# Asymptotic Law of thejth Records in the Bivariate Exponential Case

Authors: Grine Azedine
Journal: Journal of Mathematics
 (2014)

---

## Abstract

We consider a sequence(
X
i
,
Y
i
)
1
⩽
i
⩽
n of independent and identically distributed random variables with joint cumulative distribution H
(
x
,
y
), which has exponential marginals F
(
x
) and G
(
y
) with parameter λ
=
1. We also assume that X
i
(
ω
)
≠
Y
i
(
ω
), ∀
i
∈
N, and ω
∈
Ω. We denote  R
k
(
j
)
k
⩾
1 and S
k
(
j
)
k
⩾
1 by the sequences of the jth records in the sequences (
X
i
)
1
⩽
i
⩽
n, (
Y
i
)
1
⩽
i
⩽
n, respectively. The main result of of the paper is to prove the asymptotic independence of R
k
(
j
)
k
⩾
1 and S
k
(
j
)
k
⩾
1 using the property of stopping time of the jth record times and that of the exponential distribution.


---

## Body

## 1. Introduction

Let(
X
i
)
1
⩽
i
⩽
n be a sequence of independent and identically distributed (i
.
i
.
d) random variables (r.v’s.) from a distribution F. Let us consider the jth record time defined recurrently for k
=
2,3
,
… and j
⩾
1 as
(1)
L
1
(
j
)
=
j
,
L
k
(
j
)
=
min
⁡
{
m
>
L
k
-
1
(
j
)
;
X
m
-
j
+
1
,
m
>
X
L
k
-
1
(
j
)
-
j
+
1
,
L
k
-
1
(
j
)
}
,
and the kth ordinary jth record from F as
(2)
R
k
(
j
)
=
X
L
k
(
j
)
-
j
+
1
,
L
k
(
j
)
;
R
1
(
j
)
=
X
1
,
j
,
where X
i
,
m denotes the ith order statistic of a sample of size m.In 1976 Dziubdziela-Kopocinski [1] proved that
(3)
lim
⁡
n
→
∞
P
(
R
n
(
j
)
⩽
a
n
x
+
b
n
)
=
N
(
-
log
⁡
(
-
log
⁡
Θ
(
x
)
)
)
,
where Θ
(
x
) is one of the three well known limit laws of X
n
,
n
=
max
⁡
1
⩽
i
⩽
n
(
X
i
) (Frechet, Wiebull, and Gumbell), N
(
·
) is the standard normal distribution, and a
n, b
n are the constants of normalization.Taking account of these three limit laws we get those of thejth record; namely,(i)
Type 1:(4)
N
(
x
)
,
x
∈
R
,
(ii)
Type 2:(5)
N
1
α
=
{
N
(
log
⁡
x
α
)
,
x
>
0
,
α
>
0
,
0
x
⩽
0
,
(iii)
Type 3:(6)
N
2
α
=
{
N
(
log
⁡
(
-
x
)
-
α
)
,
x
<
0
,
α
>
0
,
1
x
⩾
0
.The authors have presented the expressions of the probability density function f
n
(
j
)
(
x
) and the distribution function F
n
(
j
)
(
x
) of the jth record as follows:
(7)
f
n
(
j
)
(
x
)
=
j
(
n
-
1
)
!
[
-
j
log
⁡
(
1
-
F
(
x
)
)
]
n
-
1
[
1
-
F
(
x
)
]
j
-
1
f
(
x
)
,
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
n
=
1,2
,
…
,
F
n
(
j
)
(
y
)
=
∫
0
-
j
log
⁡
(
1
-
F
(
y
)
)
e
-
u
u
n
-
1
(
n
-
1
)
!
d
u
,
n
=
1,2
,
…
.
Contrary to that of records, the theory of limit law of the maximum has been extended to the bivariate case by the works of Finkelshtein [2], Geffroy [3], de Oliveira [4], Sibuya [5], Galombos [6], Marshall and Olkin [7], Kotz and Nadarajah [8], Coles [9], and Smith [10].In this work, we assume thatX
n
(
ω
)
≠
Y
n
(
ω
), ∀
n
∈
N and ω
∈
Ω. Moreover, we will focus on the bivariater.v’s.  (
R
n
(
j
)
,
S
n
(
j
)
)
n
⩾
1 issued from pairs ofr.v’s.  (
X
n
,
Y
n
)
n
⩾
1 with joint cumulative distribution H
(
x
,
y
) whose marginals F
(
x
) and G
(
y
) are exponential with parameter λ
=
1 and prove that
(8)
lim
⁡
n
→
∞
P
(
R
n
(
j
)
⩽
a
n
x
+
b
n
,
S
n
(
j
)
≤
c
n
y
+
d
n
)
=
lim
⁡
n
→
∞
P
(
R
n
(
j
)
⩽
a
n
x
+
b
n
)
lim
⁡
n
→
∞
P
(
S
n
(
j
)
⩽
c
n
y
+
d
n
)
=
N
(
x
)
N
(
y
)
,
where N
(
·
) is the standard normal distribution, and a
n
=
c
n
=
n
/
j, b
n
=
d
n
=
n
/
j
2.


