PORTUGALIAE MATHEMATICA
Vol. 56 Fasc. 1 – 1999
MAXIMA AND MINIMA
OF STATIONARY RANDOM SEQUENCES
UNDER A LOCAL DEPENDENCE RESTRICTION
M. Graça Temido *
e n , vn ) for stationary random
Abstract: In this paper a local mixing condition D(u
sequences satisfying Davis’ condition D(un , vn ) is introduced. Under these conditions,
the asymptotic joint distribution of the maxima and minima can be calculated with
the knowledge of the crossing probabilities. An illustrative example of a 2-dependent
sequence where the maxima and minima are not asymptotically independent is also
given.
1 – Introduction
Let {Xn } be a strictly stationary random sequence with marginal distribution
function F, let {un } and {vn } be real sequences and consider the maxima Mn =
max{X1 , X2 , ..., Xn } and the minima Wn = min{X1 , X2 , ..., Xn }.
It is well known that, if {Xn } is a sequence of independent and identically
distributed (i.i.d.) random variables, the maxima and minima, with linear normalization, are asymptotically independent. Davis (1979) gives the sufficient
conditions D(un , vn ) and D 0 (un , vn ), under which the maxima and minima, both
jointly and marginally, behave as though the sequence {Xn } was i.i.d.. The condition D(un , vn ) is an asymptotic independence condition, weaker than strong
mixing, and D 0 (un , vn ) is a local dependence condition which implies the non
existence of clustering of high and low values of the sequence {Xn } above {un }
and below {vn }, respectively.
Received : March 4, 1997; Revised : July 1, 1997.
Keywords and Phrases: Nondegenerate limiting distributions; maximum and minimum
value; stationary sequence; m-dependence.
* This work was partially supported by JNICT/PRAXIS XXI/FEDER.
60
M. GRAÇA TEMIDO
Oliveira and Turkman (1992) introduce the local mixing condition D ∗ (un , vn )
which is weaker than D 0 (un , vn ) and generalizes D 00 (un ) of Leadbetter and Nandagopalan (1989). If this condition holds along with D(un , vn ) the asymptotic joint
distribution of the maxima and minima may be computed from the bivariate distribution of two consecutive random variables. Namely, the stationary sequence
{Xn } satisfies D(un , vn ) if for every n and integers 1 ≤ i1 < ... < ip < j1 ... <
jq ≤ n, such that j1 − ip > `,
¯ ³
´
¯
¯P Xi ≤ un , ..., Xi ≤ un , Xj ≤ un , ..., Xj ≤ un −
p
q
1
1
¯
³
´ ³
(1)
´¯¯
− P Xi1 ≤ un , ..., Xip ≤ un P Xj1 ≤ un , ..., Xjq ≤ un ¯¯ ≤ αn,` ,
¯ ³
´
¯
¯P Xi > vn , ..., Xi > vn , Xj > vn , ..., Xj > vn −
p
q
1
1
¯
³
´ ³
− P Xi1 > vn , ..., Xip > vn P Xj1 > vn , ..., Xjq
and
´¯¯
> vn ¯¯ ≤ αn,` ,
¯ ³
´
¯
¯P vn < Xi ≤ un , ..., vn < Xi ≤ un , vn < Xj ≤ un , ..., vn < Xj ≤ un −
p
q
1
1
¯
³
´ ³
´¯¯
−P vn < Xi1 ≤ un , ..., vn < Xip ≤ un P vn < Xj1 ≤ un , ..., vn < Xjq ≤ un ¯¯ ≤ αn,` ,
where lim αn,`n = 0 for some `n such that lim `n /n = 0.
n→+∞
n→+∞
∗
Furthermore, D ∗ (un , vn ) is satisfied by {Xn } if lim lim sup Sn,k
= 0 where
k→+∞ n→+∞
∗
Sn,k
=n
[n/k]½
X
j=1
³
´
³
P X1 > un , Xj ≤ un < Xj+1 + P X1 < vn , Xj ≥ vn > Xj+1
³
´
³
+ P X1 > un , Xj ≥ vn > Xj+1 + P X1 < vn , Xj ≤ un < Xj+1
´¾
´
.
