Qianmin Chen and Jicheng Liu
r in
THE CONDITIONAL BOREL-CANTELLI LEMMA
AND APPLICATIONS
t
J. Korean Math. Soc. 0 (0), No. 0, pp. 1–0
http://dx.doi.org/10.4134/JKMS.j160036
pISSN: 0304-9914 / eISSN: 2234-3008
fP
Abstract. In this paper, we establish some conditional versions of the
first part of the Borel-Cantelli lemma. As its applications, we study strong
limit results of F -independent random variables sequences, the convergence of sums of F -independent random variables and the conditional
version of strong limit results of the concomitants of order statistics.
1. Introduction
Let (Ω, A , P ) be a probability space and F be a sub-σ-algebra of A . A
sequence of events {An , n ≥ 1} is said to be conditionally independent given
F if
!
n
n
\
Y
(1.1)
P
Ai F =
P (Ai |F ) a.s..
i=1
Ah
ea
do
i=1
A sequence of dependent random variables {Xn , n ≥ 1} is said to be conditionally independent given F if
!
n
n
Y
\
(1.2)
P (Xi ≤ xi |F ) a.s..
P
(Xi ≤ xi )F =
i=1
i=1
If F = {∅, Ω}, the conditional independence reduces to the ordinary (unconditional) independence. The ordinary independence does not imply conditional
independence and the opposite implication is also not true.
In applied sciences, if you want to obtain a suitable model to analyze and
study a real problem, conditioning is often used. For example, the martingales
and Markov processes are well-known stochastic processes defined through conditional expectation. In particular, the past and the future of Markov processes
are conditionally independent given the present. A more extensive enumeration of models such as statistical inference and engineering literature is given
by Roussas [19] in which conditioning plays an essential role.
Received January 17, 2016; Revised August 23, 2016.
2010 Mathematics Subject Classification. 60F15.
Key words and phrases. conditional independence, conditional Borel-Cantelli lemma, Order statistics, concomitant.
c
2017
Korean Mathematical Society
1
2
Q. CHEN AND J. LIU
Ah
ea
do
fP
r in
t
The statistical perspective of conditional independence is that of a Bayesian.
A problem begins with a parameter θ with its prior probability distribution
that exists only in mind of the investigator. The statistical model that is most
commonly in use is that of a sequence {Xn , n > 1} of observable random
variables that is independent and identically distributed for each given value of
θ. A concrete example where conditional limit theorems are useful is the study
of statistical inference for non-ergodic models as discussed in [4]. For instance,
if one wants to estimate the mean off-spring for a Galton-Watson branching
process, the asymptotic properties of the maximum likelihood estimator depend
on the set of non-extinction.
In the past decade, a number of authors have obtained rich results of conditional dependence and relative limit results. For instance, Majerek et al. [13]
for conditional strong law of large number, Yuan et al. [23, 22] for a conditional version of the classical central limit theorem, a conditional version of
the extended Kolmogorov-Feller weak law of large numbers, Christofides and
Hadjikyriakou [7] for conditional demimartingale, Liu and Prakasa Rao [11]
for conditional Borel-Cantelli lemma, Ordonez Cabrera et al. [15] for conditionally negatively quadrant dependence, Wang and Wang [21] for conditional
demimartingale and conditional N-demimartingale, Yuan and Lei (2013) for
conditionally strong mixing.
The Borel-Cantelli lemma has been found to be extremely useful for the
derivations of many theorems of probability. A classical form of Borel-Cantelli
Lemma can be stated as follows. Let {An } be a sequence of events on a
probability
P∞space (Ω, F , P ). Then
(a) if n=1 P (An ) < ∞, then P (lim sup An ) = 0;
(b) if {An } are independent and
(1.3)
∞
X
P (An ) = ∞,
n=1
∞
then P (lim sup An ) = 1, where lim sup An = ∩∞
n=1 ∪k=n Ak .
In many applications, the assumption of independence in (b) fails to hold and
needs to be replaced by more relaxed assumptions. There are many attempts
to weaken the independent condition in the second part of the Borel-Cantelli
lemma [6, 9, 10, 14, 16, 17]. Under condition (1.3), the probability of event
