The asymptotic locations of the maximum and minimum of stationary sequences

Journal of Statistical Planning and Inference 104 (2002) 287–295

www.elsevier.com/locate/jspi

The asymptotic locations of the maximum and minimum of stationary sequences L. Pereira, H. Ferreira ∗ Department of Mathematics, University of Beira Interior, Convento de San Antonio, 6200-001 Covilh˜a, Portugal Received 21 June 2000; received in revised form 20 February 2001; accepted 26 June 2001

Abstract We show that the normalized 4rst location of the maximum of a stationary sequence satisfying a long-range dependence condition converges to a uniform variable on [0; 1] and is asymptotically independent of the height of the maximum. Additional conditions are given in order to obtain c 2002 Elsevier the asymptotic independence of the locations of maximum and minimum.
Science B.V. All rights reserved. MSC: 62G55; 60G70 Keywords: Extreme value theory; Dependence conditions; Location of maxima

1. Introduction For

a

sequence

of

random

Ws; t = mins6j6t {Xj } and Ln tively, id est, Ln

(1)

(1)

variables

and Ln

(1)

{Xn }n¿1 ,

let

Ms; t = maxs6j6t {Xj },

the 4rsts locations of M1; n and W1; n ; respec-

= min{1 6 j 6 n: M1; n = Xj }

and Ln (1) = min{1 6 j 6 n: W1; n = Xj };

n ¿ 1:

We shall assume that {Xn }n¿1 is strictly stationary and that there are constants {an ¿ 0}n¿1 and {bn }n¿1 such that, for each x ∈ R, P(a−1 n (M1; n − bn ) 6 x) → H (x);

(1.1)

where H is a nondegenerate distribution function. If {Xn }n¿1 is a sequence of independent and identically distributed variables or if it satis4es the long-range dependence condition D(un (x)) from Leadbetter (1974), ∗ Tel.: +351-275-319000; fax: +351-275-319057. E-mail address: [email protected] (H. Ferreira).

c 2002 Elsevier Science B.V. All rights reserved. 0378-3758/02/$ - see front matter  PII: S 0 3 7 8 - 3 7 5 8 ( 0 1 ) 0 0 2 5 9 - 2

288

L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

with un (x) = an x + bn , then the Extremal Type Theorem holds, id est; H is a Gumbel, Weibull or a FrDechet distribution (Galambos, 1987; Leadbetter et al., 1983). We say that the stationary sequence {Xn }n¿1 , with marginal distribution F, has extremal index ; 0 ¡  6 1, if for each  ¿ 0 there exists {un() }n¿1 such that n(1 − F(un() )) −→  and P(Mn 6 un() ) −→ e− (Leadbetter, 1974). When {Xn }n¿1 has n→∞