For stationary sequences satisfying D(un , vn ) and D ∗ (un , vn ), Oliveira and Turkman (1992) consider high and low levels, un and vn , verifying lim P (X2 ≤
n→+∞
un /X1 > un ) = θ1 , lim P (X2 > vn /X1 ≤ vn ) = θ2 , lim nP (X1 > un ) = τ1 (x)
n→+∞
n→+∞
and lim nP (X1 < vn ) = τ2 (y), with θ1 , θ2 in ]0, 1] and τ1 (x), τ2 (y) in ]0, +∞[.
n→+∞
The limit
³
´
lim P Mn ≤ un , Wn > vn = e−(θ1 τ1 (x)+θ2 τ2 (y))
n→+∞
is obtained and hence, the maxima and minima, are yet asymptotically independent.
MAXIMA AND MINIMA OF STATIONARY RANDOM SEQUENCES
61
The constant θ1 is called the extremal index of the stationary sequence {Xn }
and θ = (θ1 , θ2 ) is the extremal index of {Xn , −Xn }. The definition of multivariate extremal index for multivariate stationary sequences can be found in
Nandagopalan (1990). As we already said before, if {Xn } satisfies D(un , vn ) and
D0 (un , vn ) we easily deduce θ1 = θ2 = 1.
Dealing with the asymptotic behavior of the exceedance point process for
stationary sequences satisfying Leadbetter’s condition D(un ), defined by (1),
e (k) (un ), which also genFerreira (1996) introduce another mixing condition D
e (k) (un ) is satisfied by {Xn } if k is the minimum
eralizes D 00 (un ). The condition D
positive integer for which there exists a sequence of positive integers {k n }, with
lim kn = +∞,
lim kn
n→+∞
n→+∞
`n
= 0,
n
and
X
s(k)
n =n
µ
lim kn αn,`n = 0,
n→+∞
P X1 > u n ,
2≤j1 <j2 <...<jk ≤[ kn ]−1
n
k n
\
lim kn (1−F (un )) = 0
n→+∞
Xji ≤ un < Xji +1
i=1
o¶
→ 0, n → +∞ .
e (k) (un ) has
The condition D 00 (un ) is obtained for k = 1. The author of D
(2)
e
proven that, if {Xn } satisfies D (un ) and lim nP (X1 ≤ un < X2 ) = ν, with
n→+∞
ν in [0, +∞[, then
lim P (Mn ≤ un ) = e−ν+β ,
n→+∞
β≥0,
if and only if
lim kn
n→+∞
X
³
´
P Xi ≤ un < Xi+1 , Xj ≤ un < Xj+1 = β .
1≤i<j≤[ kn ]−1
n
e n , vn ),
In this paper we introduce a local mixing restriction, condition D(u
(2)
∗
e
which generalizes D (un ) and is weaker than D (un , vn ). Under D(un , vn ) and
e n , vn ) the joint limit distribution of the maxima and minima can be comD(u
puted from the mean number of four kinds of crossings of the considered levels:
upcrossings in a cluster of high values; downcrossings in a cluster of low values;
paired upcrossings, paired downcrossings and pairs with one upcrossing and one
downcrossing in representative clusters.
e n , vn ) the maxima and
It should be noticed that under D(un , vn ) and D(u
minima are not necessarily asymptotically independent.
62
M. GRAÇA TEMIDO
2 – Main result
As we said before we consider strictly stationary sequences satisfying Davis’
condition D(un , vn ). For the proof of our main result it will be convenient to
present the following lemma.
Lemma 1 (Davis (1979)). Suppose D(un , vn ) is satisfied by the stationary
sequence {Xn }. Then, for every positive integer k,
lim
n→+∞
½ ³
´
³
P Mn ≤ u n , W n > v n − P k Mn 0 ≤ u n , W n 0 > v n
´¾
=0,
where n0 = [n/k].