lim sup An can be any value between 0 and 1. The first part of the BorelCantelli lemma was generalized in Barndorff-Nielsen
(1961) as follows:
P∞
(a0 ) If lim inf n→∞ P (An ) = 0 and n=1 P (An Acn+1 ) < ∞, then
P (lim sup An ) = 0.
In this paper, we will establish some conditional versions of the first part of
the Borel-Cantelli lemma in Section 2, which extends the results of BarndorffNielsen (1961), Stepanov [20] and Majerek et al. [13] in the different directions.
As its applications, we study the limit behaviors of random variables sequence,
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
3
r in
t
which include that strong limit results of independent random variables in Section 3, the convergence of sums of F -independent random variables in Section
4 and limit results on the concomitants of order statistics in Section 5. Some
of the unconditional versions could be found in Balakrishnan and Stepanov [3].
2. Conditional Borel-Cantelli lemma
Majerek has obtained a conditional version of the first part of the BorelCantelli Lemma.
Lemma 2.1. Let {An , n ≥ 1} be a sequence of events such that
∞
X
P (An |F ) < ∞ a.s..
fP
n=1
Then P (lim sup An ) = 0.
We get the following conditional versions of the first part of the BorelCantelli Lemma.
Lemma 2.2. Let {An , n ≥ 1} be a sequence of events such that P (An |F ) → 0.
If
∞
X
(2.1)
P (An Acn+1 |F ) < ∞ a.s.,
n=1
Ah
ea
do
then P (lim sup An |F ) = 0 a.s.. In this case, P (lim sup An ) = 0.
Proof. For any m > n > 1, we have
!
m
[
P
Ak F = P (Am |F ) + P (Am−1 Acm |F ) + P (Am−2 Acm−1 Acm |F )
k=n
+ · · · + P (An Acn+1 · · · Acm |F )
≤ P (Am |F ) +
m−1
X
P (Ak Ack+1 |F ).
k=n
Therefore
∞ [
∞
\
!
P (lim sup An |F ) = P
Ak F = lim P
n→∞
n=1 k=n
!
m
[
= lim lim P
Ak F
n→∞ m→∞
∞
[
k=n
!
Ak F
k=n
≤ lim P (Am |F ) + lim
m→∞
n→∞
∞
X
P (Ak Ack+1 |F )
k=n
= 0.
In this case, P (lim sup An ) = E(P (lim sup An |F )) = 0.
4
Q. CHEN AND J. LIU
P (Acn An+1 |F ) < ∞ a.s.,
n=1
r in
∞
X
t
Lemma 2.2 holds true if (2.1) is substituted with
because for all m > n ≥ 1
!
m
[
P
Ak F = P (An |F ) + P (Acn An+1 |F ) + P (Acn Acn+1 An+2 |F )
k=n
+ · · · + P (Acn Acn+1 · · · Am |F )
≤ P (An |F ) +
m−1
X
P (Ack Ak+1 |F ).
fP
k=n
Observe that
P (An Acn+1 |F ) = P (An |F ) − P (An An+1 |F ),
we know that even if
∞
X
P (An |F ) = ∞
n=1
and
∞
X
Ah
ea
do
P (An An+1 |F ) = ∞
n=1
P∞
n=1 [P (An |F ) − P (An An+1 |F )]
hold true, as long as
< ∞, then
P (lim sup An |F ) = 0.
P∞
P∞
It is easy toP
see that condition n=1 P (An |F ) < ∞ is weaker than P n=1 P (An )
∞
∞
< ∞, thus n=1 [P (An |F )−P (An An+1 |F )] < ∞ is weaker than n=1 [P (An )
−P (An An+1 )] < ∞, namely we get the same result as Barndorff-Nielsen (1961)
and Stepanov [20] under a weaker condition.
Theorem 2.3. Let {Xn , n ≥ 1} be a sequence of random variables. If for all
small > 0,
P (|Xn − µ| ≥ |F ) → 0 as n → ∞
and
∞
X
[P (|Xn − µ| ≥ |F ) − P (|Xn − µ| ≥ , |Xn+1 − µ| ≥ | F )] < ∞,
n=1
a.s.
where µ is a constant, then Xn −−→ µ as n → ∞.
Proof. Let event An = [|Xn − µ| ≥ ], then applying Lemma 2.2 to get
a.s.
P (lim sup An ) = 0, thus Xn −−→ µ as n → ∞.
In fact, the following more general result can be obtained.
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
5
P (Acn · · · Acn+m−1 An+m |F ) < ∞,
n=1
then P (lim sup An |F ) = 0.
Proof. Since
P
k=n
!
Ak F
fP
j
[
r in
∞
X
t
Lemma 2.4. Let {An , n ≥ 1} be a sequence of events such that P (An |F ) → 0.
If for some m > 0,
= P (An |F ) + P (Acn An+1 |F ) + · · · + P (Acn · · · Acn+m−1 An+m |F )
+ P (Acn · · · Acn+m An+m+1 |F ) + · · · + P (Acn · · · Acj−1 Aj |F )
≤
n+m−1
X
P (Ak |F ) +
k=n
j−m
X
P (Ack · · · Ack+m−1 Ak+m |F ),
k=n
we have
∞
[
!
Ak F
Ah
ea
do
P (lim sup An |F )
!
∞
∞ [
\
Ak F = lim P
=P
n→∞
n=1 k=n
= lim lim P
n→∞ j→∞
≤ lim
n→∞
n+m−1
X
j
[
k=n
k=n
!
Ak F
P (Ak |F ) + lim
n→∞
k=n
∞
X
P (Ack · · · Ack+m−1 Ak+m |F )
k=n
= 0.
3. Strong limit results for maxima
In this section, we will give several examples to show the importance of
Theorem 2.3.
Example 3.1. Let {Xn , n ≥ 1} be a sequence of dependent random variables
defined for any n ≥ 1 by the Clayton copula
1
1
1
+
+ ··· +
− (n − 1)
F (x1 , x2 , . . . , xn ) =
x1
x2
xn
−1
,
6
Q. CHEN AND J. LIU
r in
t
where 0 < xi < 1, 1 ≤ i ≤ n. We know that X1 is uniform distribution on
[0, 1]. Let A = {ω : X1 ∈ (0, 1/2)} and F = {Ø, Ω, A, Ac }, then
 −1
1
1
1


