The stopped clock model

Abstract

The extreme value theory presents specific tools for modeling and predicting extreme phenomena. In particular, risk assessment is often analyzed through measures for tail dependence and high values clustering. Despite technological advances allowing an increasingly larger and more efficient data collection, there are sometimes failures in the records, which causes difficulties in statistical inference, especially in the tail where data are scarcer. In this article, we present a model with a simple and intuitive failures scheme, where each record failure is replaced by the last record available. We will study its extremal behavior with regard to local dependence and high values clustering, as well as the temporal dependence on the tail.

Full text

Research Article
Helena Ferreira and Marta Ferreira*
The stopped clock model
https://doi.org/10.1515/demo-2022-0101
received January 7, 2022; accepted March 9, 2022
Abstract: The extreme value theory presents specific tools for modeling and predicting extreme phe-
nomena. In particular, risk assessment is often analyzed through measures for tail dependence and high
values clustering. Despite technological advances allowing an increasingly larger and more efficient data
collection, there are sometimes failures in the records, which causes difficulties in statistical inference,
especially in the tail where data are scarcer. In this article, we present a model with a simple and intuitive
failures scheme, where each record failure is replaced by the last record available. We will study its
extremal behavior with regard to local dependence and high values clustering, as well as the temporal
dependence on the tail.
Keywords: extreme values, stationary sequences, failures model, extremal index, tail dependence coefficient
MSC 2020: 60G70
1 Introduction
Let 
{
}∈
Xnn and 
{
}∈
Unn be stationary sequences of real random variables on the probability space (
)
PΩ, ,
and
({})∈=
P
U0, 1
1
n.Wedefine, for
≥n
1
,
⎧
⎨
⎩
===
−
YXU
YU
,1
,0.
nnn
nn1
(1)
Sequence
{
}
≥
Y
nn
1
corresponds to a model of failures on records of 
{
}∈
Xnn replaced by the last available
record, which occurs in some random past instant, if we interpret
n
as time. Thus, if, for example, it occurs
{
}=======UU UU U U U1, 0, 1, 0, 0, 0, 1
12 34 5 6 7
, we will have
{
=====YXYXYXYXYX,,,,,
1121334353
}
==YXYX,
6377
. This constancy of some variables of 
{
}∈
Xnn for random periods of time motivates the
designation of “stopped clock model”for sequence
{
}
≥
Y
nn
1
.
Failure models studied in the literature from the point of view of extremal behavior do not consider the
stopped clock model (Hall and Hüsler [4]; Ferreira et al. [3]and references therein).
The model we will study can also be represented by
{
}
≥
X
Nn
1
n, where
{
}
≥
N
nn
1
is a sequence of positive
integer variables representable by ⎛⎝⎜()
⎞⎠⎟()
∑∏
=+ − − ≥
−≥≥ =−−−
N
nU U U n i n1,1
.
nn
ni j
i
nj ni
11 0
1
We can also state a recursive formulation for
{
}
≥
Y
nn
1
through
Helena Ferreira: Universidade da Beira Interior, Centro de Matemática e Aplicações (CMA-UBI), Departamento de Matemática,
Avenida Marquês d’Avila e Bolama, 6200-001 Covilhã, Portugal, e-mail: helena.ferreira@ubi.pt

* Corresponding author: Marta Ferreira, Centro de Matemática e Departamento de Matemática, Universidade do Minho, CEMAT,
Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal; Statistics and Applications Center (CEAUL), University of
Lisbon, Lisbon, Portugal, e-mail: msferreira@math.uminho.pt
Dependence Modeling 2022; 10: 48–57
Open Access. © 2022 Helena Ferreira and Marta Ferreira, published by De Gruyter. This work is licensed under the Creative
Commons Attribution 4.0 International License.

⎛⎝⎜()
⎞⎠⎟()
∑∏∏
=+ − +− ≥≥
−≥≥ =−−−−≥−−
YXU U UX UY n κ11,1,1
.
nnn
ni j
i
nj nini ini nκ
11 0
1
0
Under any of the three possible representations (failures model, random index sequence or recursive
sequence), we are not aware of an extremal behavior study of
{
}
≥
Y
nn
1
in the literature.
Our departure hypotheses about the base sequence 
{
}∈
Xnn and about sequence 
{
}∈
Unn are as follows:
(1)
{
}∈
Xnn is a stationary sequence of random variables almost surely distinct and, without loss of gen-
erality, such that
() () (
)
≔=−/
F
xFx xexp 1
X
n
,>
x
0, i.e., standard Fréchet distributed.
(2)
{
}∈
Xnn and 
{
}∈
Unn are independent.
(3)
{
}∈
Unn is stationary and ()
(
)
…≔ =…=
…
p
iiPUiUi,, ,,
nn s n ns,, 1 1
ss
11,{}∈i0, 1
j,
=…
js
1, ,
, is such that
()…=
+…+−
p
0, ,0
0
nn n κ,1,, 1
, for some
≥
κ1
.
The trivial case =
κ1
corresponds to =YX
nn
,
≥n
1
. Hypothesis (3)means that we are assuming that it is
almost impossible to lose
κ
or more consecutive values of 
{
}∈
Xnn . We remark that, along the paper, the
summations, produts and intersections is considered to be nonexistent whenever the end of the counter is
less than the beginning. We will also use notation
(
)
∨=
a
babmax ,
.
Example 1.1. Consider an independent and identically distributed sequence