In what follows the events {Xi ≤ un < Xi+1 } and {Xi > vn ≥ Xi+1 } are
represented by Ai and Bi , respectively.
e n , vn ) if
Definition 1.
The sequence {Xn } satisfies condition D(u
e
lim lim sup k Sn,k = 0 where
k→+∞ n→+∞
Sen,k =
(2)
X
½
P (Ai , Aj , Ak ) + P (Ai , Aj , Bk ) + P (Ai , Bj , Bk )
1≤i<j<k≤n0 −1
+ P (Bi , Bj , Bk ) + P (Bi , Aj , Bk ) + P (Ai , Bj , Ak )
+ P (Bi , Bj , Ak ) + P (Bi , Aj , Ak )
¾
.
This condition restricts the occurence of three or more level crossings in a
cluster.
The following theorem is the main result of this paper. We first present some
assumptions of the theorem. Specifically, we will consider that {Xn } satisfies
(3)
(4)
(5)
and
lim nP (A1 ) = ν1 ,
n→+∞
lim
n→+∞
lim
n→+∞
lim nP (B1 ) = ν2 ,
n→+∞
X
P (Ai , Aj ) =
β1
+ ok (1/k) ,
k
X
P (Bi , Bj ) =
β2
+ ok (1/k)
k
1≤i<j≤n0 −1
1≤i<j≤n0 −1
MAXIMA AND MINIMA OF STATIONARY RANDOM SEQUENCES
lim
(6)
n→+∞
0 −1 n0 −1
nX
X
P (Ai , Bj ) =
i=1 j=1
63
β3
+ ok (1/k) ,
k
with ν1 , ν2 , β1 , β2 and β3 in [0, +∞[. It should be remarked that, under stationarity, β1 ≤ ν1 , β2 ≤ ν2 , β3 ≤ ν1 − β1 and β3 ≤ ν2 − β2 .
Theorem 1. Suppose that the stationary sequence {Xn } satisfies D(un , vn )
e n , vn ) and that, for all positive integer k, (3), (4), (5) and (6) hold, where
and D(u
{un } and {vn } are real sequences satisfying
(7)
lim P (X1 > un ) = P (X1 ≤ vn ) = 0 .
n→+∞
Then,
´
³
lim P Mn ≤ un , Wn > vn = e−(ν1 +ν2 −β1 −β2 −β3 ) .
n→+∞
Proof: We start by observing that
{Mn0 > un } = {X1 > un } ∪
0 −1
nn[
i=1
(8)
{Wn0 ≤ vn } = {X1 ≤ vn } ∪
0 −1
nn[
o
Ai ,
Bi
i=1
o
and
³
´
P Mn0 ≤ un , Wn0 > vn = 1 − P (Mn0 > un ) − P (Wn0 ≤ vn )
(9)
³
´
+ P Mn 0 > u n , W n 0 ≤ v n .
From Bonferroni’s inequality we get
0 −1
nX
i=1
P (Ai ) −
X
P (Ai , Aj ) ≤
1≤i<j≤n0 −1
≤ P (Mn0 > un )
(10)
≤ P (X1 > un ) +
0 −1
nX
i=1
+
X
1≤i<j<k≤n0 −1
P (Ai ) −
X
1≤i<j≤n0 −1
P (Ai , Aj , Ak ) .
P (Ai , Aj )
64
M. GRAÇA TEMIDO
Using now stationarity it results
lim
n→+∞
and
0 −1
nX
n→+∞
i=1
X
lim sup
P (Ai ) = lim (n0 − 1) P (A1 ) =
ν1
k
P (Ai , Aj , Ak ) ≤ lim sup Sen,k = ok (1/k) .
n→+∞ 1≤i<j<k≤n0 −1
n→+∞
Hence, attending to (4), (7) and (10), we have
o
n
β1 − ν 1
+ ok (1/k) ≤ lim inf −P (Mn0 > un )
n→+∞
k
n
o
≤ lim sup −P (Mn0 > un )
(11)
n→+∞
β1 − ν 1
≤
+ ok (1/k) .
k
Analogously we prove
o
n
β2 − ν 2
+ ok (1/k) ≤ lim inf −P (Wn0 ≤ vn )
n→+∞
k
n
o
≤ lim sup −P (Wn0 ≤ vn )
(12)
n→+∞
β2 − ν 2
+ ok (1/k) .
