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The limit theorems on extremes for Gaussian random fields
Statistics and Probability Letters 83 (2013) 436–444
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The limit theorems on extremes for Gaussian random fields✩ Zhongquan Tan ∗ College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, PR China
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info
Article history: Received 7 January 2012 Received in revised form 16 October 2012 Accepted 19 October 2012 Available online 5 November 2012 MSC: primary 62G70 secondary 60G60
abstract Motivated by the papers of Choi (2010) and Pereira (2010), in this work, we proved two limit theorems for the maxima of Gaussian fields. First, a Cox limit theorem is established for a stationary strongly dependent Gaussian random field. Second, a Gumbel type extreme limit theorem is proved for a non-stationary Gaussian random field with covariance functions satisfying the Cesàro convergence. © 2012 Elsevier B.V. All rights reserved.
Keywords: Cox limit theorem Gumbel type extreme limit theorem Extremes Gaussian random fields Strongly dependent
1. Introduction In recent years, there have been a lot of papers on the limit properties of extremes for random sequences, but only a few papers on random fields. In this work, we are interested in the limit theorem for extremes of Gaussian random fields. It is well known that Gaussian random fields play a very important role in many applied sciences, such as in image analysis, atmospheric sciences, geostatistics, hydrology and agriculture, among others. See the monograph of Adler and Taylor (2007) for details. Firstly, we introduce some notation and notions of Gaussian random fields. Denote the set of all positive integers and all non-negative integers by Z and N, respectively. Let Zd and Nd be the d-dimensional product spaces of Z and N, respectively, where d ≥ 2. In this work, we only consider the case of d = 2 since it is notationally the simplest and the results for higher dimensions follow analogous arguments. For i = (i1 , i2 ) and j = (j1 , j2 ), i ≤ j and i − j mean ik ≤ jk , k = 1, 2, and (i1 − j1 , i2 − j2 ), respectively. |i| and n → ∞ mean (|i1 |, |i2 |) and nk → ∞, k = 1, 2, respectively. Let In = {j ∈ Z2 : 1 ≤ ji ≤ ni , i = 1, 2} and Jn = {j ∈ N2 : 0 ≤ ji ≤ ni , i = 1, 2}. Let χE be the number of elements in E for any subset E of Z2 . Let χk = i:ki ̸=0 |ki | for k = (k1 , k2 ) and χ0 = 1. Note
that χk = χIk when k ∈ Z2 . Let Φ (·) and φ(·) denote the standard Gaussian distribution function and its density function respectively. Let X = {Xn }n≥1 be a standardized Gaussian random field on Z2 . Let rij = Cov(Xi , Xj ) be the covariance functions of the Gaussian random field X = {Xn }n≥1 . Choi (2002) and Pereira (2010) studied the extremes for stationary Gaussian random fields and obtained the following Gumbel type extreme limit theorem.
✩ Research supported by National Science Foundation of China (No. 11071182).
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Tel.: +86 13957395469. E-mail address: [email protected].
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.10.025
Z. Tan / Statistics and Probability Letters 83 (2013) 436–444
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Theorem 1.1. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field. Assume that the covariance functions rm = EXj Xj+m satisfy lim r(n1 ,0) log n1 = 0,
lim r(0,n2 ) log n2 = 0,
n1 →∞
n2 →∞
(1)
for n ∈ Z2 , lim rn log χn = 0,
(2)
n→∞
and supn∈N2 −{0} |rn | < 1. (i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then,
lim P Mn ≤ un
n→∞
= exp(−τ ),
where Mn = maxi∈In Xi . (ii) Let an =
√
2 log χn and bn = an −
lim P an Mn − bn ≤ x
n→∞
log log χn +log(4π) . 2an
(3)
Then,
= exp(−e−x ).