{
}
∈
W
nn
of real random vari-
ables on (
)
PΩ, , and a Borelian set
A
. Let
()=
p
PA
n
, where {
}
=∈AWA
nn
,∈n. The sequence of
Bernoulli random variables

{} {}
{}
()
=+− ∈
⋂⋂
=−−=−−
U
n1111,
,
nAA
A
i
κni i
κni n
1111(2)
where
{}⋅
1
denotes the indicator function, defined for some fixed
≥
κ2
, is such that
()…=
+…+−
p
0, ,0 0
nn n κ,1,, 1
,
i.e., it is almost sure that after −
κ1
consecutive variables equal to zero, the next variable takes value one. We
also have
⎜⎟
⎜⎟
() ( ) ⎛⎝⎞⎠
() ⎛⎝⎞⎠()
{}{}
====⋃∩
=−⋂=−−−
⋂=−−
=−−
=−−
pP PAA
PA P A p p
1100,0
11,
nAAi
κ
ni n
ni
κ
ni κ
1
1
0
1
i
κni n
11
0 20406080100
0 20 40 60 80 100 120
Figure 1: Sample path of 100 observations simulated from
{}
Yndefined in (1)based on independent standard Fréchet
{}
X
n
and on
{
}U
ngiven in (2)where we take random variables
{
}W
n
standard exponential distributed, ]]=/
A
0, 1 2 , and thus, =p0.393
5
and
considering =κ3.
The stopped clock model 49

since the independence of random variables
W
n
implies the independence of events
A
n
, and, for >
κ
2,
⎜⎟
⎜⎟
⎜⎟⎜ ⎟
⎜⎟
⎜⎟
()⎛⎝⎛⎝⎞⎠⎛⎝⎞⎠⎞⎠
()
⎛⎝⎛⎝⎞⎠⎞⎠
()( ) ⎛⎝⎞⎠()()
=⋃∩∩⋃ ∩
=∩−∩∩⋂∪⋂
=−⋂=−−−
−=−−=−−− −
−−
=−−=−−−
−=−−
pPAAAA
PA A PA A A A
PA PA P A p p
0, 0
11,
nn i
κ
ni n i
κ
ni n
nn nn i
κ
ni i
κ
ni
nn i
κ
ni κ
1, 1
1
1
1
11
11
1
1
1
1
1
10
12
() () ()(
)
=− =−
−−
p
pp pp1, 0 0 0, 0 1
nn n nn1, 1, .
In Figure 1, we illustrate with a particular example based on independent standard Fréchet 
{
}∈
Xnn ,

{
}
∈
W
nn
with standard exponential marginals,
]]=/A0, 1 2
and thus, =
p
0.393
5
and considering
=
κ
3
.
Therefore, ()=
++
p
0, 0, 0
0
nn n,1,2
,() ()()==−
++ +
p
ppp1, 0, 0 0, 0 1
nn n nn,1,2 ,1
2
.
In the next section, we propose an estimator for probabilities
()…
…+
p
1, 0, ,0
nns,,
,≤<−sκ0
1
.In
Section 3, we analyze the existence of the extremal index for
{
}
≥
Y
nn
1
, an important measure to evaluate
the tendency to occur clusters of its high values (see, e.g., Kulik and Solier [6], and references therein).
A characterization of the tail dependence will be presented in Section 4. The results are illustrated with an
ARMAX sequence.
For the sake of simplicity, we will omit the variation of
n
in sequence notation whenever there is no
doubt, taking into account that we will keep the designation
{}
Ynfor the stopped clock model and
{}
X
n
and
{
}U
n
for the sequences that generate it.
2 Inference on
{}
Un
Assuming that
{
}U
n
is not observable, as well as the values of
{}
X
n
that are lost, it is of interest to retrieve
information about these sequences from the available sequence
{}
Yn.
Since, for
≥n
1
and ≥
s1
, we have
()()()() ( )( )
{} {} { }
== …=
≠=−−+… ≠==…=
−− −− − −+
p
EpE p E11 11,0 and 1,0,,0 ,
n Y Y n Y Y nsns n Y Y Y Y,1,,
nn nn ns ns ns n11 11
we propose to estimate these probabilities from the respective empirical counterparts of a random sample
()…YY Y
ˆ,ˆ,,
ˆ
m12
from
{}
Yn, i.e.,


