Under these conditions, the asymptotic joint distribution of the maxima and minima can be calculated with the knowledge of the crossing probabilities

MAXIMA AND MINIMA

OF STATIONARY RANDOM SEQUENCES UNDER A LOCAL DEPENDENCE RESTRICTION

M. Grac¸a Temido*

Abstract: In this paper a local mixing condition D(ue n, vn) for stationary random 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{X_n}be a strictly stationary random sequence with marginal distribution function F, let{un} and{vn} be real sequences and consider the maximaMn= max{X₁, X₂, ..., X_n} and the minimaW_n= min{X₁, X₂, ..., X_n}.

It is well known that, if {X_n} is a sequence of independent and identically distributed (i.i.d.) random variables, the maxima and minima, with linear nor- malization, are asymptotically independent. Davis (1979) gives the sufficient conditionsD(u_n, v_n) andD⁰(u_n, v_n), under which the maxima and minima, both jointly and marginally, behave as though the sequence{X_n}was i.i.d.. The con- dition D(u_n, v_n) is an asymptotic independence condition, weaker than strong mixing, and D⁰(u_n, v_n) is a local dependence condition which implies the non existence of clustering of high and low values of the sequence {X_n} above {u_n} 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.

Oliveira and Turkman (1992) introduce the local mixing conditionD^∗(u_n, v_n) which is weaker thanD⁰(un, vn) and generalizesD⁰⁰(un) of Leadbetter and Nanda- gopalan (1989). If this condition holds along withD(u_n, v_n) the asymptotic joint distribution of the maxima and minima may be computed from the bivariate dis- tribution of two consecutive random variables. Namely, the stationary sequence {Xn} satisfies D(un, vn) if for every n and integers 1 ≤ i1 < ... < ip < j1... <

j_q ≤n, such thatj₁−i_p> `,

(1)

¯¯

¯¯P^³X_i1 ≤u_n, ..., X_ip ≤u_n, X_j1 ≤u_n, ..., X_jq ≤u_n^´−

−P^³X_i1 ≤u_n, ..., X_ip ≤u_n^´P^³X_j1 ≤u_n, ..., X_jq ≤u_n^´¯¯¯_¯ ≤ α_n,` ,

¯¯

¯¯P^³X_i1 > v_n, ..., X_ip > v_n, X_j1 > v_n, ..., X_jq > v_n^´−

−P^³X_i1 > v_n, ..., X_ip> v_n^´P^³X_j1 > v_n, ..., X_jq > v_n^´¯¯¯¯ ≤ α_n,` , and

¯¯

¯¯P^³v_n< X_i1≤u_n, ..., v_n< X_ip≤u_n, v_n< X_j1≤u_n, ..., v_n< X_jq≤u_n^´−

−P^³v_n< X_i1≤u_n, ..., v_n< X_ip≤u_n^´P^³v_n< X_j1≤u_n, ..., v_n< X_jq≤u_n^´¯¯¯¯≤α_n,`, where lim

n→+∞α<sub>n,`n</sub> = 0 for some `_n such that lim

n→+∞`_n/n= 0.

Furthermore, D^∗(un, vn) is satisfied by {Xn} if lim

k→+∞lim sup

n→+∞ S_n,k^∗ = 0 where S_n,k^∗ =n

[n/k]

X

j=1

½

P^³X₁> u_n, X_j ≤u_n< X_j+1^´+P^³X₁ < v_n, X_j ≥v_n> X_j+1^´ +P^³X₁ > u_n, X_j ≥v_n> X_j+1^´+P^³X₁< v_n, X_j ≤u_n< X_j+1^´¾ . For stationary sequences satisfyingD(u_n, v_n) andD^∗(u_n, v_n), Oliveira and Turk- man (1992) consider high and low levels, un and vn, verifying lim

n→+∞P(X2 ≤ u_n/X₁ > u_n) =θ₁, lim

n→+∞P(X₂ > v_n/X₁ ≤v_n) =θ₂, lim

n→+∞nP(X₁ > u_n) =τ₁(x) and lim

n→+∞nP(X₁ < v_n) =τ₂(y), withθ₁, θ₂ in ]0,1] and τ₁(x), τ₂(y) in ]0,+∞[.

