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Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences
Journal of Multivariate Analysis 101 (2010) 2641–2647
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Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva
Note(s)
Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences Zuoxiang Peng a , Lunfeng Cao a , Saralees Nadarajah b,∗ a
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
b
School of Mathematics, University of Manchester, Manchester, United Kingdom
article
abstract
info
Let (Xn ) be a sequence of d-dimensional stationary Gaussian vectors, and let Mn denote the partial maxima of {Xk , 1 ≤ k ≤ n}. Suppose that there are missing data in each component e n denote the partial maxima of the observed variables. In this note, we study of Xk and let M e n , Mn ) where the correlation two kinds of asymptotic distributions of the random vector (M and cross-correlation satisfy some dependence conditions. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 22 December 2009 Available online 26 June 2010 AMS 2000 subject classifications: primary 60E70 secondary 60F05 60G15 Keywords: Asymptotic distribution Missing data Multivariate stationary Gaussian vector Weak and strong dependence
1. Introduction Let (Xn ) be a sequence of standard stationary Gaussian random variables. Define Mn = max1≤k≤n Xk and r (n) = EX1 Xn+1 , the partial maximum and the correlation, respectively. It is well known that the limiting distribution of the normalized maxima differs according to the rate of convergence of r (n). Berman [1] proved that the limiting distribution of Mn is similar to that for an independent and identically distributed Gaussian sequence if r (n) log n → 0, i.e. 1 lim P a− n (Mn − bn ) ≤ x = exp {− exp(−x)} ,
n→∞
where an = (2 log n)−1/2 ,
1 b n = a− n + an (log log n + log 4π ) /2.
(1.1)
For r (n) log n → γ ∈ (0, ∞), Mittal and Ylvisaker [8] proved that 1 lim P a− n (Mn − bn ) ≤ x =
n→∞
Z
∞
n
exp − exp −x − γ +
p
2γ z
o
φ(z )dz ,
−∞
where φ(x) is the probability density function (pdf) of a standard Gaussian random variable. For r (n) log n → ∞ with some regular conditions for r (n), the strongly dependent case, McCormick and Mittal [6] proved that lim P r −1/2 (n) Mn − (1 − r (n))1/2 bn ≤ x = Φ (x),
n→∞
∗
Corresponding author. E-mail address: [email protected] (S. Nadarajah).
0047-259X/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2010.06.016
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Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
where Φ (x) is the cumulative distribution function (cdf) of a standard Gaussian random variable. For more details, see Sections 4.3, 6.5 and 6.6 of Leadbetter et al. [4]. McCormick [5] introduced the following condition: n log n X
n
|r (k) − r (n)| = o(1)
(1.2)
k=1
as n → ∞ and considered the maximum centered at the sample mean Xn =
n 1X
n i=1
Xi .
For r (k) < 1 for some k and (1.2) holding, McCormick [5] showed that Mn − X n
−1
lim P an
√
n→∞
1 − r ( n)
! − bn
! ≤ x = exp {− exp(−x)} ,
x ∈ R.
McCormick and Qi [7] and Peng and Nadarajah [11] considered the joint limiting behavior of the partial sum and maximum. Hu et al. [2] and Peng [10] studied the joint limiting distribution of the partial sum and point process of exceedances. For the limiting distribution of the extremes of vector Gaussian sequences, see [3,12–14]. The aim of this note is to establish the joint limiting distribution of the maxima of complete and incomplete samples of stationary Gaussian vector sequences under some weakly and strongly dependent conditions similar to those of [9,8], where the univariate case is considered. Let {Xn = (Xn1 , Xn2 , . . . , Znd ), n ≥ 1} be a sequence of d-dimensional stationary Gaussian random vectors, i.e. EXni = 0,
Var (Xni ) = 1
(1.3)
for n ≥ 1 and 1 ≤ i ≤ d and Cov Xli , Xkj = rij (|l − k|)
(1.4) (s)
(1)
for 1 ≤ i, j ≤ d, k, l ≥ 1. Let Mn denote the sth order statistic (componentwise) and Mn constants an = (an , . . . , an ) ,
= Mn . Define the norming
bn = (bn , . . . , bn )
(1.5)
with an and bn defined by (1.1). Wiśniewski [13] proved the following main result: Theorem 1.1. Let (Xn ) satisfy rij (n) log n → ρij for 1 ≤ i, j ≤ d and max1≤i6=j≤d,n≥0 |rij | < 1. Then d
1 a− M(ns) − bn → M(s) (ρ) + Rρ Z n
for s ≥ 1, where ρ = (ρii , 1 ≤ i ≤ d) and Rρ =
√
√
2ρii , 1 ≤ i ≤ d , Z is a standard Gaussian vector with variance–covariance
matrix ρij / ρii ρjj d×d and
P M(s) (ρ) ≤ x =
d Y
exp {− exp (−xi − ρii )}
s−1 X
i =1
{exp (−xi − ρii )}j /j!.
j =0
(s)
Furthermore, M (ρ) and Z are independent of each other. Now suppose that some of the variables of Xk can be observed. Let εki denote the indicator of the event that Xki is observed. Then Sni = ε1i + ε2i + · · · + εni is the number of observed random variables from the set {X1i , X2i , . . . , Xni }, where 1 ≤ i ≤ d. In order to prove the main results, the following conditions are needed: C1 . The sequence (Sni ) satisfies Sni n
P
→ pi ∈ (0, 1]
as n → ∞ for 1 ≤ i ≤ d. C2 . The indicator random variables {εki , 1 ≤ k ≤ n, 1 ≤ i ≤ d} are independent, and also independent of {Xk }. For simplicity, we will use the following notation throughout this note: un = (un1 , . . . , und ) ,
vn = (vn1 , . . . , vnd ) ,
(1.6)
where uni = an xi + bn , vni = an yi + bn and xi < yi for 1 ≤ i ≤ d, and Mni = max {X1i , . . . , Xni } ,
max {Xki , 1 ≤ k ≤ n, εki = 1} , e Mni = −∞,
if Sni ≥ 1, if Sni = 0.