## 2. Preliminaries

In this section we recall some relevant results for future use.Theorem 1 (Deheuvels [11]).
LetX
1
,
X
2
,
…
,
X
n be a sequence of i
.
i
.
d r.v’s. with Φ
(
x
) continues repartition function.
Let{
X
k
,
n
:
1
⩽
k
⩽
n
} and {
R
k
(
j
)
,
k
⩾
1
} be, respectively, the sequences of order statistics and jth records.
∀
n
⩾
2 and for all j
⩾
1 one has the following.(i)
Forn
⩾
j
⩾
1 and Φ
(
s
)
<
1, the sequences
(9)
{
X
k
,
n
:
1
≤
k
≤
n
}
,
{
R
k
(
j
)
:
k
⩾
1
}
form a Markov chain with equal probability of transition. That means(10)
P
(
R
k
+
1
(
j
)
⩾
t
∣
R
k
(
j
)
=
s
)
=
P
(
X
n
-
j
+
1
,
n
⩾
t
∣
X
n
-
j
,
n
=
s
)
=
(
1
-
Φ
(
t
)
1
-
Φ
(
s
)
)
j
.
(ii)
The assertions{
R
k
(
j
)
a
n
d
R
k
+
1
(
j
)
-
R
k
(
j
)
i
n
d
e
p
e
n
d
e
n
t
} and {
X
n
-
j
,
n
a
n
d
X
n
-
j
+
1
,
n
-
X
n
-
j
,
n
i
n
d
e
p
e
n
d
e
n
t
} are equivalent.Corollary 2.
Let(
X
i
)
1
⩽
i
⩽
n be a sequence of i
.
i
.
d r.v’s. with repartition Φ
(
x
) continues and let, for j
⩾
1 fixed, (
R
k
(
j
)
)
k
⩾
1 be the sequence of jth records from Φ.
Ifk
⩾
1 is a fixed integer, R
k
(
j
) and R
k
+
1
(
j
)
-
R
k
(
j
) are independent if and only if Φ
(
x
)
=
1
-
e
(
-
b
(
x
-
B
)
) with x
⩾
B for B and b being positive finite constants.Corollary 3.
ForΦ
(
x
)
=
1
-
e
-
x
,
x
⩾
0 and (
R
k
(
j
)
)
k
⩾
1 being the sequence of jth records, then α
k
(
j
)
=
(
R
k
(
j
)
-
R
k
-
1
(
j
)
)
1
⩽
k
⩽
n is a sequence of i
.
i
.
d r.v’s.s with repartition function Ψ
(
x
)
=
1
-
e
-
j
x
,
x
⩾
0.Corollary 4.
ForΦ
(
x
)
=
1
-
e
-
x
,
x
⩾
0, and for all m
≤
k
-
1, R
k
(
j
)
-
R
k
-
1
(
j
) is independent of R
m.Let(
U
i
)
1
⩽
i
⩽
n be the sequence of i
.
i
.
d r.v’s. uniformly distributed on [
0,1
]; then one has the following.Theorem 5 (Deheuvels [12]).
For allA
>
0 and for all P
⩾
4  a
.
s. for n
→
∞,
(11)
U
m
,
n
⩽
1
n
{
log
⁡
2
n
+
(
m
+
1
)
log
⁡
3
n
+
log
⁡
4
n
+
⋯
+
(
1
+
A
)
log
⁡
p
n
}
.Theorem 6 (Deheuvels [12]).
For allA
>
0, and for all p
⩾
3  a
.
s. for n large enough,
(12)
U
m
,
n
>
1
n
{
(
log
⁡
n
)
(
log
⁡
2
n
)
⋯
(
log
⁡
p
n
)
1
+
A
}
1
/
m
.Corollary 7.
LetX
1
,
X
2
,
…
,
X
n be a sequence of i
.
i
.
d r.v’s. with repartition function Φ
(
x
) and Γ
(
t
)
=
inf
⁡
{
x
∈
R
∣
Φ
(
x
)
≥
t
}. Then, a.s. for n
→
∞, A
>
0, p
⩾
4,
(13)
Γ
{
1
n
{
(
log
⁡
(
n
)
)
(
log
⁡
2
(
n
)
)
⋯
(
log
⁡
p
(
n
)
)
1
+
A
}
1
/
m
}
⩽
X
m
,
n
⩽
Γ
{
+
⋯
+
(
1
+
A
)
log
⁡
p
(
n
)
}
1
n
{
+
⋯
+
(
1
+
A
)
log
⁡
p
(
n
)
log
⁡
2
(
n
)
+
(
m
+
1
)
log
⁡
3
(
n
)
+
⋯
+
(
1
+
A
)
log
⁡
p
(
n
)
}
1
n
}
.
Moreover, a
.
s., for A
=
0, both members of these two inequalities are not satisfied infinitely many times.Corollary 8.
IfΦ
(
x
)
=
1
-
e
-
x
,
x
>
0, then Γ
(
x
)
=
-
log
⁡
(
1
-
x
), x
<
1.Theorem 9 (Barndorff-Nielsen [13]).
If(14)
lim
⁡
n
→
∞
P
{
X
m
,
n
⩽
λ
n
}
=
0
,
∑
n
n
m
-
1
{
Φ
(
λ
n
)
}
n
-
m
+
1
{
1
-
Φ
(
λ
n
+
1
)
}
m
<
∞
,
then λ
n
≪
X
m
,
n  a
.
s. with {
λ
n
}
n
⩾
1 an increasing sequence of numbers.Lemma 10.
If(
X
i
,
Y
i
)
1
⩽
i
⩽
n is a sequence of i
.
i
.
d pairs ofr.v’s. with joint cumulative distribution function Φ
(
x
,
y
), then
(15)
Φ
(
x
,
y
)
=
Φ
X
(
x
)
+
Φ
Y
(
y
)
+
P
(
X
>
x
,
Y
>
y
)
-
1
.Proof.
The proof is obvious.Theorem 11 (Marshall and Olkin [7]).
If(16)
Φ
X
(
x
)
=
1
-
e
-
x
,
x
>
0
,
Φ
Y
(
y
)
=
1
-
e
-
y
,
y
>
0
,
then P
(
X
>
x
,
Y
>
y
)
=
e
-
x
-
y
-
λ
max
⁡
(
x
,
y
), λ
>
0.Theorem 12 (Galombos [6]).
Let(
u
n
)
n
≥
1 and (
v
n
)
n
≥
1 be two sequences ofr.v’s.; suppose that there exists a distribution function Π
(
x
) such that at any point x of continuity of  Π
(17)
P
(
u
n
⩽
x
)
→
n
→
∞

Π
(
x
)
.
Suppose that for all ϵ
>
0
(18)
P
(
|
v
n
|
>
ϵ
)
→
n
→
∞

0
.
Then lim
⁡
n
→
∞
P
(
u
n
+
v
n
<
x
)
=
Π
(
x
) at all points of continuity of Π.


## 3. Main Result

Throughout the following, we consider(
X
i
,
Y
i
)
1
⩽
i
⩽
n as a sequence of i
.
i
.
d r.v’s. with joint cumulative distribution function H
(
x
,
y
) whose marginals are, respectively, F
(
x
)
=
1
-
e
-
x, x
>
0 and G
(
y
)
=
1
-
e
-
y, y
>
0.