, x1 ∈ (0, 1/2),
+
+ ··· +
− (n − 1)
 2
x
x2
xn
1
−1
F (x1 , x2 , . . . , xn |A) =

1
1

 2 2+
+ ··· +
− (n − 1)
, x1 ∈ [1/2, 1),
x2
xn

0,
x ∈ (0, 1/2),
 −1 1
F (x1 , x2 , . . . , xn |Ac ) =
1
1
1
 2
, x1 ∈ [1/2, 1).
+
+ ··· +
− (n − 1)
x1
x2
xn
Ah
ea
do
fP
Let Mn = max{X1 , . . . , Xn }, then for any x ∈ (0, 1) we have

−1
1


2 n
,
x ∈ (0, 1/2),
−1 +1

x
−1
P (Mn ≤ x|A) =

1

 2 (n − 1)
, x ∈ [1/2, 1).
−1 +2
x

1


2 −
n−1 ,
x ∈ (0, 1/2),

ln
x
∼
1

−1

(n − 1) , x ∈ [1/2, 1)
 2 −
ln x
and

0,
x ∈ (0, 1/2),
 −1
c
P (Mn ≤ x|A ) =
1
 2 n
−1 +1
, x ∈ [1/2, 1).
x

0, x ∈ (0, 1/2),
 1
∼
−1
n , x ∈ [1/2, 1).
 2 −
ln x
Obviously we get P (Mn ≤ x|F ) → 0 as n → ∞, then we have
P (|Mn − 1| ≥ |F ) = P (Mn ≥ 1 + |F ) + P (Mn ≤ 1 − |F )
= 0 + P (Mn ≤ 1 − |F ) → 0.
P∞
Because
n=1 P (Mn ≤ x|F ) = ∞, we can not apply Lemma 2.1. If x ∈
(0, 1/2), let us consider
∞
X
[P (Mn ≤ x| A) − P (Mn ≤ x, Mn+1 ≤ x| A)]
n=1
∞
X
=2
n=1
∼
(n(x−1
x−1 − 1
− 1) + 1)((n + 1)(x−1 − 1) + 1)
∞
X
2
1
< ∞.
(− ln x) n=1 n2
n=1
and
∞
X
r in
Similarly, if x ∈ (0, 1), we get
∞
X
[P (Mn ≤ x| A) − P (Mn ≤ x, Mn+1 ≤ x| A)] < ∞
7
t
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
[P (Mn ≤ x| Ac ) − P (Mn ≤ x, Mn+1 ≤ x| Ac )] < ∞,
n=1
thus
∞
X
[P (Mn ≤ x| F ) − P (Mn ≤ x, Mn+1 ≤ x| F )] < ∞
n=1
a.s.
α
fP
holds true. By Theorem 2.3, we obtain Mn −−→ 1.
In fact, Theorem 2.3 allows us to obtain further conclusions, such as the
properties of
Mnn , 0 < α < 1.
First, observe that
(
nα
P Mn ≤ x|A =
Ah
ea
do
∼




−a
2[n(x−n − 1) + 1]−1
x ∈ (0, 1/2),
−n−a
−1
2[(n − 1)(x
− 1) + 2]
x ∈ [1/2, 1).
1
nα−1
x ∈ (0, 1/2),
2 −
ln
x
1
1
2 −
x ∈ [1/2, 1).
ln x n1−α − n−α