≤
k
Furthermore, using (8) and Boole’s inequality, we obtain
´
³
P M n 0 > u n , W n 0 ≤ v n , vn < X 1 ≤ u n =
=P
(13)
0 −1
µ n[
Ai ,
i=1
n0 −1
≤
0 −1
n[
B i , vn < X 1 ≤ u n
i=1
¶
n0 −1
X X
P (Ai , Bj )
i=1 j=1
and thus
³
´
lim sup P Mn0 > un , Wn0 ≤ vn =
n→+∞
(14)
³
= lim sup P Mn0 > un , Wn0 ≤ vn , vn < X1 ≤ un
n→+∞
≤
β3
+ ok (1/k) .
k
´
65
MAXIMA AND MINIMA OF STATIONARY RANDOM SEQUENCES
On the other hand, applying Bonferroni’s inequality, we have, with B =
0 −1
n[
Bi ,
i=1
´
³
P Mn 0 > u n , W n 0 ≤ v n ≥ P
(15)
0 −1
µ n[
Ai ,
≥
Bi
i=1
i=1
0 −1
nX
0 −1
n[
P (Ai , B) −
i=1
¶
X
P (Ai , Aj , B)
1≤i<j≤n0 −1
and, using again the same inequality, we get
lim inf
n→+∞
0 −1
nX
P (Ai , B) ≥
i=1
(16)
≥ lim inf
n→+∞
0 −1 n0 −1
nX
X
P (Ai , Bj ) − lim sup
0 −1
nX
X
n→+∞ i=1 1≤j<k≤n0 −1
i=1 j=1
P (Ai , Bj , Bk ) .
Moreover, since
0 −1
nX
X
P (Ai , Bj , Bk ) =
i=1 1≤j<k≤n0 −1
X
=
P (Ai , Bj , Bk ) + P (Bi , Aj , Bk ) + P (Bi , Bj , Ak )
1≤i<j<k≤n0 −1
≤ Sen,k
e n , vn ) holds, from (16) it results
and D(u
lim inf
n→+∞
0 −1
nX
i=1
P (Ai , B) ≥
β3
+ ok (1/k) .
k
Let’s recall (15). Considering again Boole’s inequality we get
lim sup
(17)
X
P (Ai , Aj , B) ≤ lim sup
n→+∞ 1≤i<j≤n0 −1
X
0 −1
nX
n→+∞ 1≤i<j≤n0 −1 k=1
P (Ai , Aj , Bk )
≤ lim sup Sen,k = ok (1/k)
n→+∞
and thus
(18)
³
´
lim inf P Mn0 > un , Wn0 ≤ vn ≥
n→+∞
β3
+ ok (1/k) .
k
66
M. GRAÇA TEMIDO
From (14) and (18), we have
´
³
β3
+ ok (1/k) ≤ lim inf P Mn0 > un , Wn0 ≤ vn
n→+∞
k
´
³
≤ lim sup P Mn0 > un , Wn0 ≤ vn
(19)
n→+∞
β3
+ ok (1/k) .