### 3.1. Study of the Number of Coincidences

Let{
n
k
(
j
)
}
k
⩾
1 and {
m
k
(
j
)
}
k
⩾
1 be the sequences of jth record times associated, respectively, to X
i and Y
i, with R
1
(
j
)
=
X
1
,
j and S
1
(
j
)
=
Y
1
,
j dependent, suggesting that every time n
k
(
j
)
=
m
k
′
(
j
), for any k and k
′, there is dependence of the jth corresponding records.Lemma 13.
The number of coincidences, which means the number of timesn
k
(
j
) and m
k
(
j
) belong, respectively, to the sets {
m
1
(
j
)
,
m
2
(
j
)
,
…
,
m
k
(
j
)
,
…
} and {
n
1
(
j
)
,
n
2
(
j
)
,
…
,
n
k
(
j
)
,
…
}, is finite.Proof.
Let us study the nature of the following series(19)
∑
n
‍
P
(
A
n
)
=
∑
n
P
{
X
n
-
j
+
1
,
n
>
X
n
-
j
,
n
-
1
;
Y
n
-
j
+
1
,
n
>
Y
n
-
j
,
n
-
1
}
,
where A
n is the event for which n
k
(
j
)
=
m
k
′
(
j
)
=
n.
As{
X
n
-
j
+
1
,
n
>
X
n
-
j
,
n
-
1
;
Y
n
-
j
+
1
,
n
>
Y
n
-
j
,
n
-
1
} is equivalent to {
X
n
>
X
n
-
j
,
n
-
1
;
Y
n
>
Y
n
-
j
,
n
-
1
}, studying the nature of the series ∑
n
P
(
A
n
) is equivalent to the study of the series ∑
n
P
(
A
n
′
) with
(20)
A
n
′
=
{
X
n
>
X
n
-
j
,
n
-
1
;
Y
n
>
Y
n
-
j
,
n
-
1
}
.
Let (
X
n
)
n
⩾
1 be a sequence of i
.
i
.
d r.v’s. with repartition Φ
(
x
) and (
U
n
)
n
≥
1 the sequence of i
.
i
.
d r.v’s. uniformly distributed on [
0,1
] with
(21)
Γ
(
t
)
=
inf
⁡
{
x
∈
R
∣
Φ
(
x
)
≥
t
}
.
{
Γ
(
U
n
)
}
n
≥
1 and (
X
n
)
n
⩾
1 are identically distributed.
Ana
.
s. bounding of X
m
,
n is equivalent to an a
.
s. bounding of U
m
,
n.
Using Theorems5 and 6, and Corollary 7 of Deheuvels and Theorem 9 of Barndorff-Nielsen we obtain the a.s. bounding of X
n
-
j
+
1
,
n.
Choosingλ
n
=
1
-
(
1
/
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
), ϵ
>
0, we obtain a lower bound of U
n
-
j
+
1.
Indeed(22)
F
U
(
λ
n
)
=
1
-
1
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
,
∑
n
n
m
-
1
(
F
U
(
λ
n
)
)
n
-
m
+
1
(
1
-
F
U
(
λ
n
+
1
)
)
m
=
∑
n
n
n
-
j
(
1
-
1
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
2
)
)
j
×
(
1
(
n
+
1
)
(
log
⁡
n
+
1
)
(
1
+
ϵ
)
/
(
n
-
j
+
2
)
)
n
-
j
+
1
.
Let S
n be the general term of the series in the right side of the above equality. It is clear that
(23)
S
n
⩽
V
n
,
with
(24)
V
n
=
n
n
-
j
n
n
-
j
+
1
1
(
log
⁡
n
)
(
1
+
ϵ
)
(
(
n
-
j
+
1
)
/
(
n
-
j
+
2
)
)
=
1
n
1
(
log
⁡
n
)
(
1
+
ϵ
)
(
(
n
-
j
+
1
)
/
(
n
-
j
+
2
)
)
.
For n large enough V
n
~
W
n with W
n
=
1
/
n
(
log
⁡
n
)
1
+
ϵ is a general term of Bertrand convergent series.
Hence we get(25)
∑
n
‍
S
n
⩽
+
∞
,
which implies that
(26)
U
n
-
j
+
1
,
n
>
1
-
1
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
.
Consequently, we have
(27)
X
n
-
j
+
1
,
n
>
Γ
(
λ
n
)
.
Using Corollary 8, we get Γ
(
λ
n
)
=
-
log
⁡
(
1
-
λ
n
)
=
log
⁡
{
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
}.
Since(28)
{
X
n
>
X
n
-
j
,
n
-
1
,
Y
n
>
Y
n
-
j
,
n
-
1
}
implies
(29)
{
X
n
>
Γ
(
λ
n
)
,
Y
n
>
Γ
(
λ
n
)
}
,
it suffices to study the nature of the series ∑
n
‍
P
(
A
n
′′
) to know that of ∑
n
‍
P
(
A
n
′
), where A
n
′′ is given by
(30)
A
n
′′
=
{
X
n
>
log
⁡
(
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
)
;
Y
n
>
log
⁡
(
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
)
}
.
Using Theorem 11 of Marshall and Olkin, we get
(31)
P
(
X
n
>
Γ
(
λ
n
)
,
Y
n
>
Γ
(
λ
n
)
)
=
1
n
2
+
λ
(
log
⁡
n
)
(
1
+
ϵ
)
(
(
2
+
λ
)
/
(
n
-
j
+
1
)
)
=
K
n
⩽
K
n
′
=
1
n
2
+
λ
,
with ∑
n
K
n
′
<
∞ (Riemann’s series).
Then(32)
∑
n
K
n
′
<
∞
⟹
∑
n
K
n
<
∞
⟹
∑
n
P
(
A
n
′′
)
<
∞
⟹
∑
n
P
(
A
n
′
)
<
∞
⟹
∑
n
P
(
A
n
)
<
∞
.
By Borel-Cantelli’s Lemma,
(33)
∑
n
P
(
A
n
)
<
∞
⟹
lim
⁡
-
n
A
n
=
0
,
which means that with probability one at most a finite number of A
n are realized simultaneously and consequently that the number of coincidences is finite a
.
s.