() ()
()
{} {}
{}
∑∑
∑
==
…=
=≠==
−−+… =+ ≠= =…=
−−
−− − −+
pmpm
pm
11
1
11,0
1and
1, 0, ,0 1,
ni
m
YY ni
m
YY
nsns n is
m
YYY Y
2
ˆˆ
2
ˆˆ
,1,, 2
ˆˆˆ ˆ
ii ii
is is is i
11
11
which are consistent by the weak law of large numbers. The value of
κ
can be inferred from

{}
=⋁⋁
=+ ≥ ≠= =…=
−− − −+
κ
s1
.
is
m
sYYY Y
21 ˆˆˆ ˆ
is is is i11
In order to evaluate the finite sample behavior of the aforementioned estimators, we have simulated
1,000 independent replicas with size =m10
0
, 1,000, 5,000 of the model in Example 1.1. The absolute
bias (abias)and root mean squared error (rmse)are presented in Table 1. The results reveal a good
performance of the estimators, even in the case of smaller sample sizes. Parameter
κ
was always estimated
with no error.
50 Helena Ferreira and Marta Ferreira

3 The extremal index of
{}
Yn
The sequence
{}
Ynis stationary because the sequences
{}
X
n
and
{
}U
n
are stationary and independent from
each other. In addition, the common distribution for
Y
n
,
≥n
1
, is also standard Fréchet, as is the common
distribution for
X
n
, since
⎜⎟
() ( ) ( )( )
()
⎛⎝() ( )⎞⎠()
∑∑
=≤===…=+≤=
=+ …=
=−−−−+
=−−…
F x PX x U U U PX xPU
Fx p p Fx
,1, 0 1
11,0,,0.
Yi
κ
ni ni ni n n n
ni
κ
ni n
1
1
1
1
1
,,
n
For any >
τ0
,ifwedefine ()≡=/
u
uτ n
τ
nn ,
≥n
1
, it turns out that ()
{}
(∑)=>⟶
=>→∞
EnPYu
τ
1
i
nYu nn
11
in and
()>⟶
→∞
nP X u
τ
nn
1
, so we refer to these levels
u
n
by normalized levels for
{}
Ynand
{}
X
n
.
In this section, in addition to the general assumptions about the model presented in Section 1, we start
by assuming that
{}
X
n
and
{
}U
n
present dependency structures such that variables sufficiently apart can be
considered approximately independent. Concretely, we assume that
{
}U
n
satisfies the strong-mixing con-
dition (Rosenblatt [9]) and
{}
X
n
satisfies condition (
)
Du
n(Leadbetter [7]) for normalized levels
u
n
.
Proposition 3.1. If
{
}U
n
is strong-mixing and
{}
X
n
satisfies condition (
)
Du
n,then
{}
Ynalso satisfies condi-
tion (
)
Du
n.
Proof. For any choice of +
p
qintegers,
≤ <…< < <…< ≤iijj
n1
pq
11
such that
≥+
j
il
p
1
, we have that
⎜⎟⎜⎟⎜⎟
⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠
⋂≤⋂≤−⋂≤ ⋂≤ ≤
== = =
PXu Xu PXuPXu α,,
s
p
in
s
q
jn s
p
in
s
q
jn nl
11 1 1 ,
ssss
with
→
α0
nl,
n,as
→∞n
, for some sequence (
)
=
l
on
n, and
∣( ) ()()∣ ()∩− ≤PA B PAPB gl,
with
()→gl 0
,as
→∞
l
, where
A
belongs to the
σ
-algebra generated by
{
}=…Ui i,1,,
ip
and
B
belongs to
the
σ
-algebra generated by
{
}=+…Ui jj,,1,
i11 . Thus, for any choice of +
p
qintegers, ≤<…<<ii
1
p1
<…< ≤
j
jn
q1
such that
≥++
j
ilκ
p
1, we will have
⎜⎟⎜⎟⎜⎟
⎜⎟⎜⎟⎜⎟
⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠
⎛⎝⎞⎠()
⎛⎝⎞⎠⎛⎝⎞⎠()()
∑
⋂≤⋂≤−⋂≤ ⋂≤
≤⋂≤⋂≤∩−⋂≤⋂≤
== = =
−<≤
−<≤ == ∗∗ ==∗∗
∗∗∗∗ ∗ ∗
PYu Yu PYuPYu
PXu XuPAB PXuPXuPAPB
,
,,
s
p
in
s
q
jn s
p
in
s
q
jn
iκii
jκj j
s
p
in
s
q
jn s
p
in
s
q
jn
11 1 1
11 1 1
ssss
sss
sss
ssss
Table 1: The absolute bias (abias)and rmse obtained from 1,000 simulated samples with size =m100, 1,000, 5,000 of the
model in Example 1.1
abias rmse
()p0
n=m100 0.0272 0.0335
=m1,000
0.0087 0.0108
=m5,000
0.0039 0.0048
(
)
p1, 0
nn−1, =m100 0.0199 0.0253
=m1,000
0.0065 0.0080
=m5,000
0.0030 0.0037
(
)
p1, 0, 0
nnn−2, −1, =m100 0.0160 0.0200
=m1,000
0.0051 0.0064
=m5,000
0.0022 0.0028
The stopped clock model 51