≤
k
Finally, putting α = ν1 + ν2 − β1 − β2 − β3 we conclude from (9), (11), (12)
and (19), that
1−
³
´
α
+ ok (1/k) ≤ lim inf P Mn0 ≤ un , Wn0 > vn
n→+∞
k
³
´
≤ lim sup P Mn0 ≤ un , Wn0 > vn
(20)
n→+∞
≤1−
α
+ ok (1/k)
k
which implies
(21)
¯
¯
¯ ³
´
α¯
¯
lim sup¯P Mn0 ≤ un , Wn0 > vn − 1 + ¯¯ = ok (1/k) .
k
n→+∞ ¯
Observe now that
¯ ³
¯
´
¯
¯
−α ¯
¯
lim sup¯P Mn ≤ un , Wn > vn − e ¯ ≤
n→+∞
¯ ³
´¯¯
´
³
¯
≤ lim sup¯¯P Mn ≤ un , Wn > vn − P k Mn0 ≤ un , Wn0 > vn ¯¯
n→+∞
¯
¯ ³
(22)
´ ³
¯
α ´k ¯¯
+ lim sup¯¯P k Mn0 ≤ un , Wn0 > vn − 1 −
k ¯
n→+∞
¯
¯
³
¯
α ´k ¯¯
+ ¯¯e−α − 1 −
.
k ¯
Using Lemma 1, the first term of the right hand side of (22) is zero. Moreover,
using the well known inequality
¯Y
¯
k
k
Y
X
¯ k
¯
¯
¯ ≤
a
−
b
|ai − bi |
i
i
¯
¯
i=1
i=1
i=1
with a1 , ..., ak , b1 , ..., bk in [0, 1], we conclude that the second term of the right
α
hand side of (22) is bounded by lim sup k|P (Mn0 ≤ un , Wn0 > vn ) − (1 − )|.
k
n→+∞
MAXIMA AND MINIMA OF STATIONARY RANDOM SEQUENCES
67
Hence, by (21) and (22), we deduce that
(23)
¯ ³
¯
´
¯
¯
−α ¯
¯
lim lim sup P Mn ≤ un , Wn > vn − e ¯ ≤
k→+∞ n→+∞ ¯
¯
¯
¯ −α ³
α ´k ¯¯
¯
=0
≤ lim ¯e − 1 −
k→+∞
k ¯
which enables us to conclude that lim P (Mn ≤ un , Wn > vn ) = e−α .
n→+∞
The following two results are important tools on the establishment of the
asymptotic independence of the maxima and minima.
Corollary 1. Suppose that {Xn } is a stationary sequence under the assumptions of Theorem 1. Then, {Mn ≤ un } and {Wn > vn } are asymptotically
independent if and only if β3 = 0.
Proof: Since D(un , vn ) holds, we obtain lim {P (Mn ≤ un )−P k (Mn0 ≤ un )}
n→+∞
= 0 and lim {P (Wn > vn ) − P k (Wn0 > vn )} = 0.
n→+∞
On the other hand, it results from (11) that
¯
¯
³
¯
ν1 − β1 ´¯¯
¯
lim sup¯P (Mn0 ≤ un ) − 1 −
¯ = ok (1/k) .
k
n→+∞
Therefore, with the arguments used in (22) and (23), we deduce that
lim P (Mn ≤ un ) = e−ν1 +β1 .
(24)
n→+∞
Similarly we prove that
lim P (Wn > vn ) = e−ν2 +β2 .
(25)
n→+∞
So {Mn ≤ un } and {Wn > vn } are asymptotically independent if and only if
β3 = 0.
The proofs of Theorem 1 and Corollary 1 enables us to establish the following
theorem. Firstly we must define another local dependence condition, weaker than
e n , vn ).
D(u
e n , vn ) if
Definition 2.
The sequence {Xn } satisfies condition C(u
lim lim sup k Cen,k = 0 where
k→+∞ n→+∞
Cen,k =
X
1≤i<j<k≤n0 −1
n
o
P (Ai , Aj , Ak ) + P (Bi , Bj , Bk ) .