### 3.2. Some Record Series

Let us consider and give some properties of the following sequences:(34)
R
k
(
j
)
=
∑
i
=
1
k
α
i
(
j
)
with
α
i
(
j
)
=
R
i
+
1
(
j
)
-
R
i
(
j
)
S
k
(
j
)
=
∑
i
=
1
k
β
i
(
j
)
with
β
i
(
j
)
=
S
i
+
1
(
j
)
-
S
i
(
j
)
R
k
*
(
j
)
=
∑
i
=
1
n
‍
α
i
(
j
)
γ
i
with
γ
i
=
{
1
if
n
i
(
j
)
∉
{
m
1
(
j
)
,
…
,
m
k
(
j
)
,
…
}
,
0
otherwise
S
k
*
(
j
)
=
∑
i
=
1
n
‍
β
i
(
j
)
δ
i
with
δ
i
=
{
1
if
m
i
(
j
)
∉
{
n
1
(
j
)
,
…
,
n
k
(
j
)
,
…
}
,
0
otherwise
.Lemma 14.
Consider the following(35)
R
n
(
j
)
-
R
n
*
(
j
)
→
n
→
∞

C
1
<
∞
a
.
s
.Proof.
Consider the following(36)
R
n
(
j
)
-
R
n
*
(
j
)
=
∑
i
=
1
n
‍
α
i
(
j
)
-
∑
i
=
1
n
‍
α
(
j
)
γ
i
=
∑
i
=
1
n
‍
(
1
-
γ
i
)
α
i
(
j
)
=
∑
i
=
1
n
‍
θ
i
α
i
(
j
)
,
with
(37)
θ
i
=
{
1
if
n
i
(
j
)
∈
{
m
1
(
j
)
,
…
,
m
k
(
j
)
,
…
}
,
0
otherwise
.
Hence
(38)
R
n
(
j
)
-
R
n
*
(
j
)
=
∑
i
∈
I
α
i
(
j
)
,
where I is the set of i for which there are coincidences; as this set is finite ∑
i
∈
I
‍
α
i
(
j
)
<
∞. By the same method we have
(39)
S
n
(
j
)
-
S
n
*
(
j
)
→
n
→
∞

C
2
<
∞
a
.
s
.Lemma 15.
Consider(40)
(
R
n
*
(
j
)
-
(
n
/
j
)
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
.Proof.
Consider the following(41)
R
n
*
(
j
)
-
(
n
/
j
)
n
/
j
2
=
R
n
*
(
j
)
-
R
n
(
j
)
+
R
n
(
j
)
-
(
j
/
n
)
n
/
j
2
=
R
n
*
(
j
)
-
R
n
(
j
)
n
/
j
2
+
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
.
Using Corollary 3 and Theorem 12 of Galombos and the central limit theorem we get
(42)
lim
⁡
n
→
∞
P
{
R
n
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
}
=
lim
⁡
n
→
∞
P
{
R
n
*
(
j
)
-
R
n
(
j
)
+
R
n
(
j
)
-
(
j
/
n
)
n
/
j
2
⩽
x
}
=
lim
⁡
n
→
∞
P
{
R
n
*
(
j
)
-
R
n
(
j
)
n
/
j
2
+
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
}
=
lim
⁡
n
→
∞
P
{
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
}
=
1
2
Π
∫
0
x
e
-
u
2
/
2
d
u
=
N
(
x
)
.
By the same method, we have
(43)
S
n
*
(
j
)
-
(
n
/
j
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
.Now, let us consider the two following sequences:(44)
R
n
*
*
(
j
)
=
∑
k
=
1
N
(
n
)
α
k
(
j
)
γ
k
,
S
n
*
*
(
j
)
=
∑
k
=
1
M
(
n
)
‍
β
k
(
j
)
δ
k
with
(45)
N
(
n
)
=
inf
⁡
{
m
⩾
1
∣
∑
k
=
1
m
γ
k
=
n
}
,
M
(
n
)
=
inf
⁡
{
m
⩾
1
∣
∑
k
=
1
m
δ
k
=
n
}
.Lemma 16.
Consider the following(46)
R
n
*
*
(
j
)
-
R
n
*
(
j
)
→
n
→
∞

k
1
<
∞
a
.
s
.Proof.
Since the number of coincidences is finite, there existsn
0 finite, such that, for all n
>
n
0
,
γ
n
=
1;
(47)
R
n
*
*
(
j
)
-
R
n
*
(
j
)
=
∑
k
=
1
N
(
n
)
‍
α
k
(
j
)
γ
k
-
∑
k
=
1
n
‍
α
k
(
j
)
γ
k
=
∑
k
=
1
n
0
‍
α
k
(
j
)
γ
k
+
∑
k
=
n
0
+
1
N
(
n
)
‍
α
k
(
j
)
γ
k
-
∑
k
=
1
n
0
‍
α
k
(
j
)
γ
k
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
γ
k
=
∑
k
=
n
0
+
1
N
(
n
)
‍
α
k
(
j
)
γ
k
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
γ
k
=
∑
k
=
n
0
+
1
N
(
n
)
‍
α
k
(
j
)
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
=
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
+
∑
k
=
n
+
1
N
(
n
)
‍
α
k
(
j
)
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
=
∑
k
=
n
+
1
N
(
n
)
‍
α
k
(
j
)
.
R
n
*
*
(
j
)
-
R
n
*
(
j
) is the sum of (
N
(
n
)
-
n
)  i
.
i
.
d r.v’s.
Between1 and n
0 we have K coincidences and r noncoincidences, so we must have (
n
-
r
) noncoincidences between (
n
0
+
1
) and N
(
n
); hence
(48)
N
(
n
)
=
K
+
r
+
n
-
r
=
K
+
n
.
This implies that N
(
n
)
-
n
=
K with K finite, so R
n
*
*
(
j
)
-
R
n
*
(
j
) is a finite sum ofr.v’s., which proves our result.
By the same method, we have(49)
S
n
*
*
(
j
)
-
S
n
*
(
j
)
→
n
→
∞

k
2
<
∞
a
.
s
.Lemma 17.
R
n
*
*
(
j
) and S
n
*
*
(
j
) are individually the sum of n  i
.
i
.
d exponentialr.v’s. with parameter j so by the central limit theorem one gets
(50)
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
,
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
.