where
{
}
=⋂ = =…= =
∗=+
∗∗
AU UU0,1
s
piii
11
sss
and {
}
=⋂ = =…= =
∗=+
∗∗
B
UUU0,1
s
qjjj
11
sssand
>−≥+
∗∗
j
jκil
p
11
.
Therefore, the aforementioned summation is upper limited by
⎜⎟⎜⎟⎜⎟
⎛⎝⎜⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠∣( ) ()()∣
⎞⎠⎟
(())
∑∑
⋂≤⋂≤−⋂≤ ⋂≤ + ∩−
≤+
−<≤
−<≤ == = = ∗∗ ∗∗
−<≤
−<≤
∗∗∗∗ ∗ ∗
∗∗
PXu Xu PXuPXu PAB PAPB
αgl
,
,
iκii
jκj j
s
p
in
s
q
jn s
p
in
s
q
jn
iκii
jκj j
nl
11 1 1
,
sss
sss
ssss
sss
sss
which allows to conclude that (
)
Du
nholds for
{}
Ynwith ()=+
l
lκ
nYn.□
The tendency for clustering of values of
{}
Ynabove
u
n
depends on the same tendency within
{}
X
n
and the
propensity of
{
}U
n
for consecutive null values. The clustering tendency can be assessed through the
extremal index (Leadbetter, [7]). More precisely,
{}
X
n
is said to have extremal index
(]∈θ0, 1
X
if
⎜⎟
⎛⎝⎞⎠
⋁≤/=
→∞ =−
PXnτ elim
.
n
i
n
iθτ
1
X(3)
If (
)
Du
nholds for
{}
X
n
, we have
⎜
⎟
⎜⎟
⎛⎝⎞⎠⎛⎝
⎞⎠
[]
⋁≤ = ⋁≤
→∞ =→∞ =/
PXu P Xulim lim
n
i
n
inn
k
i
nk
in
11
nn
for any integers sequence
{}
k
n
, such that,
→∞ /→ → →∞kkln kα n,0and0,as
.
nnn nnl,n(4)
We can therefore say that
⎜⎟
⎛⎝⎞⎠
[]
=⋁>
→∞ =/
θτ kP X ulim
.
Xnni
nk
in
1
n
Now we compare the local behavior of sequences
{}
X
n
and
{}
Yn,i.e.,of
X
i
and
Y
i
for
()
⎡
⎣⎤
⎦⎡
⎣⎤
⎦
{
}
∈− +…ij j11,,
n
kn
k
nn
,
=…
j
k1, ,
n
, with regard to the oscillations of their values in relation to
u
n
. To this end, we will use local
dependency conditions (
)
()
Du
sn. We say that
{}
X
n
satisfies (
)
()
Du
sn,
≥
s
2
, whenever
()
[]
∑
>≤<=
→∞ =/+
nPXuXuXlim , 0,
n
js
nk
nj n j11
n
for some integers sequence
{}
k
n
satisfying (4). Condition
(
)
()
Du
n
1translates into
()
[]
∑
>>=
→∞ =/
nPXuXulim , 0
.
n
j
nk
nj n
21
n
Observe that if (
)
()
Du
snholds for some ≥
s1
, then
()
()
Du
mn
also holds for >ms
. Condition
(
)
()
Du
n
1is known
as ()
′
Du
n(Leadbetter et al. [8]) and relates to a unit extremal index, i.e., absence of extreme values
clustering. In particular, this is the case of independent variables. Although
{}
X
n
satisfies ()
′
Du
n, this
condition is not generally valid for
{}
Yn. Observe that
()
()( )
[]
[]()
∑∑∑∑
>>
=>>⋅……
=/
=− =
/=∨−+ …+…
∗∗∗∗
nPYuYu
nPXuXup
,
, 0, ,0, 1, 0, ,0 .
j
nk
nj n
iκj
nk
jijκ
j
inj ni jj j
21
2
1
21 ,,1,, 1,,
n
n
For =i
1
and
=
j
κ
, we have
=
∗
j1
and the corresponding term becomes ()>→>nP X u τ
0
n1,as
→∞n
,
and this is the reason why, in general,
{}
Yndoes not satisfy ()
′
Du
neven if
{}
X
n
satisfies it.
52 Helena Ferreira and Marta Ferreira