n→∞

extremal index , then (1.1) holds if and only if 1 n(1 − F(un (x))) −→ (x) = − log H (x); x ∈ R: n→∞  If {Xn }n¿1 is a sequence of independent and identically distributed variables or if it satis4es both the conditions D(un (x)) and D (un (x)), from Leadbetter (1983), then  = 1 and the normalized 4rst location of the maximum is asymptotically uniform and independent of its height. These results can be found in Leadbetter et al. (1983) after the discussion of the limit distribution of point processes of exceedances of un by the variables X1 ; X2 ; : : : ; Xn ; when n → ∞. The object of Section 2 is to show that, for all 0 ¡  ¡ 1, the same asymptotic (1) behavior can be applied to Ln =n, if we assume an appropriate generalization of the D(un (x)) condition instead both long and local conditions D and D . In Section 3, we deal with the asymptotic joint distributions of the maximum and minimum and their locations, as n → ∞. We prove under appropriate generalizations of the conditions D(un ; vn ) and C2 (un ; vn ) of Davis (1979) that, for each k1 ; k2 ∈ (0; 1), the random vectors (M1; [nt1 ] ; M[nt1 ]+1;n ) and (W1; [nt2 ] ; W[nt2 ]+1;n ); under linear normalizations, are asymptotically independent, which, in turn, will lead to the asymptotic independence of the locations of maximum and minimum. 2. The asymptotic rst location of the maximum We shall work here with the following condition which enable us to obtain the asymptotic independence of the maxima M1; [nk] and M[nk]+1; n , for each k ∈ ]0; 1[. Such condition was 4rst introduced in Gomes (1978) and is suitable for the joint distribution of maxima of random variables taken from disjoint intervals of a univariate stationary sequence. It is also considered in Leadbetter et al. (1983) and was there denoted by Dr (un ) where un = (un1 ; : : : ; unr ) and unj ; 1 6 j 6 r; are normalized levels. We shall take r = 2. Denition 2.1. For a given vector sequence of reals {(un1 ; un2 )}n¿1 , the sequence ∗ {Xn }n¿1 satis4es the condition D2 (un1 ; un2 ) if, for any choice of uni in {un1 ; un2 };
 p  q
  ∗ ∗
P sup {Xis 6 uni }; {Xjs 6 unj } n; l = s s
16i1 ¡···¡ip ¡j1 ¡···jq 6n j1 −ip ¿l



−P

p  s=1

s=1

  q 

 ∗ P {Xjs 6 unjs }



∗ {Xis 6 uni } s

s=1

is such that n; ln −→ 0 for some sequence ln = o(n). n→∞

s=1

(2.1)

L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

289

When un1 = un2 = un we get the condition D(un ) from Leadbetter. The next lemma (see Leadbetter et al., 1983) establishes, for each k ∈ (0; 1); the asymptotic independence of the events {M1; [nk] 6 un1 } and {M[nk]+1; n 6 un2 } under the condition D2 (un1 ; un2 ); and is the key to obtain the asymptotic independence of the location and height of the maximum of {Xn }n¿1 . Lemma 2.1. Suppose the condition D2 (un1 ; un2 ) holds for {Xn }n¿1 and {kn }n¿1 is a sequence of integers for which there exists a sequence {ln }n¿1 such that kn ln =n −→ 0 n→∞

and kn n; ln −→ 0; where n; l is the coe8cient of the condition D2 (un1 ; un2 ). Then; for n→∞

each k ∈ (0; 1);

P(M1; [nk] 6 un1 ; M[nk]+1; n 6 un2 ) − P(M1; [nk] 6 un1 )P(M[nk]+1; n 6 un2 ) −→ 0: n→∞

(1)

We are now ready to prove that Ln =n and a−1 n (M1; n − bn ) are asymptotically (1) independent, Ln =n being asymptotically uniform. Proposition 2.1. Let {Xn }n¿1 be a stationary sequence with extremal index 0 ¡  6 1 and {an ¿ 0}n¿1 and {bn }n¿1 sequences of constants such that P(M1; n 6 an x + bn ) −→ G  (x) n→∞

with a nondegenerate distribution function G. If; for each x1 ; x2 ∈ R and uni = an xi + bn ; i = 1; 2; {Xn }n¿1 satis;es the condition D2 (un1 ; un2 ) then  (1)  Ln −1 6 k; an (M1; n − bn ) 6 x −→ kG  (x); k ∈ [0; 1]; x ∈ R: P n→∞ n Proof. We start from a key relation between the 4rst location of the maximum and partial maxima. For each k ∈ [0; 1], it holds P(Ln

(1)

6 nk; M1; n 6 an x + bn )

= P(M1; [nk] ¿ M[nk]+1; n ; M1; n 6 an x + bn ) −1 = P(a−1 n (M[nk]+1; n − bn ) 6 an (M1; [nk] − bn ) 6 x):

(2.2)