### 3.3. Study of thejth Record Times

Definition 18.
Let(
Ω
,
F
) be a measurable space and (
F
t
)
t
∈
I be an increasing family of sub-σ-algebra of F, defined on an interval I of R
- or N
-.
According to(
F
t
)
t
∈
I  a  r
.
v.  T defined on (
Ω
,
F
) with values in I is called a stopping time if
(51)
{
ω
:
T
(
ω
)
⩽
t
}
∈
F
t
,
∀
t
∈
I
.Interpretation. The notion of stopping time corresponds to the notion of stopping some process at a random time T. Indeed, the stopping time is a function of ω. Also the definition (51) expresses that all the contingencies that lead to the previous stopping time are an event of F
t, that means an event definable by the behavior of the phenomenon before time t.Lemma 19.
The sequence ofjth record times {
n
k
(
j
)
}
k
⩾
1 is a sequence of stopping time.Proof.
Considering the sub-σ-algebra a
n
=
σ
(
X
1
,
Y
1
,
…
,
X
n
,
Y
n
) and a
t
=
σ
(
X
1
,
Y
1
,
…
,
X
t
,
Y
t
) the increasing family of sub-σ-algebra of a
n, n
k
j is a stopping time if and only if
(52)
{
n
k
(
j
)
=
t
}
∈
a
t
∀
t
∈
N
¯
.
But n
k
(
j
)
=
t is equivalent to X
n
k
(
j
)
-
j
+
1
,
n
k
(
j
)
=
X
t
-
j
+
1
,
t
1
{
n
k
(
j
)
=
t
}.
X
t
-
j
+
1
,
t is σ
(
X
1
,
t
,
…
,
X
t
-
j
+
1
,
t
) measurable and included in σ
(
X
1
,
…
,
X
t
) because the ordered statistic is exhaustive.
Butσ
(
X
1
,
…
,
X
t
) is included in σ
(
X
1
,
Y
1
,
…
,
X
t
,
Y
t
) which is equal to a
t.
Consequently(53)
{
n
k
(
j
)
=
t
}
∈
a
t
∀
t
∈
I
⊆
N
¯
,
which implies that n
k
(
j
) is a stopping time.
By the same methodm
k
(
j
) is also a stopping time.


### 3.4. Asymptotic Independence ofR
n
(
j
) and S
n
(
j
)

In this section we will prove the asymptotic independence of{
R
n
(
j
)
}
n
⩾
1 and {
S
n
(
j
)
}
n
⩾
1 sequences of records issued from (
X
i
)
1
≤
i
≤
n and (
Y
i
)
1
≤
i
≤
n, respectively.Proposition 20.
Let(
R
n
(
j
)
,
S
n
(
j
)
)
n
⩾
1 be a bivariater.v’s. of records issued from pairs ofr.v’s.
(
X
n
,
Y
n
)
n
⩾
1 with joint cumulative distribution H
(
x
,
y
) whose marginals F
(
x
) and G
(
y
) are exponential with parameter λ
=
1; then
(54)
lim
⁡
n
→
∞
P
(
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
=
lim
⁡
n
→
∞
P
(
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
)
lim
⁡
n
→
∞
P
(
S
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
=
N
(
x
)
N
(
y
)
,
where N
(
·
) is the standard normal distribution.Proof.
Since there exists ann
0 from which there is no coincidences, throughout the following, we will consider the case where we observe, first, a record R
k
(
j
) for the X
i.
a
n
k
(
j
)
=
{
A
∣
A
∩
n
k
(
j
)
=
t
∈
a
t
∀
t
∈
I
⊆
N
¯
} being the algebra of events prior n
(
j
), all that has led to the appearance of R
k
(
j
) that depends only on what has been observed before n
k
(
j
).
Letn
k
(
j
)
=
M. Given X
1
,
M
,
…
,
X
M
-
j
+
1
,
M
,
Y
1
,
M
,
…
,
Y
M
-
j
+
1
,
M and using Corollary 4, the first record S
i
(
j
)
with
n
i
(
j
)
>
M is such that S
i
(
j
)
-
S
i
-
1
(
j
) is independent of Y
1
,
M
,
…
,
Y
M
-
j
+
1
,
M and fortiori of X
1
,
M
,
…
,
X
M
-
j
+
1
,
M.
Thus,α
i
(
j
)
=
R
i
(
j
)
-
R
i
-
1
(
j
)
,
n
i
(
j
) and β
i
(
j
)
=
S
i
(
j
)
-
S
i
-
1
(
j
)
,
m
i
(
j
)
⩾
M are independent from all what has been observed before n
k
(
j
)
=
M, including R
k
(
j
), which means it is independent of a
M.
We conclude that(i)
for alli such that n
i
(
j
)
⩾
n
k
(
j
)
,
R
i
*
*
(
j
) is asymptotically independent of S
i
1
*
*
(
j
) with m
i
1
(
j
)
<
n
k
(
j
),
(ii)
for allt such that m
t
(
j
)
⩾
n
k
(
j
)
,
S
t
*
*
(
j
) is asymptotically independent of R
i
2
*
*
(
j
) with n
i
2
(
j
)
<
n
k
(
j
)
.This sequential reasoning will be taken each time we have either a record for the X
i or a record for the Y
i.
Thus, we prove thatR
n
*
*
(
j
) and S
n
*
*
(
j
) are asymptotically independent and consequently R
n
(
j
) and S
n
(
j
) will be as well.
Indeed, if we denote, for the sake of simplicity,lim
⁡
n
→
∞
P
(
(
(
R
n
(
j
)
-
(
n
/
j
)
)
/
n
/
j
2
)
⩽
x
,
(
(
S
n
(
j
)
-
(
n
/
j
)
)
/
n
/
j
2
)
⩽
y
) by α,we get
(55)
α
=
lim
⁡
n
→
∞
P
(
R
n
(
j
)
-
R
n
*
(
j
)
+
R
n
*
(
j
)
-
R
n
*
*
(
j
)
+
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
(
j
)
-
S
n
*
(
j
)
+
S
n
*
(
j
)
-
S
n
*
*
(
j
)
+
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
.
Using (35), (39), and Theorem 12
(56)
α
=
lim
⁡
n
→
∞
P
(
R
n
*
(
j
)
-
R
n
*
*
(
j
)
+
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
*
(
j
)
-
S
n
*
*
(
j
)
+
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
.
Using (46), (49), and Theorem 12
(57)
α
=
lim
⁡
n
→
∞
P
(
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
.
Now using (50) we obtain
(58)
α
=
lim
⁡
n
→
∞
P
(
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
)
×
lim
⁡
n
→
∞
P
(
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
=
N
(
x
)
N
(
y
)
,
which proves our result.