(4)
Pereira (2010) extended Theorem 1.1 to the non-stationary Gaussian random fields by verifying the coordinatewisemixing type condition D(un,i ) and a local dependence condition D′ (un,i ) from Pereira and Ferreira (2006). However, condition (1) was excluded from the hypothesis of Choi (2002). For proving the non-stationary case, Pereira (2010) found that a similar condition was needed. Thus, he inferred from the proofs of results for the non-stationary case that condition (1) was needed for the stationary case. It is worth mentioning that conditions (1) and (2) are extensions of the well-known Berman condition from sequences to fields and also not necessary for (3) and (4). However, Choi (2010) showed that (3) and (4) still held under conditions of another type. Theorem 1.2. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field. Assume that the covariance functions rm = EXj Xj+m satisfy lim ran = 0 for any a ∈ {−1, 1}2
n→∞
(5)
and lim
n→∞
1
χn
|rk | log χk exp γ |rk | log χk = 0 for some γ > 4.
(6)
k∈In
(i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then, (3) holds. (ii) Let an and bn be defined as in Theorem 1.1. Then (4) holds. The almost sure limit versions of Theorems 1.1 and 1.2 are dealt with by Tan and Wang (in press) and Choi (2010), respectively. For further results concerning extremes in Gaussian random fields we refer the reader to Berman (1992), Piterbarg (1996), Adler and Taylor (2007), and Pereira (2010). For other related results on random fields, we refer the reader to Pereira (2009, 2012), Ferreira and Pereira (2008, 2012). In this work, we concentrate on the limit theorem on extremes of Gaussian random fields and have two goals. The first one is to extend Theorem 1.1 to the strongly dependent case. The second one is to extend Theorem 1.2 to the non-stationary case. As a by-product, we find that (3) and (4) still holds under some conditions which are weaker than condition (1). 2. The main results Now, we state our main results. The first one is the Cox limit theorem for the stationary Gaussian random fields while the second one is the Gumbel type extreme limit theorem for the non-stationary Gaussian random fields. Theorem 2.1. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field. Assume that the covariance functions rn = EXj Xj +n satisfy r(n1 ,0) log n1
and
r(0,n2 ) log n2
are bounded,
(7)
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for n ∈ Z2 , lim rn log χn = r ∈ [0, ∞),
(8)
n→∞
and supn∈N2 −{0} |rn | < 1. (i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then,
lim P Mn ≤ un
+∞
exp −τ exp(−r +
=
n→∞
√
2rx) φ(x)dx,
(9)
−∞
where Mn = maxi∈In Xi . √ (ii) Let an = 2 log χn and bn = an −
log log χn +log(4π) . 2an
lim P an Mn − bn ≤ x
+∞
exp − exp(−x − r +
=
n→∞
Then,
√
2rz ) φ(z )dz .
(10)
−∞
Remark 2.1. (i) Let r = 0; we obtain Theorem 1.1, but the condition (7) is weaker than (1). Thus, Theorem 2.1 not only extended Theorem 1.1 to the strongly dependent case, but also weakened the condition. (ii) Theorem 2.1 extended Theorem 6.5.1 of Leadbetter et al. (1983) to the random fields. (iii) We conclude that the assertions (9) and (10) still hold for non-stationary strongly dependent Gaussian fields which satisfy some conditions similar to (7) and (8). Theorem 2.2. Let X = {Xn }n≥1 be a non-stationary standardized Gaussian random field. Assume that the covariance functions rij = EXi Xj satisfy
σ = sup |rij | < 1
(11)
i̸=j
and the Cesàro convergence, i.e., for some θ > lim
n→∞
1
χn2
i,j∈In ,i≤j,i̸=j
4(1+σ ) , 1−σ
|rij | log χj−i exp θ|rij | log χj−i = 0.