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
2643
2. Main results In this section, we consider the asymptotic distribution of the maxima of complete and incomplete samples from multivariate weakly dependent and strongly dependent stationary Gaussian sequences. We obtain two theorems. Theorem 2.1 is for the weakly dependent case and Theorem 2.2 is for the strongly dependent case. Corollaries 2.1 and 2.2 are particular cases of these theorems. Theorem 2.1. Let the d-dimensional stationary Gaussian vector sequence {Xn } satisfy the conditions (1.3) and (1.4). Suppose that the conditions C1 and C2 hold. Assume further that rij (n) log n → 0,
1 ≤ i, j ≤ d
(2.1)
as n → ∞. Then, for un and vn defined by (1.6), we have d Y
d
e n ≤ un , Mn ≤ vn −→ P M
exp {−pi exp (−xi )} exp {− (1 − pi ) exp (−yi )}
(2.2)
i
as n → ∞. Corollary 2.1. Under the conditions of Theorem 2.1, for an and bn defined by (1.5), we have d e n ≤ an x + bn −→ P M
d Y
exp {−pi exp (−xi )}
i
as n → ∞, where x = (x1 , x2 , . . . , xd ). Theorem 2.2. Let the d-dimensional stationary Gaussian vector sequence {Xn } satisfy the conditions (1.3) and (1.4). Suppose that the conditions C1 and C2 hold. Assume further that rij (n) log n → ρij ∈ (0, ∞),
1 ≤ i, j ≤ d
(2.3)
as n → ∞ and sup rij (n) < 1.
(2.4)
1≤i6=j≤d n ≥0
Then, for un and vn defined by (1.6), we have
e n ≤ un , Mn ≤ vn lim P M
+∞
Z
...
=
n→∞
Z
−∞
d +∞ Y
−∞
n
exp −pi exp −xi − ρii +
p
2ρii zi
o
i
n
o p × exp − (1 − pi ) exp −yi − ρii + 2ρii zi φ (z1 , z2 , . . . , zd ) dz1 · · · dzd , where φ(z1 , z2 , . . . , zd ) is the joint pdf of a d-variate Gaussian vector X0 with zero mean and variance–covariance matrix √ ρij / ρii ρjj d×d . Corollary 2.2. Under the conditions of Theorem 2.2, for an and bn defined by (1.5), we have
Z
+∞
...
e n ≤ an x + bn = lim P M
n→∞
d +∞ Y
Z
−∞
−∞
n
exp −pi exp −xi − ρii +
p
2ρii zi
o
φ (z1 , z2 , . . . , zd ) dz1 · · · dzd
i
as n → ∞, where x = (x1 , x2 , . . . , xd ). 3. Proofs In this section, we shall prove the main results. Firstly, let {X∗k , k ≥ 1} denote a sequence of independent d-dimensional Gaussian vectors with EXki∗ = 0, Var(Xki∗ ) = 1, 1 ≤ i ≤ d, Cov(Xki∗ , Xkj∗ ) = 0 for i 6= j, 0 ≤ k ≤ n and Cov(Xki∗ , Xlj∗ ) = 0 ∗ for 0 ≤ k 6= l ≤ n. Define M∗n to be the partial maxima of the sequence {X∗k , 1 ≤ k ≤ n}, i.e. Mni = max{X1i∗ , . . . , Xni∗ } for ∗ e 1 ≤ i ≤ d. Also let Mn denote the partial maxima of the observed variables, i.e.
eni∗ = M
max Xki∗ , 1 ≤ k ≤ n, εki = 1 ,
−∞,
if Sni ≥ 1, if Sni = 0.
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Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
For v = (v1 , . . . , vd ), vi ∈ (0, ∞), define
v1 1/2 . A (v) = ..
0
··· .. . ···
(1 − v1 )1/2 .. B(v) = .
0
.. , .
vd 1/2
0
··· .. . ···
0
.. . . 1/2 (1 − vd )
For the strongly dependent case, we need the following construction. Let X0 be a d-variate Gaussian vector with zero √ mean and variance–covariance matrix ρij / ρii ρjj d×d . Suppose that X0 is independent of {X∗k , k ≥ 1}. For ρ(n) = (ρ11 (n), . . . , ρdd (n)) and ρii (n) = ρii /(log n), define the d-variate Gaussian variables Yk = X∗0 A (ρ(n)) + X∗k B (ρ(n))
(3.1)
for 1 ≤ k ≤ n, i.e. 1/2
Yki = ρii (n)X0i∗ + (1 − ρii (n))1/2 Xki∗
(3.2)
for 1 ≤ k ≤ n and 1 ≤ i ≤ d. It is easy to check that EYki = 0, Var(Yki ) = 1 and Cov(Yki , Ylj ) = ρij (n) for 1 ≤ k 6= l ≤ n, e ∗∗ where ρij (n) = ρij /(log n). Let M∗∗ n and Mn denote, respectively, the partial maxima of {Yk , 1 ≤ k ≤ n} and that of its observed variables, i.e.