## 3.1. Study of the Number of Coincidences

Let{
n
k
(
j
)
}
k
⩾
1 and {
m
k
(
j
)
}
k
⩾
1 be the sequences of jth record times associated, respectively, to X
i and Y
i, with R
1
(
j
)
=
X
1
,
j and S
1
(
j
)
=
Y
1
,
j dependent, suggesting that every time n
k
(
j
)
=
m
k
′
(
j
), for any k and k
′, there is dependence of the jth corresponding records.Lemma 13.
The number of coincidences, which means the number of timesn
k
(
j
) and m
k
(
j
) belong, respectively, to the sets {
m
1
(
j
)
,
m
2
(
j
)
,
…
,
m
k
(
j
)
,
…
} and {
n
1
(
j
)
,
n
2
(
j
)
,
…
,
n
k
(
j
)
,
…
}, is finite.Proof.
Let us study the nature of the following series(19)
∑
n
‍
P
(
A
n
)
=
∑
n
P
{
X
n
-
j
+
1
,
n
>
X
n
-
j
,
n
-
1
;
Y
n
-
j
+
1
,
n
>
Y
n
-
j
,
n
-
1
}
,
where A
n is the event for which n
k
(
j
)
=
m
k
′
(
j
)
=
n.
As{
X
n
-
j
+
1
,
n
>
X
n
-
j
,
n
-
1
;
Y
n
-
j
+
1
,
n
>
Y
n
-
j
,
n
-
1
} is equivalent to {
X
n
>
X
n
-
j
,
n
-
1
;
Y
n
>
Y
n
-
j
,
n
-
1
}, studying the nature of the series ∑
n
P
(
A
n
) is equivalent to the study of the series ∑
n
P
(
A
n
′
) with
(20)
A
n
′
=
{
X
n
>
X
n
-
j
,
n
-
1
;
Y
n
>
Y
n
-
j
,
n
-
1
}
.
Let (
X
n
)
n
⩾
1 be a sequence of i
.
i
.
d r.v’s. with repartition Φ
(
x
) and (
U
n
)
n
≥
1 the sequence of i
.
i
.
d r.v’s. uniformly distributed on [
0,1
] with
(21)
Γ
(
t
)
=
inf
⁡
{
x
∈
R
∣
Φ
(
x
)
≥
t
}
.
{
Γ
(
U
n
)
}
n
≥
1 and (
X
n
)
n
⩾
1 are identically distributed.
Ana
.
s. bounding of X
m
,
n is equivalent to an a
.
s. bounding of U
m
,
n.
Using Theorems5 and 6, and Corollary 7 of Deheuvels and Theorem 9 of Barndorff-Nielsen we obtain the a.s. bounding of X
n
-
j
+
1
,
n.
Choosingλ
n
=
1
-
(
1
/
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
), ϵ
>
0, we obtain a lower bound of U
n
-
j
+
1.
Indeed(22)
F
U
(
λ
n
)
=
1
-
1
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
,
∑
n
n
m
-
1
(
F
U
(
λ
n
)
)
n
-
m
+
1
(
1
-
F
U
(
λ
n
+
1
)
)
m
=
∑
n
n
n
-
j
(
1
-
1
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
2
)
)
j
×
(
1
(
n
+
1
)
(
log
⁡
n
+
1
)
(
1
+
ϵ
)
/
(
n
-
j
+
2
)
)
n
-
j
+
1
.
Let S
n be the general term of the series in the right side of the above equality. It is clear that
(23)
S
n
⩽
V
n
,
with
(24)
V
n
=
n
n
-
j
n
n
-
j
+
1
1
(
log
⁡
n
)
(
1
+
ϵ
)
(
(
n
-
j
+
1
)
/
(
n
-
j
+
2
)
)
=
1
n
1
(
log
⁡
n
)
(
1
+
ϵ
)
(
(
n
-
j
+
1
)
/
(
n
-
j
+
2
)
)
.
For n large enough V
n
~
W
n with W
n
=
1
/
n
(
log
⁡
n
)
1
+
ϵ is a general term of Bertrand convergent series.
Hence we get(25)
∑
n
‍
S
n
⩽
+
∞
,
which implies that
(26)
U
n
-
j
+
1
,
n
>
1
-
1
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
.
Consequently, we have
(27)
X
n
-
j
+
1
,
n
>
Γ
(
λ
n
)
.
Using Corollary 8, we get Γ
(
λ
n
)
=
-
log
⁡
(
1
-
λ
n
)
=
log
⁡
{
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
}.
Since(28)
{
X
n
>
X
n
-
j
,
n
-
1
,
Y
n
>
Y
n
-
j
,
n
-
1
}
implies
(29)
{
X
n
>
Γ
(
λ
n
)
,
Y
n
>
Γ
(
λ
n
)
}
,
it suffices to study the nature of the series ∑
n
‍
P
(
A
n
′′
) to know that of ∑
n
‍
P
(
A
n
′
), where A
n
′′ is given by
(30)
A
n
′′
=
{
X
n
>
log
⁡
(
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
)
;
Y
n
>
log
⁡
(
n
(
log
⁡
n
)
(
1
+
ϵ
)
/
(
n
-
j
+
1
)
)
}
.
Using Theorem 11 of Marshall and Olkin, we get
(31)
P
(
X
n
>
Γ
(
λ
n
)
,
Y
n
>
Γ
(
λ
n
)
)
=
1
n
2
+
λ
(
log
⁡
n
)
(
1
+
ϵ
)
(
(
2
+
λ
)
/
(
n
-
j
+
1
)
)
=
K
n
⩽
K
n
′
=
1
n
2
+
λ
,
with ∑
n
K
n
′
<
∞ (Riemann’s series).
Then(32)
∑
n
K
n
′
<
∞
⟹
∑
n
K
n
<
∞
⟹
∑
n
P
(
A
n
′′
)
<
∞
⟹
∑
n
P
(
A
n
′
)
<
∞
⟹
∑
n
P
(
A
n
)
<
∞
.
By Borel-Cantelli’s Lemma,
(33)
∑
n
P
(
A
n
)
<
∞
⟹
lim
⁡
-
n
A
n
=
0
,
which means that with probability one at most a finite number of A
n are realized simultaneously and consequently that the number of coincidences is finite a
.
s.


## 3.2. Some Record Series

Let us consider and give some properties of the following sequences:(34)
R
k
(
j
)
=
∑
i
=
1
k
α
i
(
j
)
with
α
i
(
j
)
=
R
i
+
1
(
j
)
-
R
i
(
j
)
S
k
(
j
)
=
∑
i
=
1
k
β
i
(
j
)
with
β
i
(
j
)
=
S
i
+
1
(
j
)
-
S
i
(
j
)
R
k
*
(
j
)
=
∑
i
=
1
n
‍
α
i
(
j
)
γ
i
with
γ
i
=
{
1
if
n
i
(
j
)
∉
{
m
1
(
j
)
,
…
,
m
k
(
j
)
,
…
}
,
0
otherwise
S
k
*
(
j
)
=
∑
i
=
1
n
‍
β
i
(
j
)
δ
i
with
δ
i
=
{
1
if
m
i
(
j
)
∉
{
n
1
(
j
)
,
…
,
n
k
(
j
)
,
…
}
,
0
otherwise
.Lemma 14.
Consider the following(35)
R
n
(
j
)
-
R
n
*
(
j
)
→
n
→
∞