Proposition 3.2. The following statements hold:
(i)If
{}
Ynsatisfies (
)
()
Du
sn,
≥
s
2
,then
{}
X
n
satisfies (
)
()
Du
sn.
(ii)If
{}
X
n
satisfies (
)
()
Du
sn,
≥
s
2
,then
{}
Ynsatisfies (
)
()+−
Du
sκ n
1.
(iii)If
{}
X
n
satisfies ()
′
Du
n,then
{}
Ynsatisfies (
)
()
Du
n
2.
Proof. Consider []=/rnk
nn
. We have that
()
()
()()
()( )
∑∑∑
∑∑∑
>≤<
=>≤<===…==
=>≤<⋅……
=+
=− = ++ +
=− = =+∨−+ ++…+…+
∗∗∗∗
nPYuYu Y
nPXuYu X U U UU
nPXuXuXp
,
,,1,0,1
, 1, 0, ,0, 1, 0, ,0, 1 .
js
r
nj n j
iκjs
r
injnj i i j
iκjs
r
ji jκ
j
inj nj ii jj jj
11
2
1
11 11
2
1
11 1 , 1, , 1, , 1, , , 1
n
n
n
(5)
Since
{}
Ynsatisfies (
)
()
Du
sn, with
≥
s
2
, and thus, the first summation in (5)converges to zero, as
→∞n
,
then all the terms in the last summations also converge to zero. In particular, when =i
1
and
=
∗
jj
, we have
()∑>≤<→
=+
nPXuXuX,
0
js
rnj n j11
n,as
→∞n
, which proves (i).
Conversely, writing the first summation in (5)with
j
starting at +−
s
κ
1
, we have
()
()()
()
()
∑∑∑∑
∑∑∑∑
>≤<
=>≤<⋅……
=>≤…≤>
⋅……
=+− +
=− =+− =−+ ++…+…+
=− =+− =−+= +
+… +…+
∗∗∗∗
∗∗∗ ∗∗
∗∗
nPYuYuY
nPXuXuXp
nPXuXuXuXu
p
,
, 1, 0, ,0, 1, 0, ,0, 1
,,,,
1,0, ,0,1,0, ,0,1,
jsκ
r
nj n j
iκjsκ
r
jjκ
j
inj nj ii jj jj
iκjsκ
r
jjκ
j
ij
j
inj n i nj n
ii j j jj
111
2
1
11 1 , 1, , 1, , 1, , , 1
2
1
11 1
,1,,1,, 1,,,1
n
n
n
(6)
where the least of distances between iand ∗
icorresponds to the case =i
1
and
==
∗∗
ijs
. Therefore, if
{}
X
n
satisfies (
)
()
Du
snfor some
≥
s
2
, then each term of (6)converges to zero, as
→∞n
, and thus,
{}
Ynsatisfies
(
)
()+−
Du
sκ n
1, proving (ii).
As for (iii), observe that
()
()
()( )
()
∑∑∑
∑∑ ∑∑
∑
>≤<
=>≤<===…==
≤>>=>>
≤>>
=+
=− = ++ +
=− = +=− = −+
=
nPYuYu Y
nPXuYu XU U UU
nPXuX u nPXuX u
κn P X u X u
,
,,1,0,1
,,
,.
j
r
nj n j
iκj
r
injnj i i j
iκj
r
inj niκj
r
nji n
j
r
nj n
211
2
1
211 11
2
1
212
1
212
21
n
n
nn
n
(7)
If
{}
X
n
satisfies ()
′
Du
n, then (7)converges to zero, as
→∞n
, and (
)
()
Du
n
2holds for
{}
Yn.□
Under conditions (
)
Du
nand (
)
()
Du
snwith
≥
s
2
, we can also compute the extremal index
θ
X
defined in
(3)by (Chernick et al. [1]; Corollary 1.3)(∣)=≤…≤>
→∞
θPXuXuXulim , ,
.
Xnnsn n21 (8)
The stopped clock model 53