68
M. GRAÇA TEMIDO
e n , vn ) instead
Indeed, we will prove that, if β3 = 0, it is enough to consider C(u
e n , vn ).
of D(u
Theorem 2. Suppose that the stationary sequence {Xn } satisfies D(un , vn )
e n , vn ) where {un } and {vn } are real sequences satisfying, for all positive
and C(u
integer k, (3), (4), (5), (7) and (6) with β3 = 0. Then, {Mn ≤ un } and {Wn > vn }
are asymptotically independent with
³
´
lim P Mn ≤ un , Wn > vn = e−(ν1 +ν2 −β1 −β2 ) .
n→+∞
Proof: Observe that we established (24) and (25) only using the first and
the fourth terms of Sen,k . Moreover, with β3 = 0, from (14) we deduce
´
³
lim sup P Mn0 > un , Wn0 ≤ vn = ok (1/k) .
n→+∞
Then, (20) is similarly obtained (with β3 = 0), and the result follows immediately.
3 – Example
Let {Yn } and {Zn } be independent sequences of i.i.d. random variables, with
marginal distribution functions H and G respectively. Suppose that G(0) =
H(0) = 0 and assume that there exists a real sequence {un } satisfying
lim n(1−H(un )) = τY
n→+∞
and
lim n(1−G(un )) = τZ ,
n→+∞
with τY and τZ in [0, +∞[.
Let {Tn } be an i.i.d. sequence, independent of {Yn } and {Zn }, with support
{1, 2, 3} and P (T1= i) = pi , i = 1, 2, 3.
Define

Tn = 1,
 Yn ,
Xn =



max{Yn−2 , Zn }, Tn = 2,
−Yn−1 ,
Tn = 3 .
We easily prove that {Xn } is stationary and 2-dependent with marginal distribution function
F (x) = H(x) p1 + H(x) G(x) p2 + (1 − H(−x)) p3 ,
and satisfies D(un , −un ).
x∈R,
MAXIMA AND MINIMA OF STATIONARY RANDOM SEQUENCES
69
Moreover, {Xn } does not satisfy either D ∗ (un , −un ) or D 00 (un ) once
[n/k]
lim sup n
n→+∞
X
j=2
³
P X1 > un , Xj ≤ un < Xj+1
´
→ τ Y p1 p2 ,
k → +∞ .
e n , −un ) holds. Observe first that lim nF (−un ) =
We will prove now that D(u
n→+∞
τY p3 and
(26)
lim n(1 − F (un )) = τY p1 + (τY + τZ ) p2 .
n→+∞
Indeed, since
X
P (Ai , Aj , Ak ) is bounded by
1≤i<j<k≤n0 −1
X
n
P (A1 , Ai , Aj ) ≤
k 3≤i<j≤n0 −1
≤
´
³
X
n
P X1 > u n , A i , A j
k 2≤i<j≤n0 −1
½
0
−3
´
³
n nX
≤
P X1 > un , Xi+1 > un , Xi+3 > un
k i=2
+
0 −1
nX
j=i+3
³
P X1 > un , Xi+1 > un , Xj+1 > un
´¾
0
≤
−3
n nX
P (X1 > un ) P (Xi+3 > un )
k i=2
0
0
n
−3 X
³
´
n nX
P X1 > un , Xi+1 > un P (Xj > un )
+
k i=2 j=i+4
0
nX
−3 ³
´2
´
n2 ³
n2
≤ 2 P (X1 > un ) + 2 P (X1 > un )
P X1 > un , Xi+1 > un
k
k
i=2
=
´2
n2 ³
P (X1 > un )
2
k
½
0
−2
³
´ nX
n2
+ 2 P (X1 > un ) P X1 > un , X3 > un +
P (X1 > un ) P (Xi > un )
k
i=4
≤
´2
´3
n3 ³
2n2 ³
P
(X
>
u
)
+
P
(X
>
u
)
1
n
1
n
k2
k3
¾
70
M. GRAÇA TEMIDO
using (26), we conclude that
X
lim lim sup k
P (Ai , Aj , Ak ) = 0 .
k→+∞ n→+∞
1≤i<j<k≤n0 −1
e n , −un )
Analogously we prove the same for the other terms of Sen,k . Then D(u
holds.
The 2-dependence and the stationarity shall help us again on the computation
of the parameters.
