results as the following theorem.
Theorem B For ﬁxed sample size mn =m,

ηN
N∑
n=
(Rn – ERn)
D→N(, ) as N → ∞,
lim
N→∞

logN
N∑
n=

nI
{

ηN
N∑
n=
(Rn – ERn)≤ x
}
=(x) a.s.
for all X ∈ R, where (·) denotes the distribution function of N(, ), ηn =  ∨ sup{r >
;nL(r)≥ r}, L(r) = ERnI{|Rn| ≤ r}.
In this paper, we will make a further study on the limit properties of Rnij. In the next
section, ﬁrstly, we give the expression of the density functions of Rnij for all ≤ i < j <mn,
it is more interesting that the density function is free of the sample mean λn, this allows
us to change the equipment from sample to sample as long as the underlying distribution
remains an exponential. Alsowe discuss the existence of themoments for ﬁxed sample size
mn =m. Secondly, we establish the strong law of large number for Rnij with  = i < j <m
and ≤ i < j <m, respectively. At last we give some limit theorems such as the central limit
theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure
central limit theorem for self-normalized sums of Rnij with ≤ i < j <m.
In the following, C denotes a positive constant, which may take diﬀerent values when-
ever it appears in diﬀerent expressions. an ∼ bn means that an/bn →  as n→ ∞.
2 Main results and proofs
2.1 Density functions andmoments of Rnij
The ﬁrst theorem gives the expression of the density functions.
Theorem . For ≤ i < j ≤mn, the density function of the ratios Rnij is
fnij(r) =
mn!
(i – )!(j – i – )!(mn – j)!
i–∑
k=
j–i–∑
l=
(–)j–k–l–Cki–Clj–i–
[i – k + l + r(mn – i – l)]
I{r > }. (.)
Proof It is easy to check that the joint density function of Xn(i) and Xn(j) is
f (xi,xj) =
mn!
(i – )!(j – i – )!(mn – j)!

λn
[
 – e–xi/λn
]i–[e–xi/λn – e–xj/λn
]j–i–
· e–xi/λne–(mn–j+)xj/λn I{ < xi < xj}.
Let w = xi, r = xj/xi, then the Jacobian is w, so the joint density function of w and r is
f (w, r) = mn!(i – )!(j – i – )!(mn – j)!
w
λn
[
 – e–w/λn
]i–[e–w/λn – e–rw/λn
]j–i–
· e–w/λne–(mn–j+)rw/λn I{w > , r > }.
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 3 of 8
Therefore the density function of Rnij is
fnij(r) =
∫ ∞

f (w, r)dw
= mn!(i – )!(j – i – )!(mn – j)!

λn
i–∑
k=
j–i–∑
l=
(–)j–k–l–Cki–Clj–i–
·
∫ ∞

we–(i–k+l)w/λne–(mn–i–l)rw/λn dw
= mn!(i – )!(j – i – )!(mn – j)!
i–∑
k=
j–i–∑
l=
(–)j–k–l–Cki–Clj–i–
·
∫ ∞

te–[(i–k+l)+r(mn–i–l)]tdt
= mn!(i – )!(j – i – )!(mn – j)!
i–∑
k=
j–i–∑
l=
(–)j–k–l–
Cki–Clj–i–
[i – k + l + r(mn – i – l)]
. 
The next theorem treats the moments of Rnij with ﬁxed sample size mn =m.
Theorem . For ﬁxed sample size mn =m and  = i < j ≤m, we know
ERγnj =
{
<∞,  < γ < ,
=∞, γ ≥ ,
and with ≤ i < j ≤m,
ERγnij =
{
<∞,  < γ < ,
=∞, γ ≥ .
Let L(r) = E(Rnij – ERnij)I{|Rnij – ERnij| ≤ r},  ≤ i < j ≤ m, then L(r) is a slowly varying
function at ∞.
Proof For  = i < j ≤m, by (.), it is easy to check that
fnj(r) =
m!
(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–

[ + l + r(m – l – )]
∼ cm,jr as r → ∞,
where cm,j is a constant depend only on m and j. Obviously the γ -order moment is ﬁnite
for  < γ <  and is inﬁnite for γ ≥ .
For ≤ i < j ≤m, similarly we can obtain fnij(r)∼ dm,i,jr , where dm,i,j is a constant depend
only on m, i and j, so the γ -order moment is ﬁnite for  < γ <  and is inﬁnite for γ ≥ .
Furthermore it is not diﬃcult to verify that L(r) = ERnijI{|Rnij| ≤ r} varies slowly at ∞,
then by the fact that if L(x) = E|X|I{|X| ≤ x} is a slowly varying function at∞, then La(x) =
E|X – a|I{|X – a| ≤ x} also varies slowly at ∞ for any a ∈ R, the proof is completed. 
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 4 of 8
Remark . Miao et al. [] obtained the density function for Rnj for ﬁxed sample size
mn = m, they also proved that the expectation of Rnj is ﬁnite and the truncated second
moment is slowly varying at ∞. Adler [] also claimed that all the Rnj have inﬁnite expec-
tations for ﬁxed sample size, so our theorems extended their results.
2.2 Strong law of large numbers of Rnij
From our assumptions, we know that {Rnij,n ≥ } is an independent sequence with the
same distribution for ﬁxed sample sizemn =m. As Theorem . states that the Rnj do not
have the expectation, so the strong law of large numbers with them is not typical. Here
we give the weighted strong law of large number as follows. At ﬁrst, we list the following
lemma, that is, Theorem . from De la Peña et al. [], which will be used in the proof.
Lemma . Let {Xn,n ≥ } be a sequence of independent random variables, denote Sn =∑n
i=Xi, if bn ↗ ∞, and
∑∞
i= Var(Xi)/bi <∞, then (Sn – ESn)/bn →  a.s.
Theorem . Let {an,n ≥ } be a sequence of positive real numbers and {bn,n ≥ } be a
sequence of nondecreasing positive real numbers with limn→∞ bn =∞ and
∞∑
n=
an
bn
<∞, (.)
lim
N→∞

