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A note on maxima of bivariate random vectors
S'rA11111Cl t PROBABILITY I.ETlIR$ ELSEVIER
Statistics & Probability Letters 31 (1996) 1-6
A note on maxima o f bivariate random vectors G. H o o g h i e m s t r a a, j. Hiisler b, • a Department of Technical Mathematics, Delft University, The Netherlands b Department of Statistics, University of Bern, 3012 Bern, Switzerland Received October 1995; revised February 1996
Abstract For i.i.d, bivariate normal vectors we consider the maxima of the projections with respect to two arbitrary directions. A limit theorem for these maxima is proved for the case that the angle of the two directions approaches zero. The result is generalized to a functional limit theorem.
Keywords." Maxima; Triangular array; Bivariate normal random vectors; Limiting distribution; Extreme value distribution; Process of maxima with respect to directions; Dependence
I. Introduction
Let { ( Y in, Zi.), 1 ~i 1} be a triangular array of bivariate Gaussian random vectors with means 0 and variances 1, being i.i.d, for each fixed n. Then usually the sequence of bivariate maxima M , = (M,1,M,2) is defined componentwise by M.1 = m a x Y,.. i<~n
and
Mn2 = m a x
i<~n
Zi,.
Let Pn be the correlation coefficient o f Yln and Zln. The asymptotic behaviour o f M , as n ~ c~ was analysed by Hiisler and Reiss (1989) (see also Falk et al. (1994)). They showed that the limiting distribution G o f the linearly normalized Mn has independent components A(x) = e x p ( - e x p ( - x ) ) if (1 - p n ) l o g n ~ c~, and completely dependent components if (1 p n ) l o g n ~ 0. In the intermediate case if (1 - p n ) l o g n ~ )2, some positive constant, then the limiting distribution H;, is a max stable (or a bivariate extreme value) distribution with some dependence between the components; we have
P{Mnl <~bn + x/bn,Mn2 <~bn + y/bn} d_~H~(x, y)
* Corresponding author. 0167-7152/96/$12.00 (~) 1996 Elsevier Science B.V. All fights reserved PH S01 67-71 52(96)00005-3
2
G. Hooghiemstra, J. Hfisler / Statistics & Probability Letters 31 (1996) 1 - 6
with normalization bn such that bn = n exp(-b2J2)/v/~, where
Hi~(x,y)=exp(-q~(2+~f-)e-Y-q~(2+~-)e-X). Here • denotes the standard normal distribution function, and b 2 ~ 2 log n. For 2 = 0 and 2 = oo we retrieve the asymptotic complete dependence and independence of the components, respectively,
Ho(x, y ) = min(A(x), A(y)) and
Hoo(x, y) = A(x)A(y). In this note we will show that the distribution function H;, occurs also as limit of maxima in a rather natural situation.
2. Result For this reason let {Xi, i>~ 1} be an i.i.d, sequence o f bivariate Gaussian random vectors where the components X/j have mean 0 and variance 1 for all i,j. We denote the correlation coefficient of Xll and Xl2 by p, IPl < 1. The case [Pl --=1 is simple and not interesting. Instead of defining the maximum componentswise, i.e. with respect to the two fixed directions of the coordinates, we might consider any other two directions in the plane and define the maxima with respect to these two directions. This means that we deal with two linear combinations of the given random vectors Yin = ClnXil -~- c2ngi2
(1)
gin = dlnXil q- d2ngi2
for i<~n and some 9n and din, j = 1,2 with c2n + C2n + 2pclnczn = 1 and d2n + d2n + zpaj . = 1. The usual componentwise maxima of Xi is included by cln = 1, c2n = 0, din = O, d2n = 1 for all n. Yi and Zi are the standardized projections o f Xi on the two directions. If the two directions remain fixed and are not equal, then the limiting distribution of Mn has independent components since p = Pn 74 1. From the mentioned result we get dependent limit components only if the directions gets closer and closer, as n ~ ~ . Without loss of generality we fix one direction and vary the second one. As application we might think for example of windspeed measurements with respect to certain directions. Our paper treats the asymptotic joint behaviour o f the windspeeds in several directions. Another relation exists to the convex hull of the Xi, i <<,n. We might deal with the distance o f the boundary o f the convex hull from the origin with respect to a certain direction. Taking multiple directions we can analyse their joint behaviour. This problem is somewhat related to our posed one, since the linear boundary o f the convex hull is replaced in our case by segments which are circular if we do not standardize the distances. Let us assume for simplicity that the first direction has an angle 0, i.e. ~n = X~, for all i<~n and n~> 1. Then consider the case where the second angle 0n ~ 0 as n --~ cx~. Let 6, = tan On, thus Zin is given by (1) with dln =
1
x/1 +
+ 2p6n
and
d2n =
6n
V/1 +
+ 2p6."