α
Obviously we get P (Mnn ≤ x|A) → 0 as n → ∞. And if x ∈ (0, 1/2), we have
∞
∞
X
X
α
2
(n+1)α
= ∞,
P Mnn ≤ x, Mn+1
≤ x|A =
−n−α − 1) + x−(n+1)−a
n(x
n=1
n=1
∞ h i
X
α
α
(n+1)α
P Mnn ≤ x|A − P Mnn ≤ x, Mn+1
≤ x|A
n=1
∞
X
=2
n=1
∼
−α
(n(x−n−α
x−(n+1) − 1
− 1) + 1)(n(x−n−α − 1) + x−(n+1)−a )
∞
X
2
1
< ∞.
(− ln x) n=1 n2−α
Similarly, if x ∈ (0, 1), we get
∞ h i
X
α
α
(n+1)α
≤ x|A < ∞,
P Mnn ≤ x|A − P Mnn ≤ x, Mn+1
n=1
α
P (Mnn ≤ x|Ac ) → 0 as n → ∞,
8
Q. CHEN AND J. LIU
n=1
thus
r in
∞ h i
X
α
α
(n+1)α
P Mnn ≤ x|Ac − P Mnn ≤ x, Mn+1
≤ x|Ac < ∞,
t
and
∞ h i
X
α
α
(n+1)α
P Mnn ≤ x|F − P Mnn ≤ x, Mn+1
≤ x|F < ∞
n=1
and
α
P (Mnn ≤ x|F ) → 0 as n → ∞
α
a.s.
hold true. By Theorem 2.3, we obtain Mnn −−→ 1, 0 < α < 1.
α
fP
Next, we will find a function ϕn (Mn ) which replace Mnn in Example 3.1.
Before our discussion, the following key lemma is needed.
Lemma 3.2. Let {An , n ≥ 1} be a sequence of events such that P (An |F ) → 0.
If for any k ≥ 1,
P (Acn · · · Acn+k An+k+1 |F )
≤ q ∈ [0, 1),
lim
c
n→∞ P (Ac
n · · · An+k−1 An+k |F )
then P (lim sup An |F ) = 0.
Ah
ea
do
Proof. For all large enough n and small ,
∞
∞
X
X
P (Acn · · · Acn+k−1 An+k |F ) = P (An |F ) +
P (Acn · · · Acn+k−1 An+k |F )
k=0
k=1
≤ P (An |F ) + P (Acn An+1 |F )
q+
.
1−q−
On the other hand , we have
∞
∞ [
\
P (lim sup An |F ) = P
n=1 k=n
!
Ak F = lim P
n→∞
∞
[
k=n
!
Ak F
= lim [P (An |F ) + P (Acn An+1 |F ) + · · ·
n→∞
+ P (Acn · · · Acn+k−1 An+k |F ) + · · · ]
= lim
n→∞
∞
X
P (Acn · · · Acn+k−1 An+k |F )
k=0
q+
c
≤ lim P (An |F ) + P (An An+1 |F )
.
n→∞
1−q−
We have lim P (An |F ) = 0 and P (Acn An+1 |F ) ≤ P (An |F ), thus
n→∞
lim P (Acn An+1 |F ) = 0,
n→∞
obviously lim P (lim sup An |F ) = 0.
n→∞
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
9
a.s.
Mn −−→ rF .
r in
t
Let {Xn , n ≥ 1} be a sequence of F -independent random variables with
distribution F (x). Let Mn = max{X1 , . . . , Xn }, rF = sup{x ∈ R : F (x) < 1},
where rF ≤ ∞. If rF < ∞, let An = {|Mn − rF | ≥ } = {Mn ≤ rF − }, then
n
P
n |F ) = [P (X1 ≤ rF − |F )] , obviously 0 < P (X1 ≤ rF − |F ) < 1, thus
P(A
∞
k=1 P (An |F ) < ∞. By Lemma 2.1, we have
n
If rF = ∞, by definition rF = +∞, thus P (Mn < 0|F ) = [P (Xn < 0|F )] ,
then lim P (Mn < 0) = 0. For convenient, let Xn ≥ 0. Let mn = 1/Mn ,
n→∞
An = {|mn | ≥ } = {mn ≥ } = {Mn ≤ 1/}, because 1/ < +∞, just like
a.s.
above, we have mn −−→ 0, namely
a.s.
fP
Mn −−→ rF .
Thus we have a simple conclusion as follow, for any x ∈ {x ∈ R : F (x) < 1}, if
P
∞
n=1 P (Xn ≤ x|F ) < ∞, then
a.s.
Xn −−→ rF .
Let ϕn (x) be a measurable function of n and x, which is decreasing in n and
increasing in x. Next ,we study the limit behavior of ϕn (Mn ) and obtain the
following result.
Ah
ea
do
Theorem 3.3. Let a function ϕn (x) be such that P (ϕn (Mn ) ≤ x|F ) → 0.
a.s.
Then ϕn (Mn ) −−→ rF .
Proof. Let xn = ϕ−1
n (x), we have
P (ϕn (Mn ) ≤ x|F ) = P (Mn ≤ xn |F ).
Thus P (Mn ≤ xn |F ) → 0, then P (Mn ≤ xn ) = E[P (Mn ≤ xn |F )] → 0,
namely P (xn ≤ Mn ) → 1. By definition Mn ≤ rF , thus P (xn ≤ Mn ) ≤
P (xn ≤ rF ) ≤ 1, hence lim P (xn ≤ rF ) = 1. We obtain xn → rF because xn
n→∞
is increasing and with supremum rF . Let An = {ϕn (Mn ) ≤ x}, then
P (Acn · · · Acn+k An+k+1 |F )
P (Acn · · · Acn+k−1 An+k |F )
P (xn < Mn ≤ xn+1 , . . . , xn+k < Mn+k ≤ xn+k+1 , Mn+k+1 ≤ xn+k+1 |F )
P (xn < Mn ≤ xn+1 , . . . , xn+k−1 < Mn+k−1 ≤ xn+k , Mn+k ≤ xn+k |F )
P (xn+k < Xn+k ≤ xn+k+1 |F )P (Xn+k+1 ≤ xn+k+1 |F )
=
→ 0,
P (Xn+k ≤ xn+k |F )
=
The conditions of Lemma 3.2 are fulfilled. The result follows.
Example 3.4. Let {Xn , n ≥ 1} be a sequence of independent random variables
such that
n Y
1
,
F (x1 , x2 , . . . , xn ) =
1−
xi
i=1
10
Q. CHEN AND J. LIU
r in
t
where xi ≥ 1, 1 ≤ i ≤ n. Let A = {ω : X1 ∈ [m, ∞), m > 1} and F =
{Ø, Ω, A, Ac }, then


0,
x1 ∈ [1, m),
Qn
1
1
F (x1 , x2 , . . . , xn |A) =
 i=1 (1 − )/ , x1 ∈ [m, ∞),
xi m
and

Q
1
1

 ni=1 (1 − )/(1 − ), x1 ∈ [1, m),
c
x
m
i
F (x1 , x2 , . . . , xn |A ) =
Qn
1


(1
− ),
x1 ∈ [m, ∞).
i=2
xi
Ah
ea
do
fP
Obviously {Xn , n ≥ 1} is F -independent. Let Mn = max{X1 , . . . , Xn }, then
we have
(
0,
x ∈ [1, m),
1 n
P (Mn ≤ x|A) =
m(1 − ) , x ∈ [m, ∞)
x
and