(12)
(i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then, (3) holds. (ii) Let an and bn be defined as in Theorem 1.1. Then, (4) holds. Remark 2.2. Theorem 2.2 extended Theorem 1 of Xie (1984) to the random fields. 3. Auxiliary results In this section, we state and prove several lemmas which will be used in the proofs of our main results. As usual, an ≪ bn means an = O(bn ). Let C denote positive constants whose values may vary from place to place. The first lemma is the so called Normal Comparison lemma which can be found in Leadbetter et al. (1983). A simple special form of this theorem is given here. Lemma 3.1. Let X = {Xn }n≥1 and Y = {Yn }n≥1 be standardized Gaussian random fields with covariance functions Λ1ij and Λ2ij ,
respectively. Let maxi̸=j| γij | = γ < 1, where γij = max{Λ1ij , Λ2ij }. Then, for constants {un }n≥1 , we have
u2n { X i ≤ un } − P {Yi ≤ un } ≤ K |Λ1ij − Λ2ij | exp − , P i∈I 1 + |γij | i∈I i,j∈I ,i≤j,i̸=j n
n
n
where K is some constant, depending only on γ . Lemma 3.2. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field with covariance functions rn satisfying (7), (8) and supn∈N2 −{0} |rn | < 1. Let the constants {un }n≥1 be such that χn (1 − Φ (un )) is bounded. Put ρn = r / log χn . Then, we have
χn
k∈Jn ,k̸=0
|rk − ρn | exp −
u2n 1 + ωk
as n → ∞, where ωk = max{ρn , |rk |}.
→0
(13)
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Proof. We prove this result for when χn (1 − Φ (un )) actually converges to a finite limit, from which it follows as stated. If χn (1 − Φ (un )) < K for some K > 0 and vn is defined to satisfy χn (1 − Φ (vn )) = K then (13) follows with vn for un . But since vn ≤ un , it follows at once that (13) holds for un itself. Suppose that χn (1 − Φ (un )) converges to a finite limit. Since 1 − Φ (x) ∼ φ(x)/x as x → ∞, we have that
exp −
u2n
v
2
Kun
χn
,
un v
2 log χn
(14)
for large n. Let δ = supk∈N2 −{0} ωk . Clearly, δ < 1. Let α be a positive constant satisfying α < 2(11−δ . +δ) Split the sum on the left hand side of (13) into three parts, the first for k > 0, the second for k1 = 0 and k2 > 0, and the third for k2 = 0 and k1 > 0, and denote them by Tn,i , i = 1, 2, 3, respectively. First, we consider the term Tn,1 . Split Tn,1 into two parts:
T n ,1 = χ n
k∈Jn ,k>0
=: Tn(1,1) + Tn(2,1) .
+χn
k∈Jn ,k>0
χk ≤χnα
χk >χnα
For the first term, using the facts (14), we have (1) T n ,1
≤ δχn
u2n
exp −
k∈Jn ,k>0
χk ≤χnα
1+δ
u2 ≤ δχn χn2α exp − n 1+δ 2 1+δ un ≪ δχn χn2α χn 2 1+2α− 1+δ
≪ δχn
1
(log χn ) 1+δ . (1)
−σ1
2 Since α < 2(11−δ and 0 < δ < 1, we have 1 + 2α − 1+δ < 0. Hence, there exists a constant σ1 > 0 such that Tn,1 ≪ χn +δ) (2) (1) For the term Tn,1 , letting δp = supχk >p ωk , where p = χnα , we have
(2)
Tn,1 = χn
|rk − ρn | exp −
k∈Jn ,k>0
χk >χnα
≤ χn exp −
u2n
.
1 + ωk
u2n
(1)
1 + δp
|rk − ρn |
k∈Jn ,k>0
χk >χnα
χn2 u2n = exp − (1) log χn 1 + δp
log χn
χn
k∈Jn ,k>0
|rk − ρn |.