Let us start by calculating ν1 . In fact observing that lim P (X1 ≤ un < X2 ,
n→+∞
T2 = 3) = 0 and using the Total Probability Rule, we have
n P (X1 ≤ un < X2 ) = n P (Y1 ≤ un < Y2 ) p1 p1
´
³
+ n P Y1 ≤ un , max{Y0 , Z2 } > un p1 p2
³
´
+ n P max{Y−1 , Z1 } ≤ un , Y2 > un p1 p2
(27)
³
´
+ n P max{Y−1 , Z1 } ≤ un , max{Y0 , Z2 } > un p2 p2
+ n P (Y1 > un ) p1 p3 + n P (max{Y0 , Z2 } > un ) p2 p3 .
Therefore ν1 = lim nP (X1 ≤ un < X2 ) = (τY +τZ ) p2 + τY p1 .
n→+∞
Using similar arguments and observing that
³
´
³
´
P X1 > −un ≥ X2 , T2 = 1 = P X1 > −un ≥ X2 , T2 = 2 → 0 ,
n → +∞ ,
it results ν2 = τY p3 .
In what concerns the evaluation of β1 , we have
X
P (Ai , Aj ) =
1≤i<j≤n0 −1
0 −1
nX
(n0 − j) P (A1 , Aj )
j=3
0
= (n − 3) P (A1 , A3 ) +
0 −1
nX
(n0 − j) P (A1 , Aj ) .
j=4
Since
0 −1
nX
0
(n − j) P (A1 , Aj ) ≤ n
0
0 −1
nX
P (X2 > un ) P (Xj+1 > un )
j=4
j=4
≤
´2
n2 ³
P
(X
>
u
)
,
2
n
k2
MAXIMA AND MINIMA OF STATIONARY RANDOM SEQUENCES
71
it follows that
lim
n→+∞
X
n
P (A1 , A3 ) + ok (1/k) .
n→+∞ k
P (Ai , Aj ) = lim
1≤i<j≤n0 −1
For the computation of P (A1 , A3 ) we must use again the arguments used in
(27). We first observe that nP (A1 , A3 , C) is asymptotically zero if C is one of
the events:
{T2 = 1, T4 = 1}, {T2 = 2, T4 = 1}, {T2 = 2, T4 = 2}, {T2 = 3} or {T4 = 3} .
Thus, with straightforward calculus, we deduce that β1 = τY p1 p2 .
Moreover it is very easy to obtain β2 = 0.
On the other hand, the computation of β3 follows the steps used above. In
fact, as
lim
n→+∞
0 −1 n0 −1
nX
X
P (Ai , Bj ) =
i=1 j=1
0 −1
nX
0
(n − j) P (A1 , Bj ) +
j=2
0
0 −1
nX
(n0 − j) P (B1 , Aj )
j=2
0
= (n − 2) P (A1 , B2 ) + (n − 3) P (A1 , B3 )
+ (n0 − 2) P (B1 , A2 ) + (n0 − 3) P (B1 , A3 ) + ok (1/k)
and lim nP (A1 , B3 ) = lim nP (B1 , A3 ) = 0, it results β3 = τY (p1 p3 + p2 p3 ).
n→+∞
n→+∞
Finally, we conclude that
³
´
lim P Mn ≤ un , Wn > vn = e−α
n→+∞
where α = τY + τZ p2 − τY (p1 p2 + p1 p3 + p2 p3 ).
It should be noticed that un = un (x) and vn = vn (y). Hence, the parameters
τY , τZ , ν1 , ν2 , β1 , β2 and β3 depend on the real x and y. Then, clearly α =
α(x, y).
ACKNOWLEDGEMENTS – I would like to express my gratitude to Professor Luisa
Canto e Castro and Professor Maria Ivette Gomes for theirs helpful suggestions and
comments. I also would like to thank the referee’s comments which have resulted in
improvements to this paper.
72
M. GRAÇA TEMIDO
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Maria da Graça Temido,
Departamento de Matemática, Universidade de Coimbra,
Largo D. Dinis, 3000 Coimbra – PORTUGAL