By applying Lemma 2.1, with xi ∈ R and uni = un (xi ) = an xi + bn ; i = 1; 2, the probability in (2.2) converges to P(Vk 6 Uk 6 x) where Uk and Vk are independent random variables whose distributions can be obtained as follows. Since {Xn }n¿1 has extremal index  and [nk](1 − F(un (t)) −→ − k log G(t); n→∞

t ∈R

then P(Uk 6 t) = lim P(M1; [nk] 6 un (t)) = G k (t) n→∞

(2.3)

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L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

and, by using the stationary P(Vk 6 t) = lim P(M1; [n(1−k)] 6 un (t)) = G (1−k) (t): n→∞

(2.4)

Therefore, from (2.2), (2.3) and (2.4), we get lim P(Ln

(1)

n→∞

6 nk; M1; n 6 an x + bn ) = P(Vk 6 Uk 6 x)  =

]−∞;x]

G (1−k) (t) dG k (t) = kG  (x):

An analogous result holds for the location of the second maximum, under the conditions D (uni ); i ∈ {1; 2; 3; 4}; and D4 (un1 ; un2 ; un3 ; un4 ) (Leadbetter et al., 1983). Unfortunately, we have not been yet able to carry through the necessary calculations for the asymptotic location of any order statistic, when 0 ¡  ¡ 1 and only a long-range dependence condition holds. The condition G(un ) from Nandagopalan (1990) extends condition G(un ) in Hsing et al. (1988) and leads to an asymptotic independence of partial order statistics analogous to that in Lemma 2.1. However, a value 0 ¡  ¡ 1 indicates the possibility of occurrence of clusters of high values in the sequence {Xn }n¿1 . Our result shows that this does not aJect the location of the 4rst maximum but the question of the same behavior for all order statistics is left open.

3. Asymptotic independence of the locations of maximum and minimum In a series of papers (Davis, 1979, 1982, 1983), Davis studied the joint distribution of M1; n and W1; n for stationary sequences {Xn }n¿1 under a number of dependence conditions. He proved that if a stationary sequence satis4es the conditions D(un ; vn ) and C2 (un ; vn ); de4ned below, then the normalized maximum and minimum are asymptotically independent (Proposition 3.1 in Davis (1982)). Let {un }n¿1 and {vn }n¿1 be sequences of real numbers. The condition D(un ; vn ) holds for {Xn }n¿1 when n; ln −→ 0; for some sequence ln = o(n); where n→∞

n; l = sup n;(i)l ;

(3.1)

i=1;2;3

n;(1)l =

sup 16i1 ¡···¡ip ¡j1 ¡···¡jq 6n j1 −ip ¿l

 −P

p  s=1


 p  q

P  {Xi 6 un };  {Xj 6 un } s s
s=1

s=1

  q 

 {Xis 6 un } P {Xjs 6 un }

; s=1

(3.2)

L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

n;(2)l

=

sup 16i1 ¡···¡ip ¡j1 ¡···¡jq 6n j1 −ip ¿l

 P n;(3)l =

p  s=1

−P

s=1

s=1

p  s=1

(3.3)

s=1

16i1 ¡···¡ip ¡j1 ¡···¡jq 6n j1 −ip ¿l




 p  q
 
P {Xis ¿ vn }; Xjs ¿ vn


  q 

 {Xis ¿ vn } P {Xjs ¿ vn }

; sup

291


 p  q

P  {vn ¡ Xi 6 un };  {vn ¡ Xj 6 un } s s
s=1

s=1

  q 

 {vn ¡ Xis 6 un } P {vn ¡ Xjs 6 un }

: s=1

(3.4)