G. Hooghiemstra, J. Hiisler I Statistics & Probability Letters 31 (1996) 1 - 6
For the correlation coefficient Pn of Yln and Zln we obtain easily Pn =
1 +6np V/1 + 2p6n + 62 '
which expression can be approximated for 6n ~ 0 by
2 1 _p2) (1 - Pn) = (1 + 0 ( 1 ) ) 6n( 2
It shows the influence of p on Pn and also that IPl ¢ 1 is necessary. More generally we can select any two directions with angle 01(n) and 02(n) such that An =02(n)-Ol(n)---~ 0. We have
Y~n=
cos 01
x/1 + p sin(20~ )
x,-~ +
sin 01
v/1 + p sin(201 )
x~2
and Zin =
COS 0 2
V/1 + p sin(202)
"Xil +
sin
02
V/1 + p sin(202)
X/2.
(2)
being the standardized projections on the lines with angle Oj = Oj(n), j = 1,2. Then the correlation coefficient Pn is analysed in the same way, A2(1 _ p2) (1 - Pn) = (1 + o(1 ))2(1 +---ps~n(-~l ))2"
(3)
Using the results of Hfisler and Reiss (1989), the limiting distribution of the maxima with respect to the two directions is equal to H;~ iff AZ(1 _ p2) logn ~ 22. 2(1 + p sin(201 ))2
(4)
Proposition 1. Let {Xi, i>>.1} be a sequence of i.i.d, standardized bivariate Gaussian random vectors with correlation p = E(X11X12), IPl < 1. Let the linear transformation be given by (2) with 01 = 01(n) and 02 = 02(n) = 01 + An. Then
P{Mnl <.(bn + x/bn),Mn2 <~(bn + y/bn)} & H~(x,y) as n ---~cx) iff 22 = lim An2(l°gn)(1 - p2) n ~ 2( 1 + p sin(201 ))2' Instead of considering two directions we can deal with a fixed number of directions Oi, i<<.d, and analyse the multivariate maxima with respect to these directions. This limiting distribution is then the d-dimensional extension of H~, denoted by Ha, where A is the matrix of the 2(Oi, Oj) values (cf. Hiisler and Reiss, (1989)). This indicates that it is more convenient to deal with the stochastic process ~n of linearly normalized maxima with respect to the direction 0, with 0 E [0, n) as parameter. More precisely, this global process Cn
4
G. Hooghiemstra, 3. Hfisler I Statistics & Probability Letters 31 (1996) 1 - 6
is defined by
~,(O)=bn
max \i~,
X/I cos 0 + Xt2 sin 0 V/1 + psin(20)
-b,
"~
).
For fixed directions Oi, i<~d, with Oi ~ Oj, i ~ j, the limiting distribution Ga of (~,(01) ..... ~n(Oa)) has independent components: d
Gd(x) = 1"I A(xj), j=l
since all 2(Oi, Oj)= c~. Hence, it is of more interest to analyse the local behaviour of the process ¢. as n ---, c~. Therefore we define
(n(t) = ~n(O + tAn), where
Anbn ~ 1,
(5)
and t E [-T, T], T finite. Relation (5) is equivalent to A n x / ~ n ~ 1. It is convenient to take A. = 1/b.. It is easily seen that ¢. can be written as the process of maxima (taken pointwise) of n independent Gaussian processes Xi
zi(O)
-~
(X/_z cos 0_+_~2 sin 0 ) X/1 + P sin(20) - bn
bn \
and ¢,(0) = max Zi(0). i<~n
Brown and Resnick (1977) considered some special cases of limiting processes of extreme values of independent processes. They dealt also with Gaussian processes Xi, the Brownian motion and the Ornstein Uhlenbeck process. More recently, de Haan and Pickands (1986), Gin0 et al. (1990) and Penrose (1992) treated more general cases. Penrose analysed limiting processes which are min-stable of order ct, which do not include our limiting process. De Haan and Pickands (1986) presented characterizations of min-stable processes via Poisson processes. Gin6 et al. (1990) discussed the class of max-stable processes on a compact metric space and give conditions for sample path continuity. Our limiting process ~ has continuous sample paths, thus belonging to this class, as we will see below. From the above discussion we know that all the finite dimensional distributions of (, converge to the distributions HA which are the finite dimensional distributions of the limiting process ~. Here A is the matrix of the 2(ti, tj), the 2-factor with respect to the directions 0 + tiAn and 0 + tjA,. We have from (5) and (4) that
22(ti, tj) = (ti - tj)2(1 - p2)/(2(1 + p sin(20))2). This shows that ~ is a stationary process with marginal distributions A(x).