If
{}
X
n
and
{}
Ynhave extremal indexes
θ
X
and
θ
Y
, respectively, then ≤θθ
Y
X
, since ()⋁≤/
≤
=
P
Xnτ
i
ni
1
()⋁≤/
=
P
Ynτ
i
ni
1. This corresponds to the intuitively expected, if we remember that the possible repetition of
variables
X
n
leads to larger clusters of values above
u
n
. In the following result, we establish a relationship
between
θ
X
and
θ
Y
.
Proposition 3.3. Suppose that
{
}U
n
is strong-mixing and
{}
X
n
satisfies conditions (
)
Du
nand (
)
()
Du
sn,
≥
s
2
,for
normalized levels ()≡
u
uτ
nn
.If
{}
X
n
has extremal index
θ
X
,then
{}
Ynhas extremal index
θ
Y
given by
()
∑
=…
=−…++
θθp β1, 0, ,0, 1
,
YX
j
κ
jj j
0
1
1,2, , 1, 2
where (∣ )=>≤…≤<
→∞ +−
β
PX u X u X u Xlim , ,
.
jnsj n n s n s11
Proof. By Proposition 3.1,
{}
Ynalso satisfies condition (
)
Du
n. Thus, we have
⎜⎟
⎜⎟
⎛⎝⎞⎠⎧
⎨
⎩⎛⎝⎞⎠⎫
⎬
⎭
[]
⋁≤ = − ⋁>
→∞ =→∞ =/
PYu kP Yulim exp lim
n
i
n
in nni
nk
in
11
n
and
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎛⎝⎞⎠
⎛⎝{}
⎞⎠
⎛⎝{}
⎞⎠
⎛⎝{}
⎞⎠
⎛⎝{}
⎞⎠
⎛⎝{}
⎞⎠
()()
()()
[]
[]
[]
[]
[]
[]
[]
[]
∑∑
∑∑
⋁>
=≤⋁>
=⋃≤<
=⋃≤<=
=⋃⋃≤<===…==
=⋃⋃≤<===…==
=≤…≤<>⋅…
=≤…≤<>⋅…
→∞ =/
→∞ =/
→∞ =/+
→∞ =/++
→∞ =/=−−+−−+ +
→∞ =/=−++ + + ++
→∞ =/=−+++ +…+++
→∞ =/=−−+ + ++ …++
kP Y u
kPY u Y u
kP Y u Y
kP Y u X U
kP X u X U U U U
kP X u X U U U U
kPXuXuXXup
kPXuXuXXup
lim
lim ,
lim
lim , 1
lim , 1, 0 , 1
lim , 1, 0 , 1
lim , , , 1, 0, ,0, 1
lim , , , 1, 0, ,0, 1
nni
nk
in
nnn
i
nk
in
nni
nk
ini
nni
nk
ini i
nni
nk
j
κ
ij n i ij ij i i
nni
nk
j
κ
i n ij i i ij ij
nni
nk
j
κ
niniij niiijij
nni
nk
j
κ
is n i n i ij n j j
1
11
11
111
10
1
11 1
10
1
11 1
10
1
111,1,,,1
10
1
2111,2,,1,2
n
n
n
n
n
n
n
n
(9)
since
{}
X
n
satisfies condition (
)
()
Du
snfor some
≥
s
2
. The stationarity of
{}
X
n
leads to
()()
()()
()()
[]
[]
∑∑∑∑
∑≤…≤ >⋅ …
=≤…≤<>⋅…
=≤…≤<>⋅ …
→∞ =/=−−+ <+ ++ …++
→∞ =/=−−+…++
→∞ =−−+…++
kPXuXuXXup
kPXuXuXXup
nP X u X u X X u p
lim , , , 1, 0, ,0, 1
lim , , , 1, 0, ,0, 1
lim , , , 1, 0, ,0, 1
nni
nk
j
κ
is n i i ij n j j
nni
nk
j
κ
nsnssjn jj
nj
κ
nsnssjn jj
10
1
2111,2,,1,2
10
1
11 1,2,,1,2
0
1
11 1,2,,1,2
n
n
54 Helena Ferreira and Marta Ferreira