The condition C2 (un ; vn ) holds for {Xn }n¿1 if lim sup n

r n −1

n→∞

j=1

(P(X1 ¿ un ; Xj+1 6 vn ) + P(X1 6 vn ; Xj+1 ¿ un )) = 0

where rn = [n=kn ]; [s] denotes the greatest integer not greater than s and {kn }n¿1 is a sequence of integer numbers such that kn → ∞. We shall de4ne generalizations of the above conditions of Davis in order to obtain, for every subintervals I; J; I  ; J  of {1; 2; : : : ; n} such that I ∪J = I  ∪J  = {1; 2; : : : ; n} and I ∩J = I  ∩J  = ∅; the asymptotic independence of the events {M (I ) 6 un1 ; M (J ) 6 un2 } and {W (I  ) ¿ vn1 ; W (J  ) ¿ vn2 }, where M (I ) = max{Xj : j ∈ I }; W (I ) = min{Xj : j ∈ I } and the levels uni ; vni ; 1 6 i 6 2; satisfy n(1−F(uni )) → i ; nF(vni ) → i . This, in turn, will lead to the asymptotic independence of the locations of maximum and minimum. Denition 3.1. Let {uni }n¿1 ; {vni }n¿1 ; i = 1; 2; be sequences of real numbers. The sequence {Xn }n¿1 satis4es the condition D((un1 ; un2 ); (vn1 ; vn2 )) if in (3.2) – (3.4) ∗ ∗ ∗ ∗ we consider, respectively, the events {Xi 6 uni }; {Xi ¿ vni } and {vni ¡ Xi 6 uni }; where ∗ ∗ uni ∈ {un1 ; un2 } and vni ∈ {vn1 ; vn2 }. ∗

We shall de4ne the mixing coeKcients in condition D((un1 ; un2 ); (vn1 ; vn2 )) by n;(1l ) ; ∗

∗

∗

n;(2l ) ; n;(3l ) and supi=1; 2; 3 n;(i l ) = n;∗ l . It is worth noting that the condition D((un1 ; un2 ); (vn1 ; vn2 )) is equivalent to conditions ∗ D2 (un1 ; un2 ) for {Xn }n¿1 ; D2 (−vn1 ; −vn2 ) for {−Xn }n¿1 and n;(3ln) = o(1), jointly. Denition 3.2. The sequence {Xn }n¿1 satis4es the condition C2 ((un1 ; un2 ); (vn1 ; vn2 )) if lim sup kn n→∞

[n=k n ][n=k n ] i=1 j=1

∗ ∗ P(Xi ¿ uni ; Xj 6 vnj ) = 0;

∗ ∗ where uni ∈ {un1 ; un2 }; vnj ∈ {vn1 ; vn2 } and {kn }n¿1 is a sequence of integer numbers such that kn → ∞.

If un1 = un2 = un and vn1 = vn2 = vn we obtain the conditions D(un ; vn ) and C2 (un ; vn ).

292

L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

The following lemma is a generalization of Lemma 3.1 of Davis (1979). Lemma 3.1. Let {uni }n¿1 ; {vni }n¿1 ; i = 1; 2; be sequences of real numbers such that; for each i ∈ {1; 2}; nF(vni ) → i ;

n(1 − F(uni )) → i ;

(3.5)

where 1 ; 2 ; 1 ; 2 ¡ ∞. If the stationary sequence {Xn }n¿1 satis;es the condition n;∗ ln = o(1); {kn }n¿1 is a sequence of integer numbers satisfying kn ln −→ 0; n n→∞

kn n;∗ ln −→ 0; n→∞

kn → ∞

(3.6)

and J1 ; J2 ; : : : ; Jkn are disjoint subintervals of {1; 2; : : : ; n}; then   k n ∗ ∗ {vni ¡ W (Ji ) ¡ M (Ji ) 6 uni } P i=1

−

kn i=1

∗ ∗ ) −→ 0; P(vni ¡ W (Ji ) ¡ M (Ji ) 6 uni

(3.7)

n→∞

∗ ∗ where uni ∈ {vn1 ; vn2 }. ∈ {un1 ; un2 } and vni

The proof is omitted since it follows the same lines of argument as in the proof of Lemma 2:5 of Leadbetter (1983) with the obvious modi4cations. As a consequence of Lemma 3.1, it follows that if J1 ; : : : ; Js are disjoint subintervals of {1; 2; : : : ; n}; Jk having approximately (k n members, for 4xed positive (1 ; : : : ; (s such s that i=1 (i 6 1; then   s  ∗ ∗ {vni ¡ W (Ji ) ¡ M (Ji ) 6 uni } P i=1