The limit

n→+∞lim P^³M_n≤u_n, W_n> v_n^´=e^{−(θ1τ1(x)+θ2τ2(y))}

is obtained and hence, the maxima and minima, are yet asymptotically indepen- dent.

The constantθ₁ is called the extremal index of the stationary sequence {X_n} and θ = (θ1, θ2) is the extremal index of {Xn,−Xn}. The definition of mul- tivariate extremal index for multivariate stationary sequences can be found in Nandagopalan (1990). As we already said before, if{Xn}satisfiesD(un, vn) and D⁰(u_n, v_n) we easily deduceθ₁=θ₂ = 1.

Dealing with the asymptotic behavior of the exceedance point process for stationary sequences satisfying Leadbetter’s condition D(u_n), defined by (1), Ferreira (1996) introduce another mixing condition D^e(k)(u_n), which also gen- eralizesD⁰⁰(u_n). The conditionD^e(k)(u_n) is satisfied by{X_n}ifkis the minimum positive integer for which there exists a sequence of positive integers{k_n}, with

n→+∞lim k_n= +∞, lim

n→+∞k_n`n

n = 0, lim

n→+∞k_nα_n,`n = 0, lim

n→+∞k_n(1−F(u_n)) = 0 and

s^(k)_n =n ^X

2≤j1<j2<...<jk≤[ⁿ

kn]−1

P µ

X₁> u_n,

\k i=1

nX_ji≤u_n< X_ji+1^o¶→0, n→+∞.

The condition D⁰⁰(u_n) is obtained for k = 1. The author of D^e(k)(u_n) has proven that, if{Xn} satisfies D^e(2)(un) and lim

n→+∞nP(X1 ≤un< X2) =ν, with ν in [0,+∞[, then

n→+∞lim P(Mn≤un) =e^−ν+β, β ≥0 , if and only if

n→+∞lim k_n ^X

1≤i<j≤[_knⁿ]−1

P^³X_i≤u_n< X_i+1, X_j ≤u_n< X_j+1^´=β .

In this paper we introduce a local mixing restriction, condition D(u^e _n, v_n), which generalizesD^e(2)(un) and is weaker thanD^∗(un, vn). Under D(un, vn) and D(ue _n, v_n) the joint limit distribution of the maxima and minima can be com- 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.

It should be noticed that under D(u_n, v_n) and D(u^e _n, v_n) the maxima and minima are not necessarily asymptotically independent.

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{X_n}. Then, for every positive integer k,

n→+∞lim

½

P^³M_n≤u_n, W_n> v_n^´−P^k³M_n⁰ ≤u_n, W_n⁰ > v_n^´¾= 0 , wheren⁰= [n/k].

In what follows the events {X_i ≤ u_n < X_i+1} and {X_i > v_n ≥ X_i+1} are represented byA_i and B_i, respectively.

Definition 1. The sequence {X_n} satisfies condition D(u^e _n, v_n) if

k→+∞lim lim sup

n→+∞

kS^e_n,k= 0 where

(2)

Se_n,k= ^X

1≤i<j<k≤n⁰−1

½

P(A_i, A_j, A_k) +P(A_i, A_j, B_k) +P(A_i, B_j, B_k) +P(B_i, B_j, B_k) +P(B_i, A_j, B_k) +P(A_i, B_j, A_k)

+P(Bi, Bj, A_k) +P(Bi, Aj, A_k)

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{X_n} satisfies

n→+∞lim nP(A₁) =ν₁, lim

n→+∞nP(B₁) =ν₂ , (3)

n→+∞lim X

1≤i<j≤n⁰−1

P(Ai, Aj) = β₁

k +o_k(1/k) , (4)

n→+∞lim X

1≤i<j≤n⁰−1

P(B_i, B_j) = β₂

k +o_k(1/k) (5)

and

n→lim+∞ nX⁰−1

i=1 nX⁰−1

j=1

P(Ai, Bj) = β₃

k +ok(1/k) , (6)

withν₁, ν₂, β₁, β₂ and β₃ in [0,+∞[. It should be remarked that, under station- arity,β₁ ≤ν₁,β₂ ≤ν₂,β₃≤ν₁−β₁ andβ₃≤ν₂−β₂.