bN
N∑
n=
an log
(bn
an
)
= λ ∈ [,∞). (.)
Then, for the ﬁxed sample size mn =m and ≤ j ≤m, we have
lim
N→∞

bN
N∑
n=
anRnj =
λm!
(j – )!(m – j)!
j–∑
l=
Clj–
(–)j–l–
(m – l – ) a.s. (.)
For mn → ∞,
lim
N→∞

bN
N∑
n=
anRn = λ a.s. (.)
Proof By (.) we get cn = bn/an → ∞, so without loss of generality we assume that cn ≥ 
for any n≥ . Notice that

bN
N∑
n=
anRnj =

bN
N∑
n=
an
[
RnjI{≤ Rnj ≤ cn} – ERnjI{≤ Rnj ≤ cn}
]
+ bN
N∑
n=
anRnjI{Rnj > cn}
+ bN
N∑
n=
anERnjI{≤ Rnj ≤ cn}
= I + I + I. (.)
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 5 of 8
By (.) and (.), it is easy to show
∞∑
n=
Var
( 
cn
(
RnjI{≤ Rnj ≤ cn} – ERnjI{≤ Rnj ≤ cn}
))
≤
∞∑
n=

cn
ERnjI{≤ Rnj ≤ cn}
=
∞∑
n=
m!
cn(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–
∫ cn

r
[l +  + r(m – l – )] dr
≤ C
∞∑
n=

cn
j–∑
l=
∫ cn

dr ≤ C
∞∑
n=

cn
= C
∞∑
n=
an
bn
<∞,
then by Lemma ., we have
I →  a.s. n→ ∞. (.)
For any  < ε < ,
∞∑
n=
P
{
RnjI{Rnj > cn} > ε
}
=
∞∑
n=
P{Rnj > cn} =
∞∑
n=
m!
(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–
∫ ∞
cn

[l +  + r(m – l – )] dr
≤ C
∞∑
n=
j–∑
l=
∫ ∞
cn

r dr ≤ C
∞∑
n=

cn
= C
∞∑
n=
an
bn
<∞.
Then by the Borel-Cantelli lemma, we get
RnjI{Rnj > cn} →  a.s. n→ ∞. (.)
By (.) and (.), we can obtain
lim sup
N→∞

bN
N∑
n=
an ≤ λ. (.)
Therefore combining (.) with (.), we can easily conclude
I →  a.s. n→ ∞. (.)
For I, by (.) and noting cn → ∞, we get
ERnjI{≤ Rnj ≤ cn}
= m!(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–
∫ cn

r
[l +  + r(m – l – )] dr
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 6 of 8
= m!(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–

(m – l – )
∫ l++cn(m–l–)
m
[ 
y –
i + 
y
]
dy
= m!(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–

(m – l – )
·
[
log
l +  + cn(m – l – )
m – (i + )
( 
m –

l +  + cn(m – l – )
)]
∼ m!(j – )!(m – j)!
j–∑
l=
(–)j–l–Clj–

(m – l – ) log(cn);
then combining with (.), we show
I → λm!(j – )!(m – j)!
j–∑
l=
Clj–
(–)j–l–
(m – l – ) , n→ ∞. (.)
So the proof of (.) is completed by combining (.), (.), (.), and (.).
By the same argument as in the proof of (.), we can get (.), so we omit it here. 
Remark . If we take an = (logn)
α
n , bn = (logn)α+, α > –, it is easy to check that condi-
tions (.) and (.) hold with λ = 
α+ , so Theorems . and . and . from Adler [] are
special cases of our Theorem .. There are some other sequences satisfying conditions
(.) and (.), such as (a) an = , bn = nβ , β > , λ = ; (b) an = , bn = n(logn)γ , γ > , λ = ;
(c) an = , bn = n(logn)(log logn)δ , δ > , λ = ; (d) an = (log logn)
θ
n , bn = (logn)(log logn)θ ,
θ ∈ R, λ =  , so the conditions (.) and (.) are mild conditions. At the end of this re-
mark, we point out that only when an = L(n)/n, where L(n) is a slowly varying function,
the limit value λ will be λ > , this is known as an exact strong law, one can refer to Adler
[] for more details. For the weak law, i.e., convergence in probability, one can see Feller
[] for full details.
For Rnij, i≥ , since the expectation is ﬁnite, by the classical strong law of large numbers,
we have the following.
Theorem . For ﬁxed mn =m, we have for ≤ i < j ≤m,
lim
N→∞