Proposition 2. Let {Xi, i>~l} be as in Proposition 1 with Ipl < 1. Condition (5) holds iff as random elements of the space C[-T, T], for any finite positive T.
G. Hooghiemstra, J. Hiisler I Statistics & Probability Letters 31 (1996) 1-6
5
Proof. We treat the case where 0 = 0, the general case with 0 E (0,n) follows similarly. Since the finite dimensional distributions of (n converge to those of ( we only have to prove that (5) implies tightness. Now (n(0) converges in distribution to ((0), and hence is tight, so according to Theorem 8.3 of Billingsley (1968) it is sufficient to show that for each t E I-T, T] and each e > 0, l6Z0 i m l inm~s u p l6P /
sup
kt<.s~t+6
[(n(S)
--
(n(t)[ > e } = 0.
(6)
For the proof of (6) we may restrict to t = 0, the general case t ¢ 0 follows after a simple transformation (rotation and standardization). By series expansion we get that for u ~ 0 cos u X/1 + p sin(2u)
sin u
g/1 +
X/1 + p sin(2u)
Xi2 = (1 - pu)X,.l + uXi2 + O(u2[gil [) "]- O(u21X,21).
We use the representation
x,2 = pX. + v/1 - p2r,, where {Y i, i>>.1} are independent standard normal random variables, being independent of {Xil, i t> 1}. Then we obtain / \ ~.(s) = b. {max[(1 - psAn~il + sA.Xi2] - b. ] + O(s2 A2nbnmax(lXil ], Ix,21))
/
\ i<~n
= bn (max[X/l + sAn V/1 - p2 Yi] - bn) + O(s 2A2nbnmax(IX/1 I, IX,-21)) \ i<<.n / = L(sx/1
-
p2) + O(szZZb. max(IX~l I, IX~2I)),
where
~n(s) = bn (max[Xi i<~n tk
+ sAnYi] - bn) .
Note that the variance of X,.1 + sA,Yi is 1 + s2A 2. For s < 6 the remainder term can be neglected: using An ~ (21ogn) -1/2, P I.~max > i < ~[X,n.I]
e/(O2A2nbn)} ~<2n[1- qb(e/(O2A2nbn))]=O (nAn exp ( - - -
--. 0,
(7)
as n ~ c~, and 6 < x/~. Thus it remains to show that lim lim sup - P
sup
610 n--*o¢~ ~
kO<<.s<~6
-
(8)
~ 0.