Theorem 1. Suppose that the stationary sequence{X_n}satisfiesD(u_n, v_n) andD(u^e _n, v_n)and that, for all positive integer k, (3), (4), (5) and (6) hold, where {u_n}and {v_n}are real sequences satisfying

(7) lim

n→+∞P(X₁ > u_n) =P(X₁≤v_n) = 0 . Then,

n→+∞lim P^³M_n≤u_n, W_n> v_n^´=e^{−(ν1+ν2−β1−β2−β3)} . Proof: We start by observing that

(8)

{M_n⁰ > u_n}={X₁ > u_n} ∪ⁿ

n[⁰−1 i=1

A_i^o,

{W_n⁰ ≤v_n}={X₁ ≤v_n} ∪ⁿ

n[⁰−1 i=1

B_i^o

and

(9) P^³M_n⁰ ≤u_n, W_n⁰ > v_n^´= 1−P(M_n⁰ > u_n)−P(W_n⁰ ≤v_n) +P^³M_n⁰ > un, W_n⁰ ≤vn

´.

From Bonferroni’s inequality we get

nX⁰−1 i=1

P(A_i)− ^X

1≤i<j≤n⁰−1

P(A_i, A_j)≤

(10)

≤P(M_n⁰ > un)

≤P(X₁ > u_n) +

nX⁰−1 i=1

P(A_i)− ^X

1≤i<j≤n⁰−1

P(A_i, A_j)

+ ^X

1≤i<j<k≤n⁰−1

P(A_i, A_j, A_k) .

Using now stationarity it results

n→+∞lim

nX⁰−1 i=1

P(A_i) = lim

n→+∞(n⁰−1)P(A₁) = ν₁ k

and

lim sup

n→+∞

X

1≤i<j<k≤n⁰−1

P(A_i, A_j, A_k) ≤ lim sup

n→+∞

Se_n,k = o_k(1/k) . Hence, attending to (4), (7) and (10), we have

(11)

β1−ν1

k +o_k(1/k)≤lim inf

n→+∞

n−P(M_n⁰ > un)^o

≤lim sup

n→+∞

n−P(M_n⁰ > un)^o

≤ β1−ν1

k +o_k(1/k) . Analogously we prove

(12)

β2−ν2

k +o_k(1/k)≤lim inf

n→+∞

n−P(W_n⁰ ≤vn)^o

≤lim sup

n→+∞

n−P(W_n⁰ ≤vn)^o

≤ β2−ν2

k +o_k(1/k) . Furthermore, using (8) and Boole’s inequality, we obtain

P^³M_n⁰ > u_n, W_n⁰ ≤v_n, v_n< X₁ ≤u_n^´=

(13) =P

µn[⁰−1 i=1

A_i,

n[⁰−1 i=1

B_i, v_n< X₁ ≤u_n

¶

≤

nX⁰−1 i=1

nX⁰−1 j=1

P(A_i, B_j) and thus

lim sup

n→+∞ P^³M_n⁰ > u_n, W_n⁰ ≤v_n^´=

(14) = lim sup

n→+∞ P^³M_n⁰ > u_n, W_n⁰ ≤v_n, v_n< X₁ ≤u_n^´

≤ β3

k +o_k(1/k) .

On the other hand, applying Bonferroni’s inequality, we have, withB =

n[⁰−1 i=1

Bi,

(15)

P^³M_n⁰ > u_n, W_n⁰ ≤v_n^´≥P µn[⁰−1

i=1

A_i,

n[⁰−1 i=1

B_i

¶

≥

nX⁰−1 i=1

P(A_i, B) − ^X

1≤i<j≤n⁰−1

P(A_i, A_j, B) and, using again the same inequality, we get

lim inf

n→+∞

nX⁰−1 i=1

P(A_i, B)≥ (16)