N
N∑
n=
(Rnij – ERnij) =  a.s. (.)
2.3 Other limit properties for Rnij, 2≤ i < j ≤m
By the above discussion, we know that, for ﬁxed sample size mn = m and  ≤ i < j ≤ m,
{Rnij,n≥ } is a sequence of independent and identically distributed randomvariableswith
ﬁnite mean, and L(r) = E(Rnij –ERnij)I{|Rnij –ERnij| ≤ r} is a slowly varying function at∞.
Therefore the limit properties ofRnij for ﬁxed sample size can easily be established by those
of the self-normalized sums.We list someof them, such as the central limit theorem (CLT),
the law of iterated logarithm (LIL), the moderate deviation principle (MDP), the almost
sure central limit theorem (ASCLT). Denote SN =
∑N
n=(Rnij – ERnij), V N =
∑N
n=(Rnij –
ERnij).
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 7 of 8
Theorem . (CLT) For ﬁxed sample size mn =m, we know
SN
VN
D→N(, ). (.)
Proof By Theorem . from Giné et al. [], we can obtain the CLT for Rnij. 
Theorem . (LIL) For ﬁxed sample size mn =m, we get
lim sup
N→∞
SN
VN
√ log logN =  a.s. (.)
Proof By Theorem  from Griﬃn and Kuelbs [], the LIL for Rnij holds. 
Theorem . (MDP) Let {xn,n≥ } be a sequence of positive numbers with xn → ∞ and
xn = o(
√n), as n→ ∞, then, for ﬁxed sample size mn =m, we conclude
lim
N→∞

xN
P
{ SN
VN
≥ xN
}
= –  . (.)
Proof By Theorem . from Shao [], we can prove the MDP for Rnij. 
Theorem . (ASCLT) Suppose that  ≤ α < / and set dk = exp{(logk)α}/k and Dn =∑n
k= dk . Then, for ﬁxed sample size mn =m and any x ∈ R,
lim
k→∞

Dk
k∑
N=
dNI
{ SN
VN
≤ x
}
=(x) a.s., (.)
where (·) is the distribution function of the standard normal random variable.
Proof By Corollary  from Zhang [], we know ASCLT for Rnij holds. 
Remark . It is easy to check that ηN /VN
p→ , then by the Slutsky lemma and Theo-
rem ., we can get Theorem . fromMiao et al. [].
Competing interests
The authors declares that they have no competing interests.
Authors’ contributions
Both authors contributed equally and signiﬁcantly in writing this article. Both authors read and approved the ﬁnal
manuscript.
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11101180, 11201175); the Science
and Technology Development Program of Jilin Province (Grant Nos. 20130522096JH, 20140520056JH).
Received: 4 July 2016 Accepted: 23 December 2016
References
1. Adler, A: Strong laws for ratios of order statistics from exponentials. Bull. Inst. Math. Acad. Sin. (N.S.) 10(1), 101-111
(2015)
2. Miao, Y, Wang, RJ, Adler, A: Limit theorems for order statistics from exponentials. Stat. Probab. Lett. 110, 51-57 (2016)
3. De la Peña, VH, Tai, TL, Shao, QM: Self-Normalized Process: Limit Theory and Statistical Applications. Springer, New York
(2009)
4. Adler, A: Exact strong laws. Bull. Inst. Math. Acad. Sin. 28(3), 141-166 (2000)
Zhang and Ding Journal of Inequalities and Applications  (2017) 2017:11 Page 8 of 8
5. Feller, W: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)
6. Giné, E, Götze, F, Mason, DM: When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25,
1514-1531 (1997)
7. Griﬃn, PS, Kuelbs, JD: Self-normalized laws of the iterated logarithm. Ann. Probab. 17, 1571-1601 (1989)
8. Shao, QM: Self-normalized large deviations. Ann. Probab. 25, 285-328 (1997)
9. Zhang, Y: A general result on almost sure central limit theorem for self-normalized sums for mixing sequences. Lith.
Math. J. 53(4), 471-483 (2013)


Limit properties for ratios of order statistics from exponentials

Yong Guo

Stephen D. Weigand

Bogdan F. Popescu

Vanda A. Lennon

Joseph E. Parisi

Sean J. Pittock

Natalie E. Parks

Stacey L. Clardy

Charles L. Howe

Claudia F. Lucchinetti

Pathogenic implications of cerebrospinal fluid barrier pathology in neuromyelitis optica

Who uses height-adjustable desks? - Sociodemographic, health-related, and psycho-social variables of regular users

Patients with preexisting psychiatric disorders admitted to ICU: a descriptive and retrospective cohort study

Not a member yet