Because of the independence between {Y.} and {Xnl} we have sup I~,(s)-~n(O)l= O~s~6
sup bn (max(X/1 + s Y i / b n ) - m a x X i l )
O<~s<~6
sup
O~s~6
\i<~n
i<~n
max(s~ + bn(X(n+l-i) - XCn))) , i~n
]
(9)
6
G. Hooohiemstra, J. Hiisler I Statistics & Probability Letters 31 (1996) 1-6
where )((1) ..... X(n) are the order statistics Of X l l . . . . . Xnl, Define n o w A i = bn(X(n)-S(n+l-i))~O (a.s.). By (9) it follows that
1 [I o<<. sup~<_a( ~ n ( s ) - ~ n ( O ) ) < - ~ } < ~ P { 3 s < 6 " m a x ( s Y l , s Y z - A 2 6P
..... s Y n - A n ) < - ~ }
<~~P{3s < 6 : sYI < - e } ~<~P{Y1 < -~/62} = ~ ( - g / 8 2 )
-'-* 0,
as 8---,0. If max(sY1, sYz -A2,...,sYn - A n ) > e for some s, there exists i = i(s) such that s Y i - A i > e, hence Y/ > 0 ( s i n c e Ai>~O ). It follows that 8Yi - A i > e, and also max(SYl,SY2 - A 2 ..... 8Yn - A n ) > e. This implies that
n(0) > According to (4), nlimooP{~n(0) <~x, ~n(8) <~y} = H~(~)(x, y - 82/2), where 22(8) = (8/2) 2. The density h;. of 11). equals for 2 # 0,
ha(x, y) = H).(x, y){[e-X f ( y - x) + e-X f ' ( y - x) - e-Y f ' ( x - y)] x [e-Yf(x - y) + e - Y f ' ( x - y) - e-Xft(y - x)] + [e-Yf'(x - y) + e - Y f " ( x - y) + e - X f ' ( y - x) + e - X f " ( y - x)]},
(10)
where
f ( t ) = q~(2 + t/(22)). From this explicit form it is straightforward (by considering the terms on the right-hand side of (10) individually) to conclude that for each e > 0,
lim ~ P { ( ( 8 ) - ((O) > e} = lim l f f 6~0
8,[0 ~
h a o ) ( x , y - 82/2)dxdy = O.
{(x,y):y--x>e,}
Together it implies (8) and the tightness. References Billingsley, P. (1968), Weak Convergence of Probability Measures (Wiley, New York). Brown, B.M. and S.I. Resnick (1977), Extreme values of independent stochastic processes, J. Appl. Probab. 14, 732-739. Falk, M., J. Hiisler, and R.D. Reiss, (1994), Laws of Small Numbers: Extremes and Rare Events, DMV Seminar Vol. 23 (Birkh/iuser, Basel). Gin6, E., M.G. Hahn, and P. Vatan (1990), Max-infinitely divisible and max-stable sample continuous processes, Probab. Theory Related Fields 87, 139-165. Haan, L. de and J. Pickands III (1986), Stationary min-stable processes, Probab. Theory Related Fields 72, 477-492. Hiisler, J. and R.D. Reiss (1989), Maxima of normal random vectors: between independence and complete dependence, Statist. Probab. Lett. 7, 283-286. Penrose, M.D. (1992), Semi-min-stable processes, Ann. Probab. 20, 1450-1463.
Statistics & Probability Letters 31 (1996) 1-6
A note on maxima o f bivariate random vectors G. H o o g h i e m s t r a a, j. Hiisler b, • a Department of Technical Mathematics, Delft University, The Netherlands b Department of Statistics, University of Bern, 3012 Bern, Switzerland Received October 1995; revised February 1996
Abstract For i.i.d, bivariate normal vectors we consider the maxima of the projections with respect to two arbitrary directions. A limit theorem for these maxima is proved for the case that the angle of the two directions approaches zero. The result is generalized to a functional limit theorem.
Keywords." Maxima; Triangular array; Bivariate normal random vectors; Limiting distribution; Extreme value distribution; Process of maxima with respect to directions; Dependence
I. Introduction
Let { ( Y in, Zi.), 1 ~i 1} be a triangular array of bivariate Gaussian random vectors with means 0 and variances 1, being i.i.d, for each fixed n. Then usually the sequence of bivariate maxima M , = (M,1,M,2) is defined componentwise by M.1 = m a x Y,.. i<~n
and
Mn2 = m a x
i<~n
Zi,.
Let Pn be the correlation coefficient o f Yln and Zln. The asymptotic behaviour o f M , as n ~ c~ was analysed by Hiisler and Reiss (1989) (see also Falk et al. (1994)). They showed that the limiting distribution G o f the linearly normalized Mn has independent components A(x) = e x p ( - e x p ( - x ) ) if (1 - p n ) l o g n ~ c~, and completely dependent components if (1 p n ) l o g n ~ 0. In the intermediate case if (1 - p n ) l o g n ~ )2, some positive constant, then the limiting distribution H;, is a max stable (or a bivariate extreme value) distribution with some dependence between the components; we have
P{Mnl <~bn + x/bn,Mn2 <~bn + y/bn} d_~H~(x, y)
* Corresponding author. 0167-7152/96/$12.00 (~) 1996 Elsevier Science B.V. All fights reserved PH S01 67-71 52(96)00005-3
2
G. Hooghiemstra, J. Hfisler / Statistics & Probability Letters 31 (1996) 1 - 6
with normalization bn such that bn = n exp(-b2J2)/v/~, where
Hi~(x,y)=exp(-q~(2+~f-)e-Y-q~(2+~-)e-X). Here • denotes the standard normal distribution function, and b 2 ~ 2 log n. For 2 = 0 and 2 = oo we retrieve the asymptotic complete dependence and independence of the components, respectively,
Ho(x, y ) = min(A(x), A(y)) and
Hoo(x, y) = A(x)A(y). In this note we will show that the distribution function H;, occurs also as limit of maxima in a rather natural situation.