−

s i=1

∗ ∗ P(vni ¡ W (Ji ) ¡ M (Ji ) 6 uni ) −→ 0:

(3.8)

n→∞

Lemma 3.2. Let {Xn }n¿1 and {−Xn }n¿1 be sequences with extremal indexes 1 and 2 ; respectively. If the stationary sequence {Xn }n¿1 satis;es the conditions D((un1 ; un2 ); (vn1 ; vn2 )) and C2 ((un1 ; un2 ); (vn1 ; vn2 )) where the levels un1 ; un2 ; vn1 ; vn2 satisfy (3:5) and the sequence {kn }n¿1 veri;es (3:6); then; for each t1 ; t2 ∈ (0; 1); P(M1; [nt1 ] 6 un1 ; M[nt1 ]+1;n 6 un2 ; W1; [nt2 ] ¿ vn1 ; W[nt2 ]+1;n ¿ vn2 ) −P(M1; [nt1 ] 6 un1 ; M[nt1 ]+1;n 6 un2 )P(W1; [nt2 ] ¿ vn1 ; W[nt2 ]+1;n ¿ vn2 ) −→ 0: n→∞

Proof. First, we calculate lim P(M1; [nt1 ] 6 un1 ; M[nt1 ]+1;n 6 un2 ; W1; [nt2 ] ¿ vn1 ; W[nt2 ]+1;n ¿ vn2 ): n

(3.9)

L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

293

By supposing, for example, that t1 ¡ t2 ; let I1 = {1; : : : ; [nt1 ]}; I2 = {[nt1 ] + 1; : : : ; [nt2 ]} and I3 = {[nt2 ] + 1; : : : ; n}: Then, by applying (3.8), we obtain that (3.9) is equal to lim P(vn1 ¡ W (I1 ) ¡ M (I1 ) 6 un1 )P(vn1 ¡ W (I2 ) ¡ M (I2 ) 6 un2 ) n

×P(vn2 ¡ W (I3 ) ¡ M (I3 ) 6 un2 ):

(3.10)

On the other hand, by attending that the conditions C2 ((un1 ; un2 ); (vn1 ; vn2 )) and D((un1 ; un2 ); (vn1 ; vn2 )) imply C2 (uni ; vnj ) and D(uni ; vnj ); for each i; j ∈ {1; 2}; it follows, from Proposition 3.1. (Davis, 1982), that (3.10) is equal to lim P(M (I1 ) 6 un1 )P(W (I1 ) ¿ vn1 )P(M (I2 ) 6 un2 )P(W (I2 ) ¿ vn1 ) n

×P(M (I3 ) 6 un2 )P(W (I3 ) ¿ vn2 ): Therefore, by using stationarity and attending that the sequences {Xn }n¿1 and {−Xn }n¿1 have extremal indexes 1 and 2 , respectively, we obtain lim P(M1; [nt1 ] 6 un1 ; M[nt1 ]+1;n 6 un2 ; W1; [nt2 ] ¿ vn1 ; W[nt2 ]+1;n ¿ vn2 ) n

= exp(−1 t1 1 − 1 2 (1 − t1 ) − 2 1 t2 − 2 2 (1 − t2 )): Now, since {Xn }n¿1 and {−Xn }n¿1 verify D2 (un1 ; un2 ) and D2 (−vn1 ; −vn2 ), respectively, by Lemma 2.1 we have P(M1; [nt1 ] 6 un1 ; M[nt1 ]+1;n 6 un2 )P(W1; [nt2 ] ¿ vn1 ; W[nt2 ]+1;n ¿ vn2 ) = P(M1; [nt1 ] 6 un1 )P(M[nt1 ]+1;n 6 un2 )P(W1; [nt2 ] ¿ vn1 ) ×P(W[nt2 ]+1;n ¿ vn2 ) + o(1); which also converges to exp(−1 t1 1 − 1 2 (1 − t1 ) − 2 1 t2 − 2 2 (1 − t2 )): Proposition 3.1. Let {Xn }n¿1 be a stationary sequence and 1 and 2 the extremal indexes of {Xn }n¿1 and {−Xn }n¿1 ; respectively. Let {an ¿ 0}n¿1 ; {bn }n¿1 ; {cn ¿ 0}n¿1 and {dn }n¿1 be sequences of constants such that 1 P(a−1 n (M1; n − bn ) 6 x1 ) → G (x1 )