C
1
<
∞
a
.
s
.Proof.
Consider the following(36)
R
n
(
j
)
-
R
n
*
(
j
)
=
∑
i
=
1
n
‍
α
i
(
j
)
-
∑
i
=
1
n
‍
α
(
j
)
γ
i
=
∑
i
=
1
n
‍
(
1
-
γ
i
)
α
i
(
j
)
=
∑
i
=
1
n
‍
θ
i
α
i
(
j
)
,
with
(37)
θ
i
=
{
1
if
n
i
(
j
)
∈
{
m
1
(
j
)
,
…
,
m
k
(
j
)
,
…
}
,
0
otherwise
.
Hence
(38)
R
n
(
j
)
-
R
n
*
(
j
)
=
∑
i
∈
I
α
i
(
j
)
,
where I is the set of i for which there are coincidences; as this set is finite ∑
i
∈
I
‍
α
i
(
j
)
<
∞. By the same method we have
(39)
S
n
(
j
)
-
S
n
*
(
j
)
→
n
→
∞

C
2
<
∞
a
.
s
.Lemma 15.
Consider(40)
(
R
n
*
(
j
)
-
(
n
/
j
)
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
.Proof.
Consider the following(41)
R
n
*
(
j
)
-
(
n
/
j
)
n
/
j
2
=
R
n
*
(
j
)
-
R
n
(
j
)
+
R
n
(
j
)
-
(
j
/
n
)
n
/
j
2
=
R
n
*
(
j
)
-
R
n
(
j
)
n
/
j
2
+
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
.
Using Corollary 3 and Theorem 12 of Galombos and the central limit theorem we get
(42)
lim
⁡
n
→
∞
P
{
R
n
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
}
=
lim
⁡
n
→
∞
P
{
R
n
*
(
j
)
-
R
n
(
j
)
+
R
n
(
j
)
-
(
j
/
n
)
n
/
j
2
⩽
x
}
=
lim
⁡
n
→
∞
P
{
R
n
*
(
j
)
-
R
n
(
j
)
n
/
j
2
+
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
}
=
lim
⁡
n
→
∞
P
{
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
}
=
1
2
Π
∫
0
x
e
-
u
2
/
2
d
u
=
N
(
x
)
.
By the same method, we have
(43)
S
n
*
(
j
)
-
(
n
/
j
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
.Now, let us consider the two following sequences:(44)
R
n
*
*
(
j
)
=
∑
k
=
1
N
(
n
)
α
k
(
j
)
γ
k
,
S
n
*
*
(
j
)
=
∑
k
=
1
M
(
n
)
‍
β
k
(
j
)
δ
k
with
(45)
N
(
n
)
=
inf
⁡
{
m
⩾
1
∣
∑
k
=
1
m
γ
k
=
n
}
,
M
(
n
)
=
inf
⁡
{
m
⩾
1
∣
∑
k
=
1
m
δ
k
=
n
}
.Lemma 16.
Consider the following(46)
R
n
*
*
(
j
)
-
R
n
*
(
j
)
→
n
→
∞

k
1
<
∞
a
.
s
.Proof.
Since the number of coincidences is finite, there existsn
0 finite, such that, for all n
>
n
0
,
γ
n
=
1;
(47)
R
n
*
*
(
j
)
-
R
n
*
(
j
)
=
∑
k
=
1
N
(
n
)
‍
α
k
(
j
)
γ
k
-
∑
k
=
1
n
‍
α
k
(
j
)
γ
k
=
∑
k
=
1
n
0
‍
α
k
(
j
)
γ
k
+
∑
k
=
n
0
+
1
N
(
n
)
‍
α
k
(
j
)
γ
k
-
∑
k
=
1
n
0
‍
α
k
(
j
)
γ
k
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
γ
k
=
∑
k
=
n
0
+
1
N
(
n
)
‍
α
k
(
j
)
γ
k
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
γ
k
=
∑
k
=
n
0
+
1
N
(
n
)
‍
α
k
(
j
)
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
=
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
+
∑
k
=
n
+
1
N
(
n
)
‍
α
k
(
j
)
-
∑
k
=
n
0
+
1
n
‍
α
k
(
j
)
=
∑
k
=
n
+
1
N
(
n
)
‍
α
k
(
j
)
.
R
n
*
*
(
j
)
-
R
n
*
(
j
) is the sum of (
N
(
n
)
-
n
)  i
.
i
.
d r.v’s.
Between1 and n
0 we have K coincidences and r noncoincidences, so we must have (
n
-
r
) noncoincidences between (
n
0
+
1
) and N
(
n
); hence
(48)
N
(
n
)
=
K
+
r
+
n
-
r
=
K
+
n
.
This implies that N
(
n
)
-
n
=
K with K finite, so R
n
*
*
(
j
)
-
R
n
*
(
j
) is a finite sum ofr.v’s., which proves our result.
By the same method, we have(49)
S
n
*
*
(
j
)
-
S
n
*
(
j
)
→
n
→
∞

k
2
<
∞
a
.
s
.Lemma 17.
R
n
*
*
(
j
) and S
n
*
*
(
j
) are individually the sum of n  i
.
i
.
d exponentialr.v’s. with parameter j so by the central limit theorem one gets
(50)
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
,
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
→
n
→
∞
L
N
(
0,1
)
.