()(∣)()
()
∑
∑
=≤…≤<>≤…≤<⋅ …
=…
→∞ =−−+ −…++
=−…++
nPXu X u XPX uXu X u Xp
τθ p β
lim , , , , 1, 0, ,0, 1
1, 0, ,0, 1 ,
nj
κ
nsnssjn nsns jj
Xj
κ
jj j
0
1
11 11 1,2,,1,2
0
1
1,2, , 1, 2
where the last step follows from (8).□
Observe that ()()(
)∑
…===
=−…++
ppPU1, 0, ,0, 1 1 1
j
κjj n n
0
11,2, , 1, 2 and thus,
()≤≤θθp θ1
YXn
X
, as expected.
Proposition 3.4. Suppose that
{
}U
n
is strong-mixing and
{}
X
n
satisfies conditions (
)
Du
nand ()
′
Du
n,for normal-
ized levels ()≡
u
uτ
nn
.Then,
{}
Ynhas extremal index
θ
Y
given by (
)
=θp1, 1
Y1,2 .
Proof. By condition ()
′
Du
n, the only term to consider in (9)corresponds to
=
j0
, and we obtain
⎜⎟
⎛⎝⎞⎠
()()
( )() ()
[]
[]
∑
⋁≤
=≤…≤<
=>=
→∞ =/
→∞ =/−
→∞
kP Y u
kPXuXuXp
nP X u p τp
lim
lim , , 1, 1
lim 1, 1 1, 1 . □
nni
nk
nn
nni
nk
nsns
nsn
1
1111,2
1,2 1,2
n
n
Observe that we can obtain the aforementioned result by applying Proposition 3.2 (iii)and calculating
directly
(
)
=≤<
→∞
τ
θnPYuYlim
Yn n12
. More precisely, we have that
{}
Ynsatisfies (
)
()
Du
n
2, and by applying
(8), we obtain
⎜⎟
⎜⎟
()
()
⎛⎝⎞⎠
⎛⎝⎞⎠()
()()
( )() ()
=≤<
=≤<=
=⋃≤<===…==
=⋃≤…≤<>⋅ …
=≤<
=>=
→∞
→∞
→∞ =−−−−+
→∞ =−−−− −−+…
→∞
→∞
τθ nP Y u Y
nP Y u X U
nP X u X U U U U
nP X u X u X X u p
nP X u X p
nP X u p τp
lim
lim , 1
lim , 1, 0 , 1,
lim , , , 1, 0, ,0, 1
lim 1, 1
lim 1, 1 1, 1 .
Ynn
nn
nj
κ
jn j j
nj
κ
κn jn j n jj
nn
nn
12
122
0
1
12111 12
0
1
21221,11,,1,2
121,2
21,2 1,2
The same result can also be seen as a particular case of Proposition 3.3, where, if we take =
s1
, we have
=
β0
j, for
≠
j0
, and we obtain () (
)
==θθβp p1, 1 1, 1
YX
01,2 1,2 , since =
β1
0
and under ()
′
Du
nit comes
=θ
1
X
.
Example 3.1. Consider
{}
Ynsuch that
{}
X
n
is an ARMAX sequence, i.e.,
()=∨−
−
X
ϕX ϕ Z1
nn n1
,
≥n
1
, where
{
}Z
nis an independent sequence of random variables with standard Fréchet marginal distribution and
{}
X
n
and
{
}Z
nare independent. We show that
{}
X
n
has also standard Fréchet marginal distribution, satisfies
condition (
)
()
Du
n
2and has extremal index
=−θϕ1
X(see, e.g., Ferreira and Ferreira [2]and references
therein).
Observe that, for normalized levels ≡/
u
n
τ
n,>
τ0
, we have
(∣ )
()( )( )( )
()( )
() () ( )
()
()()
()
=>≤<
=≤−≤ ≤−≤ ≤+≤ ≤ ≤
≤−≤ ≤
=−−− − −− − +− −
−−− − =
→∞
→∞
→∞
βPXuXuX
PX u PX u X u PX u X u PX u X u X u
PX u PX u X u
ϕϕ ϕ
ϕϕ
lim
lim ,,,,
,
lim 112 12 132
112 .
nnn
n
nnnnnnnn
nnn
n
τ
nτ
nτ
nτ
n
τ
nτ
n
131 2
11213123
112
2
The stopped clock model 55