≥lim inf

n→+∞

nX⁰−1 i=1

nX⁰−1 j=1

P(A_i, B_j)−lim sup

n→+∞

nX⁰−1 i=1

X

1≤j<k≤n⁰−1

P(A_i, B_j, B_k) . Moreover, since

nX⁰−1 i=1

X

1≤j<k≤n⁰−1

P(A_i, B_j, B_k) =

= ^X

1≤i<j<k≤n⁰−1

P(A_i, B_j, B_k) +P(B_i, A_j, B_k) +P(B_i, B_j, A_k)

≤S^e_n,k

andD(u^e _n, v_n) holds, from (16) it results lim inf

n→+∞

nX⁰−1 i=1

P(Ai, B)≥ β3

k +o_k(1/k) . Let’s recall (15). Considering again Boole’s inequality we get

(17)

lim sup

n→+∞

X

1≤i<j≤n⁰−1

P(A_i, A_j, B)≤lim sup

n→+∞

X

1≤i<j≤n⁰−1 nX⁰−1

k=1

P(A_i, A_j, B_k)

≤lim sup

n→+∞

Se_n,k=o_k(1/k) and thus

(18) lim inf

n→+∞P^³M_n⁰ > u_n, W_n⁰ ≤v_n^´≥ β₃

k +o_k(1/k) .

From (14) and (18), we have

(19)

β3

k +o_k(1/k)≤lim inf

n→+∞P^³M_n⁰ > un, W_n⁰ ≤vn

´

≤lim sup

n→+∞ P^³M_n⁰ > un, W_n⁰ ≤vn

´

≤ β3

k +o_k(1/k) .

Finally, putting α =ν₁+ν₂−β₁−β₂−β₃ we conclude from (9), (11), (12) and (19), that

(20)

1−α

k +o_k(1/k)≤lim inf

n→+∞P^³M_n⁰ ≤u_n, W_n⁰ > v_n^´

≤lim sup

n→+∞ P^³M_n⁰ ≤u_n, W_n⁰ > v_n^´

≤1−α

k +o_k(1/k) which implies

(21) lim sup

n→+∞

¯¯

¯¯

¯P^³M_n⁰ ≤un, W_n⁰ > vn

´−1 +α k

¯¯

¯¯=o_k(1/k) .

Observe now that lim sup

n→+∞

¯¯

¯¯P^³M_n≤u_n, W_n> v_n^´−e^−α

¯¯

¯¯≤

(22)

≤lim sup

n→+∞

¯¯

¯¯P^³M_n≤u_n, W_n> v_n^´−P^k³M_n⁰ ≤u_n, W_n⁰ > v_n^´¯¯¯_¯

+ lim sup

n→+∞

¯¯

¯¯P^k³M_n⁰ ≤u_n, W_n⁰ > v_n^´−^³1− α k

´k¯¯¯¯

+

¯¯

¯¯e^−α−^³1−α k

´k¯¯¯¯ .

Using Lemma 1, the first term of the right hand side of (22) is zero. Moreover, using the well known inequality

¯¯

¯¯ Yk i=1

ai− Yk i=1

bi

¯¯

¯¯ ≤ Xk i=1

|ai−bi|

with a1, ..., a_k, b1, ..., b_k in [0,1], we conclude that the second term of the right hand side of (22) is bounded by lim sup

n→+∞ k|P(M_n⁰ ≤u_n, W_n⁰ > v_n)−(1−α k)|.

Hence, by (21) and (22), we deduce that

k→+∞lim lim sup

n→+∞

¯¯

¯¯P^³M_n≤u_n, W_n> v_n^´−e^−α

¯¯

¯¯≤ (23)

≤ lim

k→+∞

¯¯

¯¯e^−α−^³1−α k

´k¯¯¯¯= 0

which enables us to conclude that lim

n→+∞P(M_n≤u_n, W_n> v_n) =e^−α.

The following two results are important tools on the establishment of the asymptotic independence of the maxima and minima.

Corollary 1. Suppose that {X_n} is a stationary sequence under the as- sumptions of Theorem 1. Then, {Mn ≤un} and {Wn > vn} are asymptotically independent if and only ifβ₃ = 0.