 (1 − 1 )n /(1 − 1 ), x ∈ [1, m),
c
x
m
P (Mn ≤ x|A ) =
1


(1 − )n−1 ,
x ∈ [m, ∞).
x
Mn tends in probability and with probability one to infinity because
P∞
n=1 P (Mn ≤ x|F ) < ∞ and rF = +∞. Let us study the following problem.
For which sequence an → 0 the sequence an Mn continues to tend to infinity
with probability one?
x
For large enough N, if n ≥ N , then
≥ m , observe that
an
x
|A
P (an Mn ≤ x|A) = P Mn ≤
an
−nan
an n
=m 1−
∼ me x ,
x
then we have
∞
∞
X
X
−nan
P (an Mn ≤ x|A) ∼ m
e x .
n=N
n=N
Let event An = {an Mn ≤ x}, if
∞
X
(3.1)
e
−nan
x
< ∞,
n=1
then
∞
X
n=1
P (an Mn ≤ x|A) < ∞.
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
11
P (an Mn ≤ x|F ) < ∞.
n=1
a.s.
r in
∞
X
t
Similarly, the inequality holds true for Ac , thus
By Lemma 2.1, P (lim sup An ) = 0, that is, an Mn −−→ ∞. We know that
−nan
an n
P (an Mn ≤ x|A) = m 1 −
∼ me x .
x
If nan → ∞ then P (an Mn ≤ x|A) → 0. Similarly, if nan → ∞ then P (an Mn ≤
x|Ac ) → 0, thus P (an Mn ≤ x|F ) → 0 . It follows from Theorem 3.3 that
a.s.
an Mn −−→ ∞ is valid, and the condition is weaker than (3.1).
fP
4. Convergence of sums of F -independent random variables
In this section, we study the convergence of sums of F -independent random
variables. For this, we need the following inequality.
Theorem 4.1 (Conditional Kolmogorov’s inequality). If {Xn , n ≥ 1} is a
sequence of F -independent random variables belonging to L2F , then for an
arbitrary F -measurable random variable > 0 a.s., we have
n
( + c)2
1 X 2
F
σF Xk a.s.
1 − Pn
≤
P
[
max
|S
−
E
S
|
≥
|F
]
≤
k
k
2
1≤k≤n
2
k=1 σF Xk
Ah
ea
do
k=1
where Sn = X1 + X2 + · · · + Xn , |Xk | ≤ c ≤ ∞.
Proof. The proof of the right part has been given by Majerek in [13], we only
prove the left part. Obviously, if c = ∞ the conclusion is true, thus we can
suppose c < ∞ . In the following, for the sake of convenience, let E F Sk =
0, E F Xk = 0. Let A0 = Ω, B1 = {|S1 | ≥ },
Ak = { max |Sj | < }, k = 1, 2, . . . , n
1≤j≤k
and
Bk = Ak−1 − Ak = {|Sj | < , 1 ≤ j ≤ k − 1; |Sk | ≥ }, k = 2, 3, . . . , n.
Because Bk ∈ σ{X1 , X2 , . . . , Xk }, Sk IBk is F -independent given Sn − Sk .
Thus, we have
E F Sn2 IBk = E F (Sk − (Sn − Sk ))2 IBk
= E F Sk2 IBk + E F (Sn − Sk )2 IBk .
(4.1)
Observe that on the set Bk , |Sk | ≤ |Sk−1 | + |Xk | ≤ + c and Acn =
applying equation (4.1), we obtain
E F Sn2 IAcn =
n
X
k=1
E F Sk2 IBk +
n
X
k=1
E F (Sn − Sk )2 IBk
Sn
k=1
Bk ,
≤ (c + )2
n
X
P (Bk |F ) +
k=1
≤ [(c + )2 +
(4.2)
n
n
X
X
E F Xj2 P (Bk |F )
k=1 j=k+1
n
X
E F Xk2 ]P (Acn |F ).
k=1
On the other hand, we have
E F Sn2 IAcn = E F Sn2 − E F Sn2 IAn
=
(4.3)
n
X
k=1
n
X
k=1
E F Xk2 − 2 P (An |F )
E F Xk2 − 2 + 2 P (Acn |F ).
fP
≥
t
Q. CHEN AND J. LIU
r in
12
By equations (4.2) and (4.3), we obtain the conclusion
Pn
E F Xk2 − 2
( + c)2
c
k=1P
P
≥
1
−
.
P (An |F ) ≥
n
n
2
( + c)2 + k=1 E F Xk2 − 2
k=1 σF Xk
P∞
A sufficient condition for the convergence of series n=1 Xn has been given
in [12] as follow.
Ah
ea
do
Theorem 4.2. Let {Xn , n ≥ 1} be a sequence of F -independent random variables such that E F Xn = 0, if
∞
X
E F Xn2 < ∞,
n=1
then
P∞
n=1 Xn converges almost surely.
Theorem 4.3. Let {Xn , n ≥ 1} be a sequence of F -independent random variables such that |Xn | ≤ c, cP
is a constant.
P∞
∞
(i) If E F Xn = 0 and n=1 E F Xn2 = ∞, then n=1 Xn diverges almost
surely. P
∞
2
(ii) If n=1 Xn converges almost surely, where σF
Xn is a conditional variP∞
P∞
F
2
ance of Xn , then n=1 E Xn and n=1 σF Xn converge almost surely.
Proof. (i) By the left of Kolmogorov’s inequality, we have
( + c)2
→ 1 as m → ∞,
P [ max |Xn+1 + · · · + Xn+k | ≥ |F ] ≥ 1 − Pn+m
F 2
1≤k≤m
k=n+1 E Xn
thus for any n ≥ 1, we obtain P [sup |Xn+1 + · · · + Xn+k | ≥ |F ] = 1, in this
k≥1
P∞
case P [sup |Xn+1 + · · · + Xn+k | ≥ ] = 1, obviously n=1 Xn diverges almost