max {Yki , 1 ≤ k ≤ n, εki = 1} , ∗∗ e Mni = −∞,
Mni = max{Y1i , . . . , Yni }, ∗∗
if Sni ≥ 1, if Sni = 0
for 1 ≤ i ≤ d. Noting (3.1) and (3.2), we have ∗ ∗ M∗∗ n = X0 A (ρ(n)) + Mn B (ρ(n)) ,
e ∗∗ = X∗ A (ρ(n)) + M e ∗ B (ρ(n)) , M n 0 n
(3.3)
i.e. 1/2
∗ Mni = ρii (n)X0i∗ + (1 − ρii (n))1/2 Mni ,
1/2
eni = ρ (n)X0i∗ + (1 − ρii (n))1/2 M eni∗ M ii
for 1 ≤ i ≤ d. Here, we use the convention a − ∞ = −∞. To prove our main results, we need some lemmas. The first one will be used to prove Theorem 2.1. Lemma 3.1. Let the d-dimensional stationary Gaussian vector sequence {Xk } satisfy the conditions (1.3), (1.4) and (2.1). Suppose that the conditions C1 and C2 hold. With {X∗k } defined as above, we have
e ∗ ≤ un , M∗ ≤ vn = 0. e n ≤ un , Mn ≤ vn − P M lim P M n n
n→∞
Proof. For fixed i ∈ (1, . . . , d) and component sequence (Xki ), firstly suppose that just {Xi1 ,i , . . . , Xik ,i } have been observed i from the set {X1i , . . . , Xni }, which is one case of {Sni = ki }. Let N = {1, 2, . . . , n}, Ii = {i1 , . . . , iki }, M (Ii ) = max{Xli , l ∈ Ii } and M ∗ (Ii ) = max{Xli∗ , l ∈ Ii }. Then, by the Normal Comparison lemma given in [4] and noting that uni ≤ vni for 1 ≤ i ≤ d, we have
P {M (Ii ) ≤ uni , M (N /Ii ) ≤ vni , 1 ≤ i ≤ d} − P M ∗ (Ii ) ≤ uni , M ∗ (N /Ii ) ≤ vni , 1 ≤ i ≤ d ) ( n d n X X X X u2ni + u2nj u2ni rij (k) exp − |rii (k)| exp − ≤ K1 + K2 n n 1 + |rii (k)| 2 1 + rij (k) i =1 k=1 k=0 1≤i6=j≤d for some absolute constants K1 > 0 and K2 > 0. So, by (2.1) and by arguments similar to those of the proof of Lemma 4.3.2 in [4], we obtain
( n X n rij (k) exp − k=0
)
u2ni + u2nj 2 1 + rij (k)
→0
(3.4)
and n
n X k=1
|rii (k)| exp −
u2ni 1 + |rii (k)|
→0
(3.5)
uniformly for all ki , 1 ≤ i ≤ d as n → ∞. The result follows by the condition C2 , the total probability formula and the uniform convergence of (3.4) and (3.5). The following result is useful for proving Theorem 2.2.
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
2645
Lemma 3.2. Let the d-dimensional stationary Gaussian vector sequence {Xk } satisfy the conditions (1.3) and (1.4). Let {Yk } be the Gaussian vector sequence defined above with equal correlations. Suppose that the conditions C1 and C2 hold. Assume further that both (2.3) and (2.4) hold. Then
e ∗∗ ≤ un , M∗∗ ≤ vn = 0. e n ≤ un , Mn ≤ vn − P M lim P M n n
n→∞
Proof. For fixed i ∈ (1, . . . , d) and component sequence (Xki ), firstly suppose that just {Xi1 ,i , . . . , Xik ,i } have been observed i from the set {X1i , . . . , Xni }, which is one case of {Sni = ki }. Define N = {1, 2, . . . , n}, Ii = {i1 , . . . , iki }, M (Ii ) = max{Xli , l ∈ ∗∗ ∗∗ Ii } and M (Ii ) = max{Xli , l ∈ Ii }. Then, by the Normal Comparison lemma of [4] and noting that uni ≤ vni for 1 ≤ i ≤ d, we have
P {M (Ii ) ≤ uni , M (N /Ii ) ≤ vni , 1 ≤ i ≤ d} − P M ∗∗ (Ii ) ≤ uni , M ∗∗ (N /Ii ) ≤ vni , 1 ≤ i ≤ d ( ) d n n X X X X u2ni + u2nj u2ni , |rii (k) − ρii (n)| exp − ≤ K3 n + K4 n rij (k) − ρij (n) exp − 1 + wii (k) 2 1 + wij (k) i=1 k=1 k=0 1≤i6=j≤d where wij (k) = max{|rij (k)|, ρij (n)}, and K3 and K4 are absolute constants. So, by arguments similar to those of the proofs of Lemmas 4.3.2 and 6.4.1 in [4], we obtain
( n X n rij (k) − ρij (n) exp − k=1
)
u2ni + u2nj 2 (1 + wii (k))
→0
and n
n X
|rii (k) − ρii (n)| exp −
k=1
u2ni 1 + wii (k)
→0
uniformly for all ki , 1 ≤ i ≤ d, as n → ∞. So, the result follows by the condition C2 and the total probability formula.
Proof of Theorem 2.1. By Lemma 3.1, we only need to prove
e ∗ ≤ un , M∗ ≤ vn = lim P M n n
n→∞
d Y
exp {−pi exp (−xi )} exp {− (1 − pi ) exp (−yi )} .
(3.6)
i
By using the total probability formula, we obtain
e ∗ ≤ un , M∗ ≤ vn = P M n n
d X n X
P {Sn1 = k1 , . . . , Snd = kd }
i=1 ki =1
d Y
{Φ (uni )}ki {Φ (vni )}n−ki .
i=1
Define
Σ1 =
d X
X
P {Sn1 = k1 , . . . , Snd = kd }
k ∃i: ni −pi >ε
i =1
d Y
{Φ (uni )}ki {Φ (vni )}n−ki
(3.7)
{Φ (uni )}ki {Φ (vni )}n−ki .