Proof: SinceD(u_n, v_n) holds, we obtain lim

n→+∞{P(M_n≤u_n)−P^k(M_n⁰≤u_n)}

= 0 and lim

n→+∞{P(W_n> v_n)−P^k(W_n⁰ > v_n)}= 0.

On the other hand, it results from (11) that lim sup

n→+∞

¯¯

¯¯P(M_n⁰ ≤un)−^³1−ν1−β1

k

´¯¯¯¯=o_k(1/k) .

Therefore, with the arguments used in (22) and (23), we deduce that

(24) lim

n→+∞P(Mn≤un) =e^−ν1+β1 . Similarly we prove that

(25) lim

n→+∞P(Wn> vn) =e^−ν2+β2 .

So {M_n ≤u_n} and {W_n> v_n} 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 D(ue n, vn).

Definition 2. The sequence {X_n} satisfies condition C(u^e _n, v_n) if

k→+∞lim lim sup

n→+∞ kC^e_n,k = 0 where Ce_n,k = ^X

1≤i<j<k≤n⁰−1

nP(A_i, A_j, A_k) +P(B_i, B_j, B_k)^o.

Indeed, we will prove that, ifβ₃ = 0, it is enough to considerC(u^e _n, v_n) instead ofD(u^e n, vn).

Theorem 2. Suppose that the stationary sequence{X_n}satisfiesD(u_n, v_n) andC(u^e n, vn) where{un}and {vn}are real sequences satisfying, for all positive integerk, (3), (4), (5), (7) and (6) withβ₃ = 0. Then,{M_n≤u_n}and{W_n> v_n} are asymptotically independent with

n→+∞lim P^³M_n≤u_n, W_n> v_n^´=e^{−(ν1+ν2−β1−β2)} .

Proof: Observe that we established (24) and (25) only using the first and the fourth terms ofS^e_n,k. Moreover, with β₃ = 0, from (14) we deduce

lim sup

n→+∞

P^³M_n⁰ > u_n, W_n⁰ ≤v_n^´=o_k(1/k).

Then, (20) is similarly obtained (with β₃ = 0), and the result follows imme- diately.

3 – Example

Let{Y_n}and{Z_n}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 {u_n} satisfying

n→+∞lim n(1−H(u_n)) =τ_Y and lim

n→+∞n(1−G(u_n)) =τ_Z , withτ_Y and τ_Z in [0,+∞[.

Let {T_n} be an i.i.d. sequence, independent of {Y_n} and {Z_n}, with support {1,2,3}and P(T₁=i) =p_i,i= 1,2,3.

Define

Xn=







Yn, Tn= 1,

max{Y_n−2, Z_n}, T_n= 2,

−Y_n−1, T_n= 3 .

We easily prove that {X_n} is stationary and 2-dependent with marginal dis- tribution function

F(x) =H(x)p₁+H(x)G(x)p₂+ (1−H(−x))p₃, x∈R, and satisfiesD(u_n,−u_n).

Moreover, {X_n} does not satisfy eitherD^∗(u_n,−u_n) or D⁰⁰(u_n) once lim sup

n→+∞ n

[n/k]

X

j=2

P^³X₁> u_n, X_j ≤u_n< X_j+1^´ → τ_Y p₁p₂, k→+∞ . We will prove now thatD(u^e _n,−u_n) holds. Observe first that lim

n→+∞nF(−u_n) = τ_Yp₃ and

(26) lim

n→+∞n(1−F(un)) =τY p1+ (τY +τZ)p2 . Indeed, since ^X

1≤i<j<k≤n⁰−1

P(A_i, A_j, A_k) is bounded by n

k

X

3≤i<j≤n⁰−1

P(A₁, A_i, A_j)≤

≤ n k

X

2≤i<j≤n⁰−1

P^³X₁> u_n, A_i, A_j^´

≤ n k

nX⁰−3 i=2

½

P^³X₁> u_n, X_i+1 > u_n, X_i+3> u_n^´

+

nX⁰−1 j=i+3

P^³X₁ > u_n, X_i+1> u_n, X_j+1> u_n^´¾

≤ n k

nX⁰−3 i=2

P(X1 > un)P(Xi+3 > un) +n

k

nX⁰−3 i=2

n⁰

X

j=i+4

P^³X₁> u_n, X_i+1 > u_n^´P(X_j > u_n)