k≥1
surely.
(ii) Let Xn0 be a new sequence of random variables such that {Xn , Xn0 ; n ≥ 1}
are F -independent, and for a fixed n, Xn and Xn0 are identically distributed.
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
13
r in
t
Let Xn00 = Xn − Xn0 , thus Xn00 is a sequence of F -independent random variables
00
F 00
2
2
such that |XP
= 0, σF
Xn00 = 2σF
Xn .
n | ≤ 2c, E XnP
P∞
∞
∞
00
0
X
converge
almost surely,
X
and
Because
n=1 Xn conn=1 n
n=1 n
P∞
P
∞
2
00
2
verges almost surely. By (i), we have n=1 σF Xn < ∞, thus n=1 σF
Xn
P∞
F
< ∞. By Theorem 4.2,
(X
−
E
X
)
converges
almost
surely,
thus
n
n
n=1
P∞
F
n=1 E Xn converges almost surely.
5. Limit results for concomitants of order statistics
Ah
ea
do
fP
In this section, we will apply our conditional versions of Borel-Cantelli lemma
to discuss the strong limit results for concomitants of order statistics, which
are the conditional versions of the results in [3].
The theory of the concomitants of order statistics plays an important role in
the process of solving practical problems. For example, Bhattacharya [5] has
given an instance to show it’s importance as follows: Studying two numerical
characteristics X and Y defined for each individual in a population. If we want
to select individuals according to their ranks in Y that can not be obtained, we
can just do it by their ranks in a related variate X. In this case, the problems
about concomitants of order statistics arise naturally.
The concept of concomitants of order statistics was introduced by David
(1973) and Bhattacharya [5]. Suppose that (X, Y ), (X1 , Y1 ), (X2 , Y2 ), . . .,
(Xn , Yn ), . . . are F -independent and identically distributed random vectors
with continuous bivariate distribution function F (x, y), marginal distributions
H(x) and G(y), and corresponding probability density functions h(x) and g(y),
Let X1,n , X2,n , . . . Xn,n be the order statistics of the sample X1 , X2 , . . . , Xn and
Y[1,n] , Y[2,n] , . . . , Y[n,n] be the corresponding concomitants of the defined order
statistics, relating to the sample Y1 , Y2 , . . . , Yn . Suppose that lH = inf{x ∈ R :
H(x) > 0} and rH = sup{x ∈ R : H(x) < 1} are the left and right extremities
of H.
5.1. Limit results for Y[n−k,n] when k is fixed
Lemma 5.1. Suppose
(5.1)
lim−
x→rH
G(y|F ) − F (x, y|F )
= β(y|F ) ∈ [0, 1],
1 − H(x|F )
then, for n → ∞, k ≥ 0,
P (Y[n−k,n] ≤ y|F ) → β(y|F ).
Proof. Obviously, for a fixed k, Xn−k,n is increasing in n and Xn−k,n ≤ rH ,
then we have Xn−k,n → rH , n → ∞. Thus for some small enough non-negative
number ,
lim P (Xn−k,n ≥ rH − |F ) = 1.
n→∞
14
Q. CHEN AND J. LIU
t
We have
P (Y[n−k,n] ≤ y, rH − ≤ Xn−k,n ≤ rH |F )
P (rH − ≤ Xn−k,n ≤ rH |F )
P (Y ≤ y, rH − ≤ X ≤ rH |F )
= lim
→0
P (rH − ≤ X ≤ rH |F )
F (dx, y|F )
= lim−
.
h(x|F )
x→rH
lim P (Y[n−k,n] ≤ y|F ) = lim
n→∞
Observing that lim−
x→rH
r in
n→∞
G(y|F ) − F (x, y|F )
F (dx, y|F )
= lim−
, then we ob1 − H(x|F )
h(x|F )
x→rH
tain
G(y|F ) − F (x, y|F )
.
1 − H(x|F )
fP
lim P (Y[n−k,n] ≤ y|F ) = lim−
n→∞
x→rH
Suppose that for some c ∈ [−∞, +∞],
β(y|F ) = 0(y < c) and β(y|F ) = 1(y > c).
Then the distribution F (x, y) is called a c-stable maximum-concomitants conditional distribution. If such a c does not exist, then F (x, y) is called an unstable
maximum-concomitants conditional distribution.
Ah
ea
do
Theorem 5.2. Let for all y < rG , β(y|F ) = 0 holds true, and
Z
G(y|F ) − F (x, y|F )
[h(x|F ) − F (dx, y|F )] < ∞.
(1 − H(x|F ))2
R
a.s.
Then Y[n−k,n] −−→ rG as n → ∞.
Proof. Suppose event An = {Y[n−k,n] ≤ y}, then P (An |F ) → 0. By symmetry
and F -independence, we have
Z Z y
n!
P (Y[n−k,n] ≤ y|F ) =
P (X2 ≤ x, . . . ,
(n − k − 1)!k! R lG
Xn−k ≤ x, Xn−k+1 > x, . . . , Xn > x|F )F (dx, dy|F ).
It follows that
(5.2)
P (Y[n−k,n] ≤ y|F )