(3.8)
i=1
and
Σ2 =
d X
X
P{Sn1 = k1 , . . . , Snd = kd }
k i =1 ∀i: ni −pi ≤ε
d Y i=1
By the condition C1 , we have
Σ1 ≤
d X
P (|Sni /n − pi | ≥ ε) → 0
(3.9)
i=1
as n → ∞. By (3.8), the inequalities
Σ2 ≤
d Y
{Φ (uni )}n(pi −ε) {Φ (vni )}n(1−pi −ε)
i =1
d X i=1
X
P (Sn1 = k1 , . . . , Snd = kd )
(3.10)
P (Sn1 = k1 , . . . , Snd = kd )
(3.11)
k ∀i: ni −pi ≤ε
and
Σ2 ≥
d Y i =1
{Φ (uni )}n(pi +ε) {Φ (vni )}n(1−pi +ε)
d X
X
k i=1 ∀i: ni −pi ≤ε
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Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
hold. By (3.7)–(3.11) and the condition C1 , we have d Y
e ∗ ≤ un , M∗ ≤ vn ≤ lim sup P M n n
n→∞
exp {− (pi − ε) exp (−xi )} exp {− (1 − pi − ε) exp (−yi )}
(3.12)
i=1
and
e ∗ ≤ un , M∗ ≤ vn ≥ lim inf P M n n
n→∞
d Y
exp {− (pi + ε) exp (−xi )} exp {− (1 − pi + ε) exp (−yi )}
(3.13)
i =1
for every ε ∈ (0, min{pi , 1 ≤ i ≤ d}). So, by letting ε ↓ 0 in (3.12) and (3.13), we obtain (3.6). The proof is complete.
Proof of Theorem 2.2. By Lemma 3.2, in order to prove (2.2), it sufficient to show that
e ∗∗ ≤ un , M∗∗ ≤ vn = lim P M n n
Z
n→∞
+∞
...
d +∞ Y
Z
−∞
−∞
n
exp −pi exp −xi − ρii +
2ρii zi
p
o
i
n
o p × exp − (1 − pi ) exp −yi − ρii + 2ρii zi φ (z1 , z2 , . . . , zd ) dz1 · · · dzd . (3.14) By (3.3) and the total probability formula, we obtain
e ∗∗ ≤ un , M∗∗ ≤ vn = P X∗ A (ρ(n)) + M e ∗ B (ρ(n)) ≤ un , X∗ A (ρ(n)) + M∗ B (ρ(n)) ≤ vn P M n n 0 n 0 n
Z
+∞
...
=
Z
−∞
+∞ h
e ∗ ≤ (un − ZA (ρ(n))) B−1 (ρ(n)) , M∗ ≤ (vn − ZA (ρ(n))) B−1 (ρ(n)) P M n n
i
−∞
× ϕ (z1 , z2 , . . . , zd ) dz1 dz2 · · · dzd Z +∞ Z +∞ X n d Y {Φ (qni )}ki {Φ (tni )}n−ki ϕ (z1 , z2 , . . . , zd ) dz1 dz2 · · · dzd , = ... P (Sn1 = k1 , . . . , Snd = kd ) −∞
−∞
ki =1 1≤i≤d
i
(3.15) where Z = (z1 , z2 , . . . , zd ) and
√ vni − zi ρii (n) tni = √ , 1 ≤ i ≤ d. 1 − ρii (n) √ √ Note that qni = an xi + ρii − 2ρii zi + bn + o(an ) and tni = an yi + ρii − 2ρii zi + bn + o(an ) from the proof of Theorem √ ρii (n) , qni = √ 1 − ρii (n) uni − zi
6.5.1 in [4]. So, one can check that
n
p
n
p
lim {Φ (qni )}n = exp − exp −xi − ρii +
n→∞
2ρii zi
o
and lim {Φ (tni )}n = exp − exp −yi − ρii +
n→∞
2ρii zi
o
.
By arguments similar to those of (3.6), we have n X
P (Sn1 = k1 , . . . , Snd = kd )
ki =1 1≤i≤d
→
d Y
{Φ (qni )}ki {Φ (tni )}n−ki
i d Y
n
exp −pi exp −xi − ρii +
2ρii zi
p
o
n
exp − (1 − pi ) exp −yi − ρii +
p
2ρii zi
o
i
as n → ∞. Combining this with (3.15), we obtain (3.14) by the dominated convergence theorem. The proof is complete.
Acknowledgments The authors would like to thank the Editor-in-Chief and the referee for careful reading and for their comments which greatly improved the paper.
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
2647
References [1] S. Berman, Limit theorems for the maximum term in stationary sequences, Annals of Mathematical Statistics 35 (1964) 502–516. [2] A. Hu, Z. Peng, Y. Qi, Joint behavior of point process of exceedances and partial sum from a Gaussian sequence, Metrika 70 (2009) 279–295. [3] B. James, K. James, Y. Qi, Limit distribution of the sum and maximum from multivariate Gaussian sequences, Journal of Multivariate Analysis 98 (2007) 517–532. [4] M.R. Leadbetter, G. Lindgren, H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer, New York, 1983. [5] W.P. McCormick, Weak convergence for the maxima of stationary Gaussian processes using random normalization, Annals of Probability 8 (1980) 483–497. [6] W.P. McCormick, Y. Mittal, On weak convergence of the maximum, Technical Report No. 81, Department of Statistics, Stanford University, 1976. [7] W.P. McCormick, Y. Qi, Asymptotic distribution for the sum and the maximum of Gaussian processes, Journal of Applied Probability 37 (2000) 958–971. [8] Y. Mittal, D. Ylvisaker, Limit distribution for the maximum of stationary Gaussian processes, Stochastic Processes and their Applications 3 (1975) 1–18. [9] P. Mladenović, V. Piterbarg, On asymptotic distribution of maxima of complete and incomplete samples from stationary sequences, Stochastic Processes and their Applications 116 (2006) 1977–1991. [10] Z. Peng, Joint asymptotic distributions of exceedances point process and partial sum of strong dependent Gaussian sequences, Acta Mathematica Sinica, Chinese Series 3 (1999) 362–367. [11] Z. Peng, S. Nadarajah, On the joint limiting distribution of sums and maxima of stationary normal sequence, Theory of Probability and its Applications 47 (2002) 817–820. [12] M. Wiśniewski, Extreme order statistics in equally corrected Gaussian array, Applicationes Mathematicae 22 (1994) 193–200. [13] M. Wiśniewski, On extreme order statistics and point processes of exceedances in multivariate stationary Gaussian sequences, Statistics and Probability Letters 29 (1996) 55–59. [14] M. Wiśniewski, Extremes in multivariate stationary normal sequences, Applicationes Mathematicae 25 (1998) 375–379.