(15)
χk >χnα
(1)
Since rn log χn → r, there is a constant C such that rn log χn ≤ C , n ≥ 1. Hence also δp log p ≤ C , so by (14) we have
χn2 u2n χn2 u2n exp − ≤ exp − (1) log χn log χn 1 + C / log χnα 1 + δp 2/(1+C / log χnα ) χn2 un ≪ log χn χn α
α
≤ (χn )(2C / log χn )/(1+C / log χn ) = O(1)
(16) α
as n → ∞. Moreover, adding and subtracting ρn log χn / log χk = r / log χk and using the fact that log χk ≥ log χn for k > p gives log χn
χn
k∈Jn ,k>0
χk >χnα
|rk − ρn | ≤
1
αχn
k∈Jn ,k>0
χk >χnα
|rk log χk − r | + r
1 − log χn . χn k∈Jn ,k>0 log χk 1
χk >χnα
(17)
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Here the first term on the right tends to 0 by (8). Furthermore, estimating the second sum by an integral, we obtain
1 1 − log χn ≤ χn k∈Jn ,k>0 log χk α log χn
log χk 1 χ χ
1
χk >χnα
≪
χk >χnα
1
1
α log χn
n
k∈Jn ,k>0
n
1
| log x1 x2 |dx1 dx2 . 0
0
Hence the left hand side of (17) tends to 0. Since by (16), the first factor on the right of (15) is bounded, we thus conclude (2) that Tn,1 = o(1) as n → ∞. This completes the proof of Tn,1 . Second, we consider the term Tn,2 . We will discuss it for two cases, the first for χnα > n2 , and the second for χnα ≤ n2 . For the first case χnα > n2 , we have u2n
Tn,2 ≤ δχn
exp −
1+δ
k∈Jn ,k1 =0
u2n
≤ δχn n2 exp − ≪ δχn n2
un
1+δ
2 1+δ
χn
2 1+α− 1+δ
≤ δχn
1
(log χn ) 1+δ . −σ2
.
−σ3
.
2 Since α < 2(11−δ and 0 < δ < 1, we have 1 + α − 1+δ < 0. Hence, there exists a constant σ2 > 0 such that Tn,2 ≪ χn +δ) α For the second case χn ≤ n2 , split Tn,2 into two parts:
Tn,2 = χn
k∈Jn ,k1 =0 α k2 ≤χn
=: Tn(1,2) + Tn(2,2) .
+χn
k∈Jn ,k1 =0 α k2 >χn
For the first term, using the facts (14), we have (1)
Tn,2 ≤ δχn
u2n
exp −
k∈Jn ,k1 =0 α k2 ≤χn
1+δ
u2 ≤ δχn χnα exp − n 1+δ 2 1+δ un ≪ δχn χnα χn 2 1+α− 1+δ
≤ δχn
1
(log χn ) 1+δ . (1)
2 Since α < 2(11−δ and 0 < δ < 1, we have 1 + α − 1+δ < 0. Hence, there exists a constant σ3 > 0 such that Tn,2 ≪ χn +δ) (2) (2) For the term Tn,2 , letting δp = supk2 >p ω(0,k2 ) = supk2 >p max{ρn , |r(0,k2 ) |}, where p = χnα , we have
(2)
Tn,2 = χn
|rk − ρn | exp −
k∈Jn ,k1 =0 α k2 >χn
≤ χn exp −
u2n
1 + ωk
u2n
(2)
1 + δp
|rk − ρn |
k∈Jn ,k1 =0 α k2 >χn
u2n log χn χn2 = exp − (2) log χn χn 1 + δp
|rk − ρn |.
(18)
k∈Jn ,k1 =0 α k2 >χn
(2)
Since r(0,n2 ) log n2 is bounded, there is a constant C such that r(0,n2 ) log n2 ≤ C , n2 ≥ 1. Hence also δp log p ≤ C , so by (14), like in the proof of (16), we conclude that
χn2 u2n exp − = O(1) (2) log χn 1 + δp
(19)
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441
as n → ∞. Since r(0,n2 ) log n2 ≤ C , we have log χn
χn
log χn
|rk − ρn | ≤
χn
k∈Jn ,k1 =0 α k2 >χn
log χn
≤
χn
χnα
log χn
≪
(|r(0,k2 ) | + ρn )
k∈Jn ,k1 =0 α k2 >χn
(C / log k2 + r / log χn )
n2
χn log χn
= o(1),
(20)
∞. Substituting (19) and (20) into (18), we obtain Tn(2,2)
as n → that Tn,2 tends to 0 as n → ∞. Likewise we can bound the third term Tn,3 .