Z
n!
=
(1 − H(x|F ))k H n−k−1 (x|F )F (dx, y|F ).
(n − k − 1)!k! R
Further, we will study that
P (Y[n−k,n] > y, Y[n−k+1,n+1] ≤ y|F )
= P (Y[n−k,n] > y, Y[n−k+1,n] ≤ y, Y[n−k+1,n+1] ≤ y|F )
+ P (Y[n−k,n] > y, Y[n−k+1,n] > y, Y[n−k+1,n+1] ≤ y|F ).
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
15
r in
P (Y[n−k,n] > y, Y[n−k+1,n] ≤ y, Y[n−k+1,n+1] ≤ y|F )
Z Z rG Z rH Z y
n!
P (X3 ≤ x, . . . , Xn−k+1 ≤ x,
=
(n − k − 1)!(k − 1)! R y
lG
x
t
Again, by symmetry and F independence, we have
Xn−k+2 > x1 , . . . , Xn > x1 , En+1 |F )F (dx1 , dv1 |F )F (dx, dv|F ),
where
n
o
[
En+1 = (X, Y ) ∈ Π(x,rH ]×[lG ,y] Π(x1 ,rH ]×(y,rG ] and
[
Π(x,rH ]×[lG ,y] Π(x1 ,rH ]×(y,rG ]
fP
are half-open rectangles with vertices a, b, c, d. In the same way
P (Y[n−k,n] > y, Y[n−k+1,n] > y, Y[n−k+1,n+1] ≤ y|F )
Z Z rG Z rH Z y
n!
=
P (X3 ≤ x, . . . , Xn−k+1 ≤ x,
(n − k − 1)!(k − 1)! R y
lG
x
∼
Xn−k+2 > x1 , . . . , Xn > x1 , E n+1 |F )F (dx1 , dv1 |F )F (dx, dv|F ),
∼
where E n+1 = (Xn+1 , Yn+1 ) ∈ Π(x,x1 ]×[lG ,y] . It can be shown that
Ah
ea
do
P (Y[n−k,n] > y, Y[n−k+1,n] ≤ y, Y[n−k+1,n+1] ≤ y|F )
Z Z rG
Z rH Z y
n!
=
[H(x|F )]n−k−1
(1 − H(x1 |F ))k−1
(n − k − 1)!(k − 1)! R y
x
lG
× [1 − H(x1 |F ) + F (x1 , y|F ) − F (x, y|F )]F (dx1 , dv1 |F )F (dx, dv|F )
and
P (Y[n−k,n] > y, Y[n−k+1,n] > y, Y[n−k+1,n+1] ≤ y|F )
Z Z rG
Z rH Z y
n!
n−k−1
[H(x|F )]
(1 − H(x1 |F ))k−1
=
(n − k − 1)!(k − 1)! R y
x
lG
× [F (x1 , y|F ) − F (x, y|F )]F (dx1 , dv1 |F )F (dx, dv|F ).
Evaluating P (Y[n−k,n] > y, Y[n−k+1,n] ≤ y, Y[n−k+1,n+1] ≤ y|F ), we get
Z rHZ y
(1 − H(x1 |F ))k−1 [1 − H(x1 |F ) + F (x1 , y|F ) − F (x, y|F )]
x
lG
× F (dx1 , dv1 |F )
Z rH Z
≤ (1 − H(x1 |F ))k
x
y
F (dx1 , dv1 |F )
lG
≤ (1 − H(x1 |F ))k [G(y|F ) − F (x, y|F )].
Evaluating P (Y[n−k,n] > y, Y[n−k+1,n] > y, Y[n−k+1,n+1] ≤ y|F ), we get
Z rHZ y
(1 − H(x1 |F ))k−1 [F (x1 , y|F ) − F (x, y|F )]F (dx1 , dv1 |F )
x
lG
Q. CHEN AND J. LIU
≤ (1 − H(x1 |F ))k
Z
rH
Z
y
[G(y) − F (x, y)]F (dx1 , dv1 |F )
x
lG
r in
≤ (1 − H(x1 |F ))k [G(y|F ) − F (x, y|F )].
t
16
Then
fP
2n!
P (Y[n−k,n] > y, Y[n−k+1,n+1] ≤ y|F ) ≤
(n − k − 1)!(k − 1)!
Z Z rG
[H(x|F )]n−k−1 [G(y|F ) − F (x, y|F )]F (dx, dy|F )
R y
Z
2n!
=
[H(x|F )]n−k−1 (1 − H(x1 |F ))k
(n − k − 1)!(k − 1)! R
[G(y|F ) − F (x, y|F )][h(x|F ) − F (dx, y|F )].
P∞ (n+k+1)! n P∞
1
n!
n−k−1
Because (1−x)
, we ob2+k =
n=0 (k+1)!n! x =
n=k+1 (n−k−1)!(k+1)! x
tain the following inequality
∞
X
P (Y[n−k,n] > y, Y[n−k+1,n+1] ≤ y|F ) ≤ 2k(k + 1)
n=k+1
Z
×
R
G(y|F ) − F (x, y|F )
[h(x|F ) − F (dx, y|F )] < ∞.
(1 − H(x|F ))2
Ah
ea
do
P∞
Namely n=1 P (Acn An+1 |F ) < ∞, by Lemma 2.2, we have P (lim sup An ) = 0,
a.s.
thus the conclusion Y[n−k,n] −−→ rG as n → ∞ is true.
Theorem 5.3. Let for all > 0, β(c−|F ) = 0, β(c+|F ) = 1, the inequalities
Z
G(c + |F ) − F (x, c + |F )
(5.3)
[h(x|F ) − F (dx, c + |F )] < ∞
(1 − H(x|F ))2
R
and
Z
(5.4)
R
G(c − |F ) − F (x, c − |F )
[h(x|F ) − F (dx, c − |F )] < ∞
(1 − H(x|F ))2
a.s.
hold true, then Y[n−k,n] −−→ c as n → ∞.
Proof. Suppose eventAn = {Y[n−k,n] ≤ c − }, because β(c − |F ) = 0, then
P∞
P (An |F ) → 0. From(5.3), we get n=1 P (Acn An+1 |F ) < ∞. By Lemma 2.2,
a.s.
P (lim sup An ) = 0, thus the conclusion Y[n−k,n] −−→ c as n → ∞ is true. Theorem 5.3 is a slight extension of Theorem 5.2.
5.2. Limit results for Y[n−k,n] when k tends to infinity
In this part, we suppose that the distribution F (x, y) is absolutely continuous
and Fx (x, y) is the density-distribution of F (x, y). Let for some y < rG
(5.5)
lim−
x→rH
Fx (x, y|F )
= β(y|F ).
h(x|F )
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
17
r in
t
Observe that the limit in (5.5) equals the limit in (5.1) for the same y. When
F (x, y) is absolutely continuous, the limit in (5.1) boils down to the limit in