2. Result For this reason let {Xi, i>~ 1} be an i.i.d, sequence o f bivariate Gaussian random vectors where the components X/j have mean 0 and variance 1 for all i,j. We denote the correlation coefficient of Xll and Xl2 by p, IPl < 1. The case [Pl --=1 is simple and not interesting. Instead of defining the maximum componentswise, i.e. with respect to the two fixed directions of the coordinates, we might consider any other two directions in the plane and define the maxima with respect to these two directions. This means that we deal with two linear combinations of the given random vectors Yin = ClnXil -~- c2ngi2
(1)
gin = dlnXil q- d2ngi2
for i<~n and some 9n and din, j = 1,2 with c2n + C2n + 2pclnczn = 1 and d2n + d2n + zpaj . = 1. The usual componentwise maxima of Xi is included by cln = 1, c2n = 0, din = O, d2n = 1 for all n. Yi and Zi are the standardized projections o f Xi on the two directions. If the two directions remain fixed and are not equal, then the limiting distribution of Mn has independent components since p = Pn 74 1. From the mentioned result we get dependent limit components only if the directions gets closer and closer, as n ~ ~ . Without loss of generality we fix one direction and vary the second one. As application we might think for example of windspeed measurements with respect to certain directions. Our paper treats the asymptotic joint behaviour o f the windspeeds in several directions. Another relation exists to the convex hull of the Xi, i <<,n. We might deal with the distance o f the boundary o f the convex hull from the origin with respect to a certain direction. Taking multiple directions we can analyse their joint behaviour. This problem is somewhat related to our posed one, since the linear boundary o f the convex hull is replaced in our case by segments which are circular if we do not standardize the distances. Let us assume for simplicity that the first direction has an angle 0, i.e. ~n = X~, for all i<~n and n~> 1. Then consider the case where the second angle 0n ~ 0 as n --~ cx~. Let 6, = tan On, thus Zin is given by (1) with dln =
1
x/1 +
+ 2p6n
and
d2n =
6n
V/1 +
+ 2p6."
G. Hooghiemstra, J. Hiisler I Statistics & Probability Letters 31 (1996) 1 - 6
For the correlation coefficient Pn of Yln and Zln we obtain easily Pn =
1 +6np V/1 + 2p6n + 62 '
which expression can be approximated for 6n ~ 0 by
2 1 _p2) (1 - Pn) = (1 + 0 ( 1 ) ) 6n( 2
It shows the influence of p on Pn and also that IPl ¢ 1 is necessary. More generally we can select any two directions with angle 01(n) and 02(n) such that An =02(n)-Ol(n)---~ 0. We have
Y~n=
cos 01
x/1 + p sin(20~ )
x,-~ +
sin 01
v/1 + p sin(201 )
x~2
and Zin =
COS 0 2
V/1 + p sin(202)
"Xil +
sin
02
V/1 + p sin(202)
X/2.
(2)
being the standardized projections on the lines with angle Oj = Oj(n), j = 1,2. Then the correlation coefficient Pn is analysed in the same way, A2(1 _ p2) (1 - Pn) = (1 + o(1 ))2(1 +---ps~n(-~l ))2"
(3)
Using the results of Hfisler and Reiss (1989), the limiting distribution of the maxima with respect to the two directions is equal to H;~ iff AZ(1 _ p2) logn ~ 22. 2(1 + p sin(201 ))2
(4)
Proposition 1. Let {Xi, i>>.1} be a sequence of i.i.d, standardized bivariate Gaussian random vectors with correlation p = E(X11X12), IPl < 1. Let the linear transformation be given by (2) with 01 = 01(n) and 02 = 02(n) = 01 + An. Then
P{Mnl <.(bn + x/bn),Mn2 <~(bn + y/bn)} & H~(x,y) as n ---~cx) iff 22 = lim An2(l°gn)(1 - p2) n ~ 2( 1 + p sin(201 ))2' Instead of considering two directions we can deal with a fixed number of directions Oi, i<<.d, and analyse the multivariate maxima with respect to these directions. This limiting distribution is then the d-dimensional extension of H~, denoted by Ha, where A is the matrix of the 2(Oi, Oj) values (cf. Hiisler and Reiss, (1989)). This indicates that it is more convenient to deal with the stochastic process ~n of linearly normalized maxima with respect to the direction 0, with 0 E [0, n) as parameter. More precisely, this global process Cn
4
G. Hooghiemstra, 3. Hfisler I Statistics & Probability Letters 31 (1996) 1 - 6
is defined by
~,(O)=bn
max \i~,
X/I cos 0 + Xt2 sin 0 V/1 + psin(20)
-b,
"~
).