## 3.3. Study of thejth Record Times

Definition 18.
Let(
Ω
,
F
) be a measurable space and (
F
t
)
t
∈
I be an increasing family of sub-σ-algebra of F, defined on an interval I of R
- or N
-.
According to(
F
t
)
t
∈
I  a  r
.
v.  T defined on (
Ω
,
F
) with values in I is called a stopping time if
(51)
{
ω
:
T
(
ω
)
⩽
t
}
∈
F
t
,
∀
t
∈
I
.Interpretation. The notion of stopping time corresponds to the notion of stopping some process at a random time T. Indeed, the stopping time is a function of ω. Also the definition (51) expresses that all the contingencies that lead to the previous stopping time are an event of F
t, that means an event definable by the behavior of the phenomenon before time t.Lemma 19.
The sequence ofjth record times {
n
k
(
j
)
}
k
⩾
1 is a sequence of stopping time.Proof.
Considering the sub-σ-algebra a
n
=
σ
(
X
1
,
Y
1
,
…
,
X
n
,
Y
n
) and a
t
=
σ
(
X
1
,
Y
1
,
…
,
X
t
,
Y
t
) the increasing family of sub-σ-algebra of a
n, n
k
j is a stopping time if and only if
(52)
{
n
k
(
j
)
=
t
}
∈
a
t
∀
t
∈
N
¯
.
But n
k
(
j
)
=
t is equivalent to X
n
k
(
j
)
-
j
+
1
,
n
k
(
j
)
=
X
t
-
j
+
1
,
t
1
{
n
k
(
j
)
=
t
}.
X
t
-
j
+
1
,
t is σ
(
X
1
,
t
,
…
,
X
t
-
j
+
1
,
t
) measurable and included in σ
(
X
1
,
…
,
X
t
) because the ordered statistic is exhaustive.
Butσ
(
X
1
,
…
,
X
t
) is included in σ
(
X
1
,
Y
1
,
…
,
X
t
,
Y
t
) which is equal to a
t.
Consequently(53)
{
n
k
(
j
)
=
t
}
∈
a
t
∀
t
∈
I
⊆
N
¯
,
which implies that n
k
(
j
) is a stopping time.
By the same methodm
k
(
j
) is also a stopping time.


## 3.4. Asymptotic Independence ofR
n
(
j
) and S
n
(
j
)

In this section we will prove the asymptotic independence of{
R
n
(
j
)
}
n
⩾
1 and {
S
n
(
j
)
}
n
⩾
1 sequences of records issued from (
X
i
)
1
≤
i
≤
n and (
Y
i
)
1
≤
i
≤
n, respectively.Proposition 20.
Let(
R
n
(
j
)
,
S
n
(
j
)
)
n
⩾
1 be a bivariater.v’s. of records issued from pairs ofr.v’s.
(
X
n
,
Y
n
)
n
⩾
1 with joint cumulative distribution H
(
x
,
y
) whose marginals F
(
x
) and G
(
y
) are exponential with parameter λ
=
1; then
(54)
lim
⁡
n
→
∞
P
(
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
=
lim
⁡
n
→
∞
P
(
R
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
)
lim
⁡
n
→
∞
P
(
S
n
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
=
N
(
x
)
N
(
y
)
,
where N
(
·
) is the standard normal distribution.Proof.
Since there exists ann
0 from which there is no coincidences, throughout the following, we will consider the case where we observe, first, a record R
k
(
j
) for the X
i.
a
n
k
(
j
)
=
{
A
∣
A
∩
n
k
(
j
)
=
t
∈
a
t
∀
t
∈
I
⊆
N
¯
} being the algebra of events prior n
(
j
), all that has led to the appearance of R
k
(
j
) that depends only on what has been observed before n
k
(
j
).
Letn
k
(
j
)
=
M. Given X
1
,
M
,
…
,
X
M
-
j
+
1
,
M
,
Y
1
,
M
,
…
,
Y
M
-
j
+
1
,
M and using Corollary 4, the first record S
i
(
j
)
with
n
i
(
j
)
>
M is such that S
i
(
j
)
-
S
i
-
1
(
j
) is independent of Y
1
,
M
,
…
,
Y
M
-
j
+
1
,
M and fortiori of X
1
,
M
,
…
,
X
M
-
j
+
1
,
M.
Thus,α
i
(
j
)
=
R
i
(
j
)
-
R
i
-
1
(
j
)
,
n
i
(
j
) and β
i
(
j
)
=
S
i
(
j
)
-
S
i
-
1
(
j
)
,
m
i
(
j
)
⩾
M are independent from all what has been observed before n
k
(
j
)
=
M, including R
k
(
j
), which means it is independent of a
M.
We conclude that(i)
for alli such that n
i
(
j
)
⩾
n
k
(
j
)
,
R
i
*
*
(
j
) is asymptotically independent of S
i
1
*
*
(
j
) with m
i
1
(
j
)
<
n
k
(
j
),
(ii)
for allt such that m
t
(
j
)
⩾
n
k
(
j
)
,
S
t
*
*
(
j
) is asymptotically independent of R
i
2
*
*
(
j
) with n
i
2
(
j
)
<
n
k
(
j
)
.This sequential reasoning will be taken each time we have either a record for the X
i or a record for the Y
i.
Thus, we prove thatR
n
*
*
(
j
) and S
n
*
*
(
j
) are asymptotically independent and consequently R
n
(
j
) and S
n
(
j
) will be as well.
Indeed, if we denote, for the sake of simplicity,lim
⁡
n
→
∞
P
(
(
(
R
n
(
j
)
-
(
n
/
j
)
)
/
n
/
j
2
)
⩽
x
,
(
(
S
n
(
j
)
-
(
n
/
j
)
)
/
n
/
j
2
)
⩽
y
) by α,we get
(55)
α
=
lim
⁡
n
→
∞
P
(
R
n
(
j
)
-
R
n
*
(
j
)
+
R
n
*
(
j
)
-
R
n
*
*
(
j
)
+
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
(
j
)
-
S
n
*
(
j
)
+
S
n
*
(
j
)
-
S
n
*
*
(
j
)
+
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
.
Using (35), (39), and Theorem 12
(56)
α
=
lim
⁡
n
→
∞
P
(
R
n
*
(
j
)
-
R
n
*
*
(
j
)
+
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
*
(
j
)
-
S
n
*
*
(
j
)
+
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
.
Using (46), (49), and Theorem 12
(57)
α
=
lim
⁡
n
→
∞
P
(
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
,
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
.
Now using (50) we obtain
(58)
α
=
lim
⁡
n
→
∞
P
(
R
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
x
)
×
lim
⁡
n
→
∞
P
(
S
n
*
*
(
j
)
-
(
n
/
j
)
n
/
j
2
⩽
y
)
=
N
(
x
)
N
(
y
)
,
which proves our result.


## 4. Conclusion

In this paper we proved the asymptotic independence of thejth records R
k
(
j
) and S
k
(
j
) in the exponential case (λ
=
1), from sequences (
X
i
)
1
⩽
i
⩽
n and (
Y
i
)
1
⩽
i
⩽
n, respectively. In this case we know that X
n
,
n
=
max
⁡
1
⩽
i
⩽
n
(
X
i
) and Y
n
,
n
=
max
⁡
1
⩽
i
⩽
n
(
Y
i
) are the type Gumbell, respectively. It would be interesting to see if we can obtain the same result when X
n
,
n and Y
n
,
n are Frechet or Weibull type.



---

*Source: 458914-2014-07-08.xml*