= o(1) as n → ∞. This completes the proof of the assertion
Lemma 3.3. Let X = {Xn }n≥1 be a non-stationary standardized Gaussian random field with covariance functions rij satisfying (11) and (12). Let the constants {un }n≥1 be such that χn (1 − Φ (un )) is bounded. Then, we have u2n
|rij | exp −
i,j∈In ,i≤j,i̸=j
→0
1 + |rij |
(21)
as n → ∞. Proof. Like in the proof of Lemma 3.2, we suppose that χn (1 − Φ (un )) converges to a finite limit. Let β = θ2 < 2(11−σ . Split +σ ) the sum into two parts:
+
i,j∈In ,i≤j,i̸=j
=: Sn,1 + Sn,2 .
i,j∈In ,i≤j,i̸=j
β χ|i−j| ≤χn
β χ|i−j| >χn
For the first term, using the facts (14), we have
|rij | exp −
Sn,1 =
i,j∈In ,i≤j,i̸=j
u2n
1 + |rij |
β χ|i−j| ≤χn
u2n
≤ σ
exp −
i,j∈In ,i≤j,i̸=j
1+σ
β χ|i−j| ≤χn
1+2β n
≤ σχ
1+2β n
≤ σχ
≪ σ χn1+2β
u2n
exp −
1+σ
exp −
un
u2n
2 1+σ
2
2 1+σ
χn
2 1− 1+σ +2β
≪ σ χn
1
(log χn ) 1+σ . −σ3
2 Since β = θ2 < 2(11−σ , we have 1 − 1+σ + 2β < 0. Hence, there exists a constant σ3 > 0 such that Sn,1 ≪ χn +σ )
For the second term, using the facts (14) again, we have Sn,2 =
|rij | exp −
i,j∈In ,i≤j,i̸=j
u2n
1 + |rij |
β χ|i−j| >χn
≪
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
|rij |
log χn
χn
1+|2r
ij |
.
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≤
|rij |
log χn
χ
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
ij |
2 n
2|rij |
≤ χn−2
1+|1r
|rij |χn
log χn
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
|rij |(χ|i−j| )θ|rij | log(χ|i−j| )
≤ β −1 χn−2
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
≪
≪
1
χ
2 n i,j∈In ,i≤j,i̸=j β χ|i−j| >χn
1
χ
|rij | exp θ|rij | log χj−i log χj−i
|rij | exp θ|rij | log χj−i log χj−i .
2 n i,j∈In ,i≤j,i̸=j
Hence Sn,2 = o(1) as n → ∞, by condition (12). 4. Proof of the main results In this section, we give the proofs of our main results. The proof of Theorem 2.1. For case (i), we split the discussion into two cases. (1) Case τ ∈ [0, ∞). Let Z = {Zn }n≥1 be an independent standardized Gaussian random field. Define
ξi =
1 − ρn Zi +
√ ρn V ,
i ∈ In ,
where V is a standard Gaussian random variable independent of Z = {Zn }n≥1 . Note that the covariance function of
ξ = {ξi }i∈In is equal to ρn . Let Mn (ξ ) = maxi∈In ξi . First, we show that
|P (Mn ≤ un ) − P (Mn (ξ ) ≤ un )| → 0
(22)
as n → ∞. By Lemmas 3.1 and 3.2, we have
|P (Mn ≤ un ) − P (Mn (ξ ) ≤ un )| ≤ K χn
|rk − ρn | exp −
k∈Jn ,k̸=0
u2n
1 + ωk
= o(1) as n → ∞. Second, in view of the definition of ξ = {ξi }i∈In , we have
P Mn (ξ ) ≤ un
= P max ξi ≤ un i∈In √ =P 1 − ρn max Zi + ρn V ≤ un i∈In +∞ √ = P 1 − ρn max Zi + ρn x ≤ un φ(x)dx i∈In
−∞
+∞
= −∞
√
ρn x P max Zi ≤ √ φ(x)dx. i∈In 1 − ρn un −
It is easy to check that
√ ρn x 1 √ = (un − ρn x) 1 + ρn + o(ρn ) 2 1 − ρn √ r − 2rx 1 + o(u− = un + n ).
u(nx) :=
un −
√
un
(23)
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443
Thus, it follows from limn→∞ χn (1 − Φ (un )) = τ ∈ (0, ∞) that lim χn (1 − Φ (un(x) )) = τ e−r +
√
n→∞
2rx
.