Analogous calculations lead to =
β
ϕ
22. Considering
=
κ
3
, we have ()(() ()=− + +θϕp ϕp11,1 1,0,1
Y1,2 1,2,3
()
)
ϕp 1, 0, 0, 1
21,2,3,4
.
The observed sequence is
{}
Yn, and therefore, results that allow retrieving information about the
extreme behavior of the initial sequence
{}
X
n
, subject to the failures determined by
{
}U
n
, may be of interest.
If we assume that
{}
Ynsatisfies (
)
()
Du
sn, then
{}
X
n
also satisfies (
)
()
Du
snby Proposition 3.2 (i), thus
coming ()
(∣)
(∣)
=≤…≤<
=≤…≤<=…==
=≤…≤<≠≠…≠
→∞ −
→∞ −
→∞ −
τθ nP X u X u X
nP Y u Y u Y U U
nP Y u Y u Y Y Y Y
lim , ,
lim , , 1
lim , , .
Xnnsns
nns ns s
nns ns s
11
111
1101
Thereby, we can write
(∣)
()
=≤…≤< ≠≠…≠
>
→∞ −
θPY u Y u YY Y Y
PY u
lim ,,
.
Xn
ns ns s
n
1101
1
4 Tail dependence
Now we will analyze the effect of this failure mechanism on the dependency between two variables,
Y
n
and
+
Yn
m
,
≥m
1
. More precisely, we are going to evaluate the lag-
m
tail dependence coefficient
(∣) ( ∣ )=>>
+→∞ +
λY Y PY xY xlim ,
nm n xnm n
which incorporates the tail dependence between
X
n
and +
X
n
j
, with
j
regulated by the maximum number of
failures −
κ1
and by the relation between
m
and
κ
. In particular, independent variables present null tail
dependence coefficients. If
=m
1
, we obtain the tail dependence coefficient in Joe [5]. For results related to
lag-
m
tail dependence in the literature, see, e.g., Zhang [10,11].
Proposition 4.1. Sequence
{}
Ynhas lag-
m
tail dependence coefficient, with
≥m
1
,
(∣) ( ) ∣
()
{} ()()
∑∑
=…+ ⋅
⋅……
+… ≤−
=∨−+ =
−++
…+++++…++
∗∗
∗∗ ∗ ∗
λY Y p λ X X
p
10, ,0
1,0, ,0,1,0, ,0,
nm n m mκ imκ
m
i
κ
nii n
ii ii i i m
1, , 1 110
1
1,2,,1,1,2,,1
(10)
provided all coefficients
∣
(
)
++
∗
λX X
nii n
exist.
Proof. Observe that
()
()( )
() () ()
{} ()
{}
()()
∑
∑∑∑
>>
= > ==…= + > > = ==…=
=> …
+>> ……
+++≤−
=∨−+ ++++ +
=
−− −−−+…+ ≤−
=∨−+ =
−− + − −+… +++…+
∗∗∗∗
PY xY x
PY xU U PY x X x U U U
PX x p
PX xX xp
1
1
,
,0 , ,1, 0
1, 0, ,0
, 1, 0, ,0, 1, 0, ,0
nnm
nn nmmκ
imκ
m
nni nini nm
i
κm
ni nini nm mκ
imκ
m
i
κ
ni ni nini nnini nm
11
11 1
0
1
,1,, 1
110
1
,1,,,,1,,
and
()()
∑
…= …
=−− …++ …
pp1, 0, ,0 0, ,0
i
κm mi m
0
11,2, , 1 1, ,
.□
56 Helena Ferreira and Marta Ferreira

Taking
=m
1
in (10), we obtain the tail dependence coefficient
(∣) () ( ∣) ( )
∑
=+ …
+=−++ …++
λY Y p λX X p01,0,,0,1
,
nn n i
κ
nin ii10
1
11,2,,1,2
provided all coefficients
(∣)
++
λX X
nin1
exist.
If
{}
X
n
is lag-∗
mtail independent for all integer (
)
≥∨−+
∗
mmκ11
, we have ∣
()
=
++∗
λX X
0
nii n in the
second ter of (10), and thus, (∣) ( )
{
}
=…
+… ≤−
λY Y p 10, ,0
nm n m mκ1, , 1 and
{}
Ynis lag-
m
tail independent for all
integer
≥mκ
.
Example 4.1. Consider again
{}
Ynbased on ARMAX sequence
{}
X
n
as in Example 3.1. We have that
{}
X
n
has
lag-
m
tail dependence coefficient (∣)=
+
λX X ϕ
nm n
m
(Ferreira and Ferreira [2]), and thus,
(∣) ( ) ( )
{} ()
∑∑
=…+ ……
+… ≤−
=∨−+ =
−+…+++++…++
∗∗∗∗ ∗ ∗
λY Y p ϕ p10, ,0 1, 0, ,0, 1, 0, ,0
.
nm n m mκ imκ
m
i
κii ii ii i i m1, , 1 110
1
1,2,,1,1,2,,1
Acknowledgements: The authors thank the reviewers for the comments that contributed to the improve-
ment of this work. Helena Ferreira was partially supported by the research unit Centre of Mathematics and
Applications of University of Beira Interior UIDB/00212/2020 -FCT (Fundação para a Ciência e a
Tecnologia). Marta Ferreira was financed by Portuguese Funds through FCT -Fundação para a Ciência e
a Tecnologia within the Projects UIDB/00013/2020 and UIDP/00013/2020 of Centre of Mathematics of the
University of Minho, UIDB/00006/2020 of Centre of Statistics and its Applications of University of Lisbon,
UIDB/04621/2020 and UIDP/04621/2020 of Center for Computational and Stochastic Mathematics and
PTDC/MAT-STA/28243/2017.
Conflict of interest: The authors state no conflict of interest.
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The stopped clock model 57