Contents lists available at ScienceDirect
Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva
Note(s)
Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences Zuoxiang Peng a , Lunfeng Cao a , Saralees Nadarajah b,∗ a
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
b
School of Mathematics, University of Manchester, Manchester, United Kingdom
article
abstract
info
Let (Xn ) be a sequence of d-dimensional stationary Gaussian vectors, and let Mn denote the partial maxima of {Xk , 1 ≤ k ≤ n}. Suppose that there are missing data in each component e n denote the partial maxima of the observed variables. In this note, we study of Xk and let M e n , Mn ) where the correlation two kinds of asymptotic distributions of the random vector (M and cross-correlation satisfy some dependence conditions. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 22 December 2009 Available online 26 June 2010 AMS 2000 subject classifications: primary 60E70 secondary 60F05 60G15 Keywords: Asymptotic distribution Missing data Multivariate stationary Gaussian vector Weak and strong dependence
1. Introduction Let (Xn ) be a sequence of standard stationary Gaussian random variables. Define Mn = max1≤k≤n Xk and r (n) = EX1 Xn+1 , the partial maximum and the correlation, respectively. It is well known that the limiting distribution of the normalized maxima differs according to the rate of convergence of r (n). Berman [1] proved that the limiting distribution of Mn is similar to that for an independent and identically distributed Gaussian sequence if r (n) log n → 0, i.e. 1 lim P a− n (Mn − bn ) ≤ x = exp {− exp(−x)} ,
n→∞
where an = (2 log n)−1/2 ,
1 b n = a− n + an (log log n + log 4π ) /2.
(1.1)
For r (n) log n → γ ∈ (0, ∞), Mittal and Ylvisaker [8] proved that 1 lim P a− n (Mn − bn ) ≤ x =
n→∞
Z
∞
n
exp − exp −x − γ +
p
2γ z
o
φ(z )dz ,
−∞
where φ(x) is the probability density function (pdf) of a standard Gaussian random variable. For r (n) log n → ∞ with some regular conditions for r (n), the strongly dependent case, McCormick and Mittal [6] proved that lim P r −1/2 (n) Mn − (1 − r (n))1/2 bn ≤ x = Φ (x),
n→∞
∗
Corresponding author. E-mail address: [email protected] (S. Nadarajah).
0047-259X/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2010.06.016
2642
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
where Φ (x) is the cumulative distribution function (cdf) of a standard Gaussian random variable. For more details, see Sections 4.3, 6.5 and 6.6 of Leadbetter et al. [4]. McCormick [5] introduced the following condition: n log n X
n
|r (k) − r (n)| = o(1)
(1.2)
k=1
as n → ∞ and considered the maximum centered at the sample mean Xn =
n 1X
n i=1
Xi .
For r (k) < 1 for some k and (1.2) holding, McCormick [5] showed that Mn − X n
−1
lim P an
√
n→∞
1 − r ( n)
! − bn
! ≤ x = exp {− exp(−x)} ,
x ∈ R.
McCormick and Qi [7] and Peng and Nadarajah [11] considered the joint limiting behavior of the partial sum and maximum. Hu et al. [2] and Peng [10] studied the joint limiting distribution of the partial sum and point process of exceedances. For the limiting distribution of the extremes of vector Gaussian sequences, see [3,12–14]. The aim of this note is to establish the joint limiting distribution of the maxima of complete and incomplete samples of stationary Gaussian vector sequences under some weakly and strongly dependent conditions similar to those of [9,8], where the univariate case is considered. Let {Xn = (Xn1 , Xn2 , . . . , Znd ), n ≥ 1} be a sequence of d-dimensional stationary Gaussian random vectors, i.e. EXni = 0,
Var (Xni ) = 1
(1.3)
for n ≥ 1 and 1 ≤ i ≤ d and Cov Xli , Xkj = rij (|l − k|)
(1.4) (s)
(1)
for 1 ≤ i, j ≤ d, k, l ≥ 1. Let Mn denote the sth order statistic (componentwise) and Mn constants an = (an , . . . , an ) ,
= Mn . Define the norming
bn = (bn , . . . , bn )
(1.5)
with an and bn defined by (1.1). Wiśniewski [13] proved the following main result: Theorem 1.1. Let (Xn ) satisfy rij (n) log n → ρij for 1 ≤ i, j ≤ d and max1≤i6=j≤d,n≥0 |rij | < 1. Then d
1 a− M(ns) − bn → M(s) (ρ) + Rρ Z n
for s ≥ 1, where ρ = (ρii , 1 ≤ i ≤ d) and Rρ =
√
√
2ρii , 1 ≤ i ≤ d , Z is a standard Gaussian vector with variance–covariance
matrix ρij / ρii ρjj d×d and
P M(s) (ρ) ≤ x =
d Y
exp {− exp (−xi − ρii )}
s−1 X
i =1
{exp (−xi − ρii )}j /j!.