Now, noting that Z = {Zn }n≥1 is an independent standardized Gaussian random field, we have
(x)
P max Zi ≤ un
=
i∈In
(x)
P Z i ≤ un
i∈In
= χn (1 − Φ (u(nx) )) = τ e −r +
√
2rx
(1 + o(1)).
Combining the last result with (22), (23) and applying the dominated convergence theorem, we obtain the desired result. (2) Case τ = ∞. Since Φ (·) is continuous, we know that for arbitrarily large τ ′ < ∞, there exists a real sequence vn such that limn→∞ χn (1 − Φ (vn )) = τ ′ . Clearly, for n sufficiently large, un ≤ vn , and hence
P Mn ≤ un
√ ≤ Mn ≤ vn → E exp −τ ′ exp(−r + 2rV ) ,
n → ∞.
Since this holds for arbitrarily large τ ′ < ∞, on letting τ ′ → ∞ we see that
lim
n→∞
Mn ≤ un
= 0,
which completes the proof.
For case (ii), we proceed as follows. Note that in this case, un = un (x) = x/an + bn . It is easy to check that
χn (1 − Φ (un )) = χn (1 − Φ (x/an + bn )) → e−x as n → ∞. Thus, case (ii) follows from case (i) with τ = e−x . The proof of Theorem 2.2. For case (i), we also split the discussion into two cases. (1) Case τ ∈ [0, ∞). Let Y = {Yn }n≥1 be an independent standardized Gaussian random field. It is easy to see that
P (Yi ≤ un ) − exp(−τ ) . |P (Mn ≤ un ) − exp(−τ )| ≤ P (Mn ≤ un ) − P (Yi ≤ un ) + i∈I i∈I n
(24)
n
By Lemmas 3.1 and 3.3, we have
u2n |rij | exp − P Y i ≤ un ≤ K P Mn ≤ un − 1 + |rij | i,j∈I ,i≤j,i̸=j i∈I n
n
= o(1). Now, note that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞). It is easy to check that the second term on the right hand side of (24) is also o(1). The proof is complete. (2) Case τ = ∞. The proof is similar to that for case τ = ∞ of Theorem 2.1. (ii) Case (ii) follows from case (i) with τ = e−x . Acknowledgments The author would like to thank the two referees for several corrections and important suggestions which led to significant improvement of this work. References Adler, R., Taylor, J.E., 2007. Random Fields and Geometry. Springer, New York. Berman, S., 1992. Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Boston. Choi, H., 2002. Central limit theory and extremes of random fields. Ph.D. Dissertation in Univ. of North Carolina at Chapel Hill. Choi, H., 2010. Almost sure limit theorem for stationary Gaussian random fields. Journal of the Korean Statistical Society 39, 449–454. Ferreira, H., Pereira, L., 2008. How to compute the extremal index of stationary random fields. Statistics and Probability Letters 78, 1301–1304. Ferreira, H., Pereira, L., 2012. Point processes of exceedances by random fields. Journal of Statistical Planning and Inference 142, 773–779. Leadbetter, M.R., Lindgren, G., Rootzén, H., 1983. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York. Pereira, L., 2009. The asymptotic location of the maximum of a stationary random field. Statistics and Probability Letters 79, 2166–2169. Pereira, L., 2010. On the extremal behavior of a nonstationary normal random field. Journal of Statistical Planning and Inference 140, 3567–3576.