and P(cn−1 (W1; n − dn ) 6 y1 ) → 1 − (1 − H (y1 ))2 ; where G and H are nondegenerate distribution functions. If; for each x1 ; x2 ; y1 ; y2 ∈ R and uni = an xi + bn ; vni = cn yi + dn ; i ∈ {1; 2}; the sequence {Xn }n¿1 satis;es the conditions D((un1 ; un2 ); (vn1 ; vn2 )) and C2 ((un1 ; un2 );

294

L. Pereira, H. Ferreira / Journal of Statistical Planning and Inference 104 (2002) 287–295

(vn1 ; vn2 )) with {kn }n¿1 a sequence of integer numbers verifying (3:6); then; for each t1 ; t2 ∈ (0; 1); P(Ln

(1)

(1) ¿ nt2 ; cn−1 (W1; n − dn ) ¿ y1 ) 6 nt1 ; a−1 n (M1; n − bn ) 6 x1 ; Ln

−→ t1 (1 − t2 )G 1 (x1 )(1 − H (y1 ))2 :

n→∞

Proof. For each t1 ; t2 ∈ [0; 1]; we have P(Ln

(1)

6 nt1 ; M1; n 6 an x1 + bn ; Ln (1) ¿ nt2 ; W1; n ¿ cn y1 + dn ) = P(M1; [nt1 ] ¿ M[nt1 ]+1;n ; M1; n 6 an x1 + bn ; W1; [nt2 ] ¿ W[nt2 ]+1;n ; W1; n ¿ cn y1 + dn ) −1 = P(a−1 n (M[nt1 ]+1;n − bn ) 6 an (M1; [nt1 ] − bn ) 6 x1 ;

cn−1 (W1; [nt2 ] − dn ) ¿ cn−1 (W[nt2 ]+1;n − dn ) ¿ y1 ):

(3.11)

So, by using Lemma 3.2, the probability in (3.11) converges to P(Vt1 6 Ut1 6 x1 ; Ut2 ¿ Vt2 ¿ y1 ) where (Ut1 ; Vt1 ) is independent of (Ut2 ; Vt2 ). Moreover both vectors have independent margins and P(Ut1 6 t) = G 1 t1 (t); P(Ut2 6 t) = 1 − (1 − H (t))2 t2 ; P(Vt1 6 t) = G (1−t1 )1 (t); and P(Vt2 6 t) = 1 − (1 − H (t))2 (1−t2 ) : Therefore lim P(Ln

n→∞

(1)

6 nt1 ; M1; n 6 an x1 + bn ; Ln (1) ¿ nt2 ; W1; n ¿ cn y1 + dn ) = P(Vt1 6 Ut1 6 x1 )P(Ut2 ¿ Vt2 ¿ y1 )  = G 1 (1−t1 ) (t) dG t1 1 (t) ]−∞; x1 ]



×

]y1 ;+∞]

(1 − H (t))2 t2 d(1 − (1 − H (t))2 (1−t2 ) )

= t1 (1 − t2 )G 1 (x1 )(1 − H (y1 ))2 : Acknowledgements We are grateful to the referee for his rigorous report, suggestions and corrections which helped in improving the 4nal form of this paper and pointed out new challenges. This work was partially supported by FCT=POCTI=FEDER.

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295

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