j =0
(s)
Furthermore, M (ρ) and Z are independent of each other. Now suppose that some of the variables of Xk can be observed. Let εki denote the indicator of the event that Xki is observed. Then Sni = ε1i + ε2i + · · · + εni is the number of observed random variables from the set {X1i , X2i , . . . , Xni }, where 1 ≤ i ≤ d. In order to prove the main results, the following conditions are needed: C1 . The sequence (Sni ) satisfies Sni n
P
→ pi ∈ (0, 1]
as n → ∞ for 1 ≤ i ≤ d. C2 . The indicator random variables {εki , 1 ≤ k ≤ n, 1 ≤ i ≤ d} are independent, and also independent of {Xk }. For simplicity, we will use the following notation throughout this note: un = (un1 , . . . , und ) ,
vn = (vn1 , . . . , vnd ) ,
(1.6)
where uni = an xi + bn , vni = an yi + bn and xi < yi for 1 ≤ i ≤ d, and Mni = max {X1i , . . . , Xni } ,
max {Xki , 1 ≤ k ≤ n, εki = 1} , e Mni = −∞,
if Sni ≥ 1, if Sni = 0.
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
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2. Main results In this section, we consider the asymptotic distribution of the maxima of complete and incomplete samples from multivariate weakly dependent and strongly dependent stationary Gaussian sequences. We obtain two theorems. Theorem 2.1 is for the weakly dependent case and Theorem 2.2 is for the strongly dependent case. Corollaries 2.1 and 2.2 are particular cases of these theorems. Theorem 2.1. Let the d-dimensional stationary Gaussian vector sequence {Xn } satisfy the conditions (1.3) and (1.4). Suppose that the conditions C1 and C2 hold. Assume further that rij (n) log n → 0,
1 ≤ i, j ≤ d
(2.1)
as n → ∞. Then, for un and vn defined by (1.6), we have d Y
d
e n ≤ un , Mn ≤ vn −→ P M
exp {−pi exp (−xi )} exp {− (1 − pi ) exp (−yi )}
(2.2)
i
as n → ∞. Corollary 2.1. Under the conditions of Theorem 2.1, for an and bn defined by (1.5), we have d e n ≤ an x + bn −→ P M
d Y
exp {−pi exp (−xi )}
i
as n → ∞, where x = (x1 , x2 , . . . , xd ). Theorem 2.2. Let the d-dimensional stationary Gaussian vector sequence {Xn } satisfy the conditions (1.3) and (1.4). Suppose that the conditions C1 and C2 hold. Assume further that rij (n) log n → ρij ∈ (0, ∞),
1 ≤ i, j ≤ d
(2.3)
as n → ∞ and sup rij (n) < 1.
(2.4)
1≤i6=j≤d n ≥0
Then, for un and vn defined by (1.6), we have
e n ≤ un , Mn ≤ vn lim P M
+∞
Z
...
=
n→∞
Z
−∞
d +∞ Y
−∞
n
exp −pi exp −xi − ρii +
p
2ρii zi
o
i
n
o p × exp − (1 − pi ) exp −yi − ρii + 2ρii zi φ (z1 , z2 , . . . , zd ) dz1 · · · dzd , where φ(z1 , z2 , . . . , zd ) is the joint pdf of a d-variate Gaussian vector X0 with zero mean and variance–covariance matrix √ ρij / ρii ρjj d×d . Corollary 2.2. Under the conditions of Theorem 2.2, for an and bn defined by (1.5), we have
Z
+∞
...
e n ≤ an x + bn = lim P M
n→∞
d +∞ Y
Z
−∞
−∞
n
exp −pi exp −xi − ρii +
p
2ρii zi
o
φ (z1 , z2 , . . . , zd ) dz1 · · · dzd
i
as n → ∞, where x = (x1 , x2 , . . . , xd ). 3. Proofs In this section, we shall prove the main results. Firstly, let {X∗k , k ≥ 1} denote a sequence of independent d-dimensional Gaussian vectors with EXki∗ = 0, Var(Xki∗ ) = 1, 1 ≤ i ≤ d, Cov(Xki∗ , Xkj∗ ) = 0 for i 6= j, 0 ≤ k ≤ n and Cov(Xki∗ , Xlj∗ ) = 0 ∗ for 0 ≤ k 6= l ≤ n. Define M∗n to be the partial maxima of the sequence {X∗k , 1 ≤ k ≤ n}, i.e. Mni = max{X1i∗ , . . . , Xni∗ } for ∗ e 1 ≤ i ≤ d. Also let Mn denote the partial maxima of the observed variables, i.e.
eni∗ = M
max Xki∗ , 1 ≤ k ≤ n, εki = 1 ,
−∞,
if Sni ≥ 1, if Sni = 0.
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Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
For v = (v1 , . . . , vd ), vi ∈ (0, ∞), define
v1 1/2 . A (v) = ..
0
··· .. . ···
(1 − v1 )1/2 .. B(v) = .