≤ n² k²

³P(X₁ > u_n)^´2+n²

k² P(X₁ > u_n)

nX⁰−3 i=2

P^³X₁> u_n, X_i+1 > u_n^´

= n² k²

³P(X₁ > u_n)^´2

+n²

k² P(X₁ > u_n)

½

P^³X₁> u_n, X₃ > u_n^´+

nX⁰−2 i=4

P(X₁ > u_n)P(X_i> u_n)

¾

≤ 2n² k²

³P(X1 > un)^´2+n³ k³

³P(X1> un)^´3

using (26), we conclude that

k→+∞lim lim sup

n→+∞ k ^X

1≤i<j<k≤n⁰−1

P(A_i, A_j, A_k) = 0.

Analogously we prove the same for the other terms ofS^e_n,k. ThenD(u^e _n,−u_n) 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

n→+∞P(X₁≤u_n< X₂, T2= 3) = 0 and using the Total Probability Rule, we have

(27)

n P(X1≤un< X2) =n P(Y1 ≤un< Y2)p1p1

+n P^³Y₁ ≤u_n,max{Y₀, Z₂}> u_n^´p₁p₂ +n P^³max{Y₋₁, Z₁} ≤u_n, Y₂> u_n^´p₁p₂

+n P^³max{Y₋₁, Z₁} ≤u_n,max{Y₀, Z₂}> u_n^´p₂p₂ +n P(Y1> un)p1p3+n P(max{Y0, Z2}> un)p2p3 . Thereforeν1= lim

n→+∞nP(X1≤un< X2) = (τY+τZ)p2+τY p1. Using similar arguments and observing that

P^³X₁ >−u_n≥X₂, T₂= 1^´=P^³X₁>−u_n≥X₂, T₂ = 2^´→0, n→+∞, it resultsν₂ =τ_Yp₃.

In what concerns the evaluation ofβ₁, we have X

1≤i<j≤n⁰−1

P(Ai, Aj) =

nX⁰−1 j=3

(n⁰−j)P(A1, Aj)

= (n⁰−3)P(A₁, A₃) +

nX⁰−1 j=4

(n⁰−j)P(A₁, A_j). Since

nX⁰−1 j=4

(n⁰−j)P(A₁, A_j)≤n⁰

nX⁰−1 j=4

P(X₂ > u_n)P(X_j+1 > u_n)

≤ n² k²

³P(X2 > un)^´2 ,

it follows that

n→+∞lim

X

1≤i<j≤n⁰−1

P(A_i, A_j) = lim

n→+∞

n

kP(A₁, A₃) +o_k(1/k) .

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 thatβ1=τ_Y p1p2.

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

n→+∞lim

nX⁰−1 i=1

nX⁰−1 j=1

P(A_i, B_j) =

nX⁰−1 j=2

(n⁰−j)P(A₁, B_j) +

nX⁰−1 j=2

(n⁰−j)P(B₁, A_j)

= (n⁰−2)P(A₁, B₂) + (n⁰−3)P(A₁, B₃)

+ (n⁰−2)P(B₁, A₂) + (n⁰−3)P(B₁, A₃) +o_k(1/k) and lim

n→+∞nP(A₁, B₃) = lim

n→+∞nP(B₁, A₃) = 0,it resultsβ₃ =τ_Y(p₁p₃+p₂p₃).

Finally, we conclude that

n→+∞lim P^³M_n≤u_n, W_n> v_n^´=e^−α where α =τ_Y +τ_Zp₂−τ_Y(p₁p₂+p₁p₃+p₂p₃).

It should be noticed that u_n =u_n(x) and v_n=v_n(y). Hence, the parameters τ_Y, τ_Z, 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.

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Maria da Gra¸ca Temido,

Departamento de Matem´atica, Universidade de Coimbra, Largo D. Dinis, 3000 Coimbra – PORTUGAL