(5.5). Let k = kn be such that kn → ∞ as n → ∞, under which condition
we will study the sequence Y[n−k,n] is convergent in probability. Before the
discussion, we need the following lemma from Dembinska et al. (2007).
Lemma 5.4. Let kn depend on n in such way that kn → ∞, n − kn → ∞ and
kn /n → 0 as n → ∞. Then, for any fixed 0 < d < 1,
n!
dn−kn → 0 as n → ∞.
(n − kn )!(kn − 1)!
P
fP
Theorem 5.5. Let F (x, y) be absolutely continues, the limit in (5.5) exists and
be equal to 0. for all y < rG , Let kn > 0 be an increasing sequence of positive
integers such that kn → ∞, n − kn → ∞ and kn /n → 0 as n → ∞. Then
Y[n−k,n] −
→ rG as n → ∞.
Proof. Because the limit in (5.5) exists and be equal to 0, then for any small
> 0, we choose x such that for any x > x , y < rG ,
(5.6)
Fx (x, y|F ) < h(x|F ).
We have
Ah
ea
do
P (Y[n−k,n] ≤ y|F ) = I1 (y|F ) + I2 (y|F ),
where
Z rH
n!
I1 (y|F ) =
(1 − H(x|F ))kn H n−kn −1 (x|F )F (dx, y|F ),
(n − kn − 1)!kn ! x
Z x
n!
I2 (y|F ) =
(1 − H(x|F ))kn H n−kn −1 (x|F )F (dx, y|F ).
(n − kn − 1)!kn ! lH
It follows from (5.6) that for any y < rG ,
Z rH
n!
(1 − H(x|F ))kn H n−kn −1 (x|F )h(x|F )dx
I1 (y|F ) <
(n − kn − 1)!kn ! x
Z rH
<
h(x|F )dx = .
lH
For any y < rG , we have
I2 (y|F ) <
n!
H n−kn −1 (x |F ).
(n − kn − 1)!kn !
Because 0 < H(x |F ) < 1, by Lemma 5.4, I2 (y|F ) → 0 for any y < rG . Thus,
P
we obtain for any y < rG , P (Y[n−k,n] ≤ y|F ) → 0, namely Y[n−k,n] −
→ rG as
n → ∞.
18
Q. CHEN AND J. LIU
t
5.3. Limit results for Cn Y[n−k,n]
r in
In this part, we will study the limit of Cn Y[n−k,n] , where Cn is a nonstochastic sequence.
Theorem 5.6. Let F (x, y) be a 0-stable maximum-concomitants conditional
distribution
(1) Let Cn → +∞ be such that for any y > 0
n[G(y/Cn |F ) − G(−y/Cn |F )] → 0 (n → ∞),
then for any fixed k ≥ 0,
P
Cn |Y[n−k,n] | −
→ ∞ as n → ∞.
n=1
then for any fixed k ≥ 0,
a.s.
fP
(2) Let Cn → +∞ be such that for any y > 0
∞
X
n[G(y/Cn |F ) − G(−y/Cn |F )] < ∞,
Cn |Y[n−k,n] | −−→ ∞ as n → ∞.
Proof. First, observe that even if lG and rG are both finite, the support for the
limiting distribution of Cn Y[n−k,n] is unbounded. It follows from (5.2) that
P (a < Y[n−k,n] < b|F ) = n[G(b|F ) − G(a|F )],
Ah
ea
do
n−1
X
k=0
then we have
P (a < Y[n−k,n] < b|F ) ≤ n[G(b|F ) − G(a|F )],
thus
P (|Cn Y[n−k,n] | ≤ y|F ) = P (−y < Cn Y[n−k,n] < y|F )
≤ n[G(y/Cn |F ) − G(−y/Cn |F )] → 0 as n → ∞,
where y > 0, Cn > 0. The first statement of Theorem 5.6 readily follows.
The second statement can be similarly obtained if we apply the Lemma 2.1 in
Section 2.
By slightly modifying the arguments in Theorem 5.6, we get the following
limit result.
Theorem 5.7. Let F (x, y) be a +∞ (or −∞)-stable maximum-concomitants
conditional distribution.
(1) Let Cn → 0 be such that for any y > 0
n[1 − G(y/Cn |F ) + G(−y/Cn |F )] → 0 as n → ∞,
then for any fixed k ≥ 0,
P
Cn Y[n−k,n] −
→ 0 as n → ∞.
THE CONDITIONAL BOREL-CANTELLI LEMMA AND APPLICATIONS
19
n[1 − G(y/Cn |F ) + G(−y/Cn |F )] < ∞,
n=1
then for any fixed k ≥ 0,
a.s.
Cn Y[n−k,n] −−→ 0 as n → ∞.
r in
∞
X
t
(2) Let Cn → +∞ be such that for any y > 0
Acknowledgements. The authors acknowledge the support provided by
NSFs of China (No.11271013, 11471340, 10901065) and the Fundamental Research Funds for the Central Universities, HUST: 2016YXMS003, 2014TS066.
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ea
do
fP
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20
Q. CHEN AND J. LIU
fP
Qianmin Chen
School of Mathematics and Statistics
Huazhong University of Science and Technology
Wuhan, 430074, P. R. China
E-mail address: [email protected]
r in
t
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Jicheng Liu
School of Mathematics and Statistics
Huazhong University of Science and Technology
Wuhan, 430074, P. R. China
E-mail address: [email protected]