0
.. , .
vd 1/2
0
··· .. . ···
0
.. . . 1/2 (1 − vd )
For the strongly dependent case, we need the following construction. Let X0 be a d-variate Gaussian vector with zero √ mean and variance–covariance matrix ρij / ρii ρjj d×d . Suppose that X0 is independent of {X∗k , k ≥ 1}. For ρ(n) = (ρ11 (n), . . . , ρdd (n)) and ρii (n) = ρii /(log n), define the d-variate Gaussian variables Yk = X∗0 A (ρ(n)) + X∗k B (ρ(n))
(3.1)
for 1 ≤ k ≤ n, i.e. 1/2
Yki = ρii (n)X0i∗ + (1 − ρii (n))1/2 Xki∗
(3.2)
for 1 ≤ k ≤ n and 1 ≤ i ≤ d. It is easy to check that EYki = 0, Var(Yki ) = 1 and Cov(Yki , Ylj ) = ρij (n) for 1 ≤ k 6= l ≤ n, e ∗∗ where ρij (n) = ρij /(log n). Let M∗∗ n and Mn denote, respectively, the partial maxima of {Yk , 1 ≤ k ≤ n} and that of its observed variables, i.e.
max {Yki , 1 ≤ k ≤ n, εki = 1} , ∗∗ e Mni = −∞,
Mni = max{Y1i , . . . , Yni }, ∗∗
if Sni ≥ 1, if Sni = 0
for 1 ≤ i ≤ d. Noting (3.1) and (3.2), we have ∗ ∗ M∗∗ n = X0 A (ρ(n)) + Mn B (ρ(n)) ,
e ∗∗ = X∗ A (ρ(n)) + M e ∗ B (ρ(n)) , M n 0 n
(3.3)
i.e. 1/2
∗ Mni = ρii (n)X0i∗ + (1 − ρii (n))1/2 Mni ,
1/2
eni = ρ (n)X0i∗ + (1 − ρii (n))1/2 M eni∗ M ii
for 1 ≤ i ≤ d. Here, we use the convention a − ∞ = −∞. To prove our main results, we need some lemmas. The first one will be used to prove Theorem 2.1. Lemma 3.1. Let the d-dimensional stationary Gaussian vector sequence {Xk } satisfy the conditions (1.3), (1.4) and (2.1). Suppose that the conditions C1 and C2 hold. With {X∗k } defined as above, we have
e ∗ ≤ un , M∗ ≤ vn = 0. e n ≤ un , Mn ≤ vn − P M lim P M n n
n→∞
Proof. For fixed i ∈ (1, . . . , d) and component sequence (Xki ), firstly suppose that just {Xi1 ,i , . . . , Xik ,i } have been observed i from the set {X1i , . . . , Xni }, which is one case of {Sni = ki }. Let N = {1, 2, . . . , n}, Ii = {i1 , . . . , iki }, M (Ii ) = max{Xli , l ∈ Ii } and M ∗ (Ii ) = max{Xli∗ , l ∈ Ii }. Then, by the Normal Comparison lemma given in [4] and noting that uni ≤ vni for 1 ≤ i ≤ d, we have
P {M (Ii ) ≤ uni , M (N /Ii ) ≤ vni , 1 ≤ i ≤ d} − P M ∗ (Ii ) ≤ uni , M ∗ (N /Ii ) ≤ vni , 1 ≤ i ≤ d ) ( n d n X X X X u2ni + u2nj u2ni rij (k) exp − |rii (k)| exp − ≤ K1 + K2 n n 1 + |rii (k)| 2 1 + rij (k) i =1 k=1 k=0 1≤i6=j≤d for some absolute constants K1 > 0 and K2 > 0. So, by (2.1) and by arguments similar to those of the proof of Lemma 4.3.2 in [4], we obtain
( n X n rij (k) exp − k=0
)
u2ni + u2nj 2 1 + rij (k)
→0
(3.4)
and n
n X k=1
|rii (k)| exp −
u2ni 1 + |rii (k)|
→0
(3.5)
uniformly for all ki , 1 ≤ i ≤ d as n → ∞. The result follows by the condition C2 , the total probability formula and the uniform convergence of (3.4) and (3.5). The following result is useful for proving Theorem 2.2.
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
2645
Lemma 3.2. Let the d-dimensional stationary Gaussian vector sequence {Xk } satisfy the conditions (1.3) and (1.4). Let {Yk } be the Gaussian vector sequence defined above with equal correlations. Suppose that the conditions C1 and C2 hold. Assume further that both (2.3) and (2.4) hold. Then
e ∗∗ ≤ un , M∗∗ ≤ vn = 0. e n ≤ un , Mn ≤ vn − P M lim P M n n
n→∞
Proof. For fixed i ∈ (1, . . . , d) and component sequence (Xki ), firstly suppose that just {Xi1 ,i , . . . , Xik ,i } have been observed i from the set {X1i , . . . , Xni }, which is one case of {Sni = ki }. Define N = {1, 2, . . . , n}, Ii = {i1 , . . . , iki }, M (Ii ) = max{Xli , l ∈ ∗∗ ∗∗ Ii } and M (Ii ) = max{Xli , l ∈ Ii }. Then, by the Normal Comparison lemma of [4] and noting that uni ≤ vni for 1 ≤ i ≤ d, we have
P {M (Ii ) ≤ uni , M (N /Ii ) ≤ vni , 1 ≤ i ≤ d} − P M ∗∗ (Ii ) ≤ uni , M ∗∗ (N /Ii ) ≤ vni , 1 ≤ i ≤ d ( ) d n n X X X X u2ni + u2nj u2ni , |rii (k) − ρii (n)| exp − ≤ K3 n + K4 n rij (k) − ρij (n) exp − 1 + wii (k) 2 1 + wij (k) i=1 k=1 k=0 1≤i6=j≤d where wij (k) = max{|rij (k)|, ρij (n)}, and K3 and K4 are absolute constants. So, by arguments similar to those of the proofs of Lemmas 4.3.2 and 6.4.1 in [4], we obtain
( n X n rij (k) − ρij (n) exp − k=1
)
u2ni + u2nj 2 (1 + wii (k))
→0
and n
n X
|rii (k) − ρii (n)| exp −
k=1
u2ni 1 + wii (k)
→0
uniformly for all ki , 1 ≤ i ≤ d, as n → ∞. So, the result follows by the condition C2 and the total probability formula.