For fixed directions Oi, i<~d, with Oi ~ Oj, i ~ j, the limiting distribution Ga of (~,(01) ..... ~n(Oa)) has independent components: d
Gd(x) = 1"I A(xj), j=l
since all 2(Oi, Oj)= c~. Hence, it is of more interest to analyse the local behaviour of the process ¢. as n ---, c~. Therefore we define
(n(t) = ~n(O + tAn), where
Anbn ~ 1,
(5)
and t E [-T, T], T finite. Relation (5) is equivalent to A n x / ~ n ~ 1. It is convenient to take A. = 1/b.. It is easily seen that ¢. can be written as the process of maxima (taken pointwise) of n independent Gaussian processes Xi
zi(O)
-~
(X/_z cos 0_+_~2 sin 0 ) X/1 + P sin(20) - bn
bn \
and ¢,(0) = max Zi(0). i<~n
Brown and Resnick (1977) considered some special cases of limiting processes of extreme values of independent processes. They dealt also with Gaussian processes Xi, the Brownian motion and the Ornstein Uhlenbeck process. More recently, de Haan and Pickands (1986), Gin0 et al. (1990) and Penrose (1992) treated more general cases. Penrose analysed limiting processes which are min-stable of order ct, which do not include our limiting process. De Haan and Pickands (1986) presented characterizations of min-stable processes via Poisson processes. Gin6 et al. (1990) discussed the class of max-stable processes on a compact metric space and give conditions for sample path continuity. Our limiting process ~ has continuous sample paths, thus belonging to this class, as we will see below. From the above discussion we know that all the finite dimensional distributions of (, converge to the distributions HA which are the finite dimensional distributions of the limiting process ~. Here A is the matrix of the 2(ti, tj), the 2-factor with respect to the directions 0 + tiAn and 0 + tjA,. We have from (5) and (4) that
22(ti, tj) = (ti - tj)2(1 - p2)/(2(1 + p sin(20))2). This shows that ~ is a stationary process with marginal distributions A(x).
Proposition 2. Let {Xi, i>~l} be as in Proposition 1 with Ipl < 1. Condition (5) holds iff as random elements of the space C[-T, T], for any finite positive T.
G. Hooghiemstra, J. Hiisler I Statistics & Probability Letters 31 (1996) 1-6
5
Proof. We treat the case where 0 = 0, the general case with 0 E (0,n) follows similarly. Since the finite dimensional distributions of (n converge to those of ( we only have to prove that (5) implies tightness. Now (n(0) converges in distribution to ((0), and hence is tight, so according to Theorem 8.3 of Billingsley (1968) it is sufficient to show that for each t E I-T, T] and each e > 0, l6Z0 i m l inm~s u p l6P /
sup
kt<.s~t+6
[(n(S)
--
(n(t)[ > e } = 0.
(6)
For the proof of (6) we may restrict to t = 0, the general case t ¢ 0 follows after a simple transformation (rotation and standardization). By series expansion we get that for u ~ 0 cos u X/1 + p sin(2u)
sin u
g/1 +
X/1 + p sin(2u)
Xi2 = (1 - pu)X,.l + uXi2 + O(u2[gil [) "]- O(u21X,21).