Proof of Theorem 2.1. By Lemma 3.1, we only need to prove
e ∗ ≤ un , M∗ ≤ vn = lim P M n n
n→∞
d Y
exp {−pi exp (−xi )} exp {− (1 − pi ) exp (−yi )} .
(3.6)
i
By using the total probability formula, we obtain
e ∗ ≤ un , M∗ ≤ vn = P M n n
d X n X
P {Sn1 = k1 , . . . , Snd = kd }
i=1 ki =1
d Y
{Φ (uni )}ki {Φ (vni )}n−ki .
i=1
Define
Σ1 =
d X
X
P {Sn1 = k1 , . . . , Snd = kd }
k ∃i: ni −pi >ε
i =1
d Y
{Φ (uni )}ki {Φ (vni )}n−ki
(3.7)
{Φ (uni )}ki {Φ (vni )}n−ki .
(3.8)
i=1
and
Σ2 =
d X
X
P{Sn1 = k1 , . . . , Snd = kd }
k i =1 ∀i: ni −pi ≤ε
d Y i=1
By the condition C1 , we have
Σ1 ≤
d X
P (|Sni /n − pi | ≥ ε) → 0
(3.9)
i=1
as n → ∞. By (3.8), the inequalities
Σ2 ≤
d Y
{Φ (uni )}n(pi −ε) {Φ (vni )}n(1−pi −ε)
i =1
d X i=1
X
P (Sn1 = k1 , . . . , Snd = kd )
(3.10)
P (Sn1 = k1 , . . . , Snd = kd )
(3.11)
k ∀i: ni −pi ≤ε
and
Σ2 ≥
d Y i =1
{Φ (uni )}n(pi +ε) {Φ (vni )}n(1−pi +ε)
d X
X
k i=1 ∀i: ni −pi ≤ε
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Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
hold. By (3.7)–(3.11) and the condition C1 , we have d Y
e ∗ ≤ un , M∗ ≤ vn ≤ lim sup P M n n
n→∞
exp {− (pi − ε) exp (−xi )} exp {− (1 − pi − ε) exp (−yi )}
(3.12)
i=1
and
e ∗ ≤ un , M∗ ≤ vn ≥ lim inf P M n n
n→∞
d Y
exp {− (pi + ε) exp (−xi )} exp {− (1 − pi + ε) exp (−yi )}
(3.13)
i =1
for every ε ∈ (0, min{pi , 1 ≤ i ≤ d}). So, by letting ε ↓ 0 in (3.12) and (3.13), we obtain (3.6). The proof is complete.
Proof of Theorem 2.2. By Lemma 3.2, in order to prove (2.2), it sufficient to show that
e ∗∗ ≤ un , M∗∗ ≤ vn = lim P M n n
Z
n→∞
+∞
...
d +∞ Y
Z
−∞
−∞
n
exp −pi exp −xi − ρii +
2ρii zi
p
o
i
n
o p × exp − (1 − pi ) exp −yi − ρii + 2ρii zi φ (z1 , z2 , . . . , zd ) dz1 · · · dzd . (3.14) By (3.3) and the total probability formula, we obtain
e ∗∗ ≤ un , M∗∗ ≤ vn = P X∗ A (ρ(n)) + M e ∗ B (ρ(n)) ≤ un , X∗ A (ρ(n)) + M∗ B (ρ(n)) ≤ vn P M n n 0 n 0 n
Z
+∞
...
=
Z
−∞
+∞ h
e ∗ ≤ (un − ZA (ρ(n))) B−1 (ρ(n)) , M∗ ≤ (vn − ZA (ρ(n))) B−1 (ρ(n)) P M n n
i
−∞
× ϕ (z1 , z2 , . . . , zd ) dz1 dz2 · · · dzd Z +∞ Z +∞ X n d Y {Φ (qni )}ki {Φ (tni )}n−ki ϕ (z1 , z2 , . . . , zd ) dz1 dz2 · · · dzd , = ... P (Sn1 = k1 , . . . , Snd = kd ) −∞
−∞
ki =1 1≤i≤d
i
(3.15) where Z = (z1 , z2 , . . . , zd ) and
√ vni − zi ρii (n) tni = √ , 1 ≤ i ≤ d. 1 − ρii (n) √ √ Note that qni = an xi + ρii − 2ρii zi + bn + o(an ) and tni = an yi + ρii − 2ρii zi + bn + o(an ) from the proof of Theorem √ ρii (n) , qni = √ 1 − ρii (n) uni − zi
6.5.1 in [4]. So, one can check that
n
p
n
p
lim {Φ (qni )}n = exp − exp −xi − ρii +
n→∞
2ρii zi
o
and lim {Φ (tni )}n = exp − exp −yi − ρii +
n→∞
2ρii zi
o
.
By arguments similar to those of (3.6), we have n X
P (Sn1 = k1 , . . . , Snd = kd )
ki =1 1≤i≤d
→
d Y
{Φ (qni )}ki {Φ (tni )}n−ki
i d Y
n
exp −pi exp −xi − ρii +
2ρii zi
p
o
n
exp − (1 − pi ) exp −yi − ρii +
p
2ρii zi
o
i
as n → ∞. Combining this with (3.15), we obtain (3.14) by the dominated convergence theorem. The proof is complete.
Acknowledgments The authors would like to thank the Editor-in-Chief and the referee for careful reading and for their comments which greatly improved the paper.
Z. Peng et al. / Journal of Multivariate Analysis 101 (2010) 2641–2647
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