We use the representation
x,2 = pX. + v/1 - p2r,, where {Y i, i>>.1} are independent standard normal random variables, being independent of {Xil, i t> 1}. Then we obtain / \ ~.(s) = b. {max[(1 - psAn~il + sA.Xi2] - b. ] + O(s2 A2nbnmax(lXil ], Ix,21))
/
\ i<~n
= bn (max[X/l + sAn V/1 - p2 Yi] - bn) + O(s 2A2nbnmax(IX/1 I, IX,-21)) \ i<<.n / = L(sx/1
-
p2) + O(szZZb. max(IX~l I, IX~2I)),
where
~n(s) = bn (max[Xi i<~n tk
+ sAnYi] - bn) .
Note that the variance of X,.1 + sA,Yi is 1 + s2A 2. For s < 6 the remainder term can be neglected: using An ~ (21ogn) -1/2, P I.~max > i < ~[X,n.I]
e/(O2A2nbn)} ~<2n[1- qb(e/(O2A2nbn))]=O (nAn exp ( - - -
--. 0,
(7)
as n ~ c~, and 6 < x/~. Thus it remains to show that lim lim sup - P
sup
610 n--*o¢~ ~
kO<<.s<~6
-
(8)
~ 0.
Because of the independence between {Y.} and {Xnl} we have sup I~,(s)-~n(O)l= O~s~6
sup bn (max(X/1 + s Y i / b n ) - m a x X i l )
O<~s<~6
sup
O~s~6
\i<~n
i<~n
max(s~ + bn(X(n+l-i) - XCn))) , i~n
]
(9)
6
G. Hooohiemstra, J. Hiisler I Statistics & Probability Letters 31 (1996) 1-6
where )((1) ..... X(n) are the order statistics Of X l l . . . . . Xnl, Define n o w A i = bn(X(n)-S(n+l-i))~O (a.s.). By (9) it follows that
1 [I o<<. sup~<_a( ~ n ( s ) - ~ n ( O ) ) < - ~ } < ~ P { 3 s < 6 " m a x ( s Y l , s Y z - A 2 6P
..... s Y n - A n ) < - ~ }
<~~P{3s < 6 : sYI < - e } ~<~P{Y1 < -~/62} = ~ ( - g / 8 2 )
-'-* 0,
as 8---,0. If max(sY1, sYz -A2,...,sYn - A n ) > e for some s, there exists i = i(s) such that s Y i - A i > e, hence Y/ > 0 ( s i n c e Ai>~O ). It follows that 8Yi - A i > e, and also max(SYl,SY2 - A 2 ..... 8Yn - A n ) > e. This implies that
n(0) > According to (4), nlimooP{~n(0) <~x, ~n(8) <~y} = H~(~)(x, y - 82/2), where 22(8) = (8/2) 2. The density h;. of 11). equals for 2 # 0,
ha(x, y) = H).(x, y){[e-X f ( y - x) + e-X f ' ( y - x) - e-Y f ' ( x - y)] x [e-Yf(x - y) + e - Y f ' ( x - y) - e-Xft(y - x)] + [e-Yf'(x - y) + e - Y f " ( x - y) + e - X f ' ( y - x) + e - X f " ( y - x)]},
(10)
where
f ( t ) = q~(2 + t/(22)). From this explicit form it is straightforward (by considering the terms on the right-hand side of (10) individually) to conclude that for each e > 0,
lim ~ P { ( ( 8 ) - ((O) > e} = lim l f f 6~0
8,[0 ~
h a o ) ( x , y - 82/2)dxdy = O.
{(x,y):y--x>e,}
Together it implies (8) and the tightness. References Billingsley, P. (1968), Weak Convergence of Probability Measures (Wiley, New York). Brown, B.M. and S.I. Resnick (1977), Extreme values of independent stochastic processes, J. Appl. Probab. 14, 732-739. Falk, M., J. Hiisler, and R.D. Reiss, (1994), Laws of Small Numbers: Extremes and Rare Events, DMV Seminar Vol. 23 (Birkh/iuser, Basel). Gin6, E., M.G. Hahn, and P. Vatan (1990), Max-infinitely divisible and max-stable sample continuous processes, Probab. Theory Related Fields 87, 139-165. Haan, L. de and J. Pickands III (1986), Stationary min-stable processes, Probab. Theory Related Fields 72, 477-492. Hiisler, J. and R.D. Reiss (1989), Maxima of normal random vectors: between independence and complete dependence, Statist. Probab. Lett. 7, 283-286. Penrose, M.D. (1992), Semi-min-stable processes, Ann. Probab. 20, 1450-1463.