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Pereira, L., 2012. Asymptotic location of largest values of a stationary random field. Communications in Statistics—Theory and Methods http://dx.doi.org/ 10.1080/03610926.2011.650270. Pereira, L., Ferreira, H., 2006. Limiting crossing probabilities of random fields. Journal of Applied Probability 43, 884–891. Piterbarg, V.I., 1996. Asymptotic Methods in the Theory of Gaussian Processes and Fields. AMS, Providence. Tan, Z., Wang, Y., 2012. Almost sure asymptotics for extremes of non-stationary Gaussian random fields. Chinese Annals of Mathematics, Series B (in press). Xie, S., 1984. Asymptotic distributions of extremes in non-stationary Gaussian sequences. Acta Mathematicae Applicatae Sinica 7, 96–100.
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The limit theorems on extremes for Gaussian random fields✩ Zhongquan Tan ∗ College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, PR China
article
info
Article history: Received 7 January 2012 Received in revised form 16 October 2012 Accepted 19 October 2012 Available online 5 November 2012 MSC: primary 62G70 secondary 60G60
abstract Motivated by the papers of Choi (2010) and Pereira (2010), in this work, we proved two limit theorems for the maxima of Gaussian fields. First, a Cox limit theorem is established for a stationary strongly dependent Gaussian random field. Second, a Gumbel type extreme limit theorem is proved for a non-stationary Gaussian random field with covariance functions satisfying the Cesàro convergence. © 2012 Elsevier B.V. All rights reserved.
Keywords: Cox limit theorem Gumbel type extreme limit theorem Extremes Gaussian random fields Strongly dependent
1. Introduction In recent years, there have been a lot of papers on the limit properties of extremes for random sequences, but only a few papers on random fields. In this work, we are interested in the limit theorem for extremes of Gaussian random fields. It is well known that Gaussian random fields play a very important role in many applied sciences, such as in image analysis, atmospheric sciences, geostatistics, hydrology and agriculture, among others. See the monograph of Adler and Taylor (2007) for details. Firstly, we introduce some notation and notions of Gaussian random fields. Denote the set of all positive integers and all non-negative integers by Z and N, respectively. Let Zd and Nd be the d-dimensional product spaces of Z and N, respectively, where d ≥ 2. In this work, we only consider the case of d = 2 since it is notationally the simplest and the results for higher dimensions follow analogous arguments. For i = (i1 , i2 ) and j = (j1 , j2 ), i ≤ j and i − j mean ik ≤ jk , k = 1, 2, and (i1 − j1 , i2 − j2 ), respectively. |i| and n → ∞ mean (|i1 |, |i2 |) and nk → ∞, k = 1, 2, respectively. Let In = {j ∈ Z2 : 1 ≤ ji ≤ ni , i = 1, 2} and Jn = {j ∈ N2 : 0 ≤ ji ≤ ni , i = 1, 2}. Let χE be the number of elements in E for any subset E of Z2 . Let χk = i:ki ̸=0 |ki | for k = (k1 , k2 ) and χ0 = 1. Note
that χk = χIk when k ∈ Z2 . Let Φ (·) and φ(·) denote the standard Gaussian distribution function and its density function respectively. Let X = {Xn }n≥1 be a standardized Gaussian random field on Z2 . Let rij = Cov(Xi , Xj ) be the covariance functions of the Gaussian random field X = {Xn }n≥1 . Choi (2002) and Pereira (2010) studied the extremes for stationary Gaussian random fields and obtained the following Gumbel type extreme limit theorem.
✩ Research supported by National Science Foundation of China (No. 11071182).
∗
Tel.: +86 13957395469. E-mail address: [email protected].
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.10.025
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Theorem 1.1. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field. Assume that the covariance functions rm = EXj Xj+m satisfy lim r(n1 ,0) log n1 = 0,
lim r(0,n2 ) log n2 = 0,
n1 →∞
n2 →∞
(1)
for n ∈ Z2 , lim rn log χn = 0,
(2)
n→∞
and supn∈N2 −{0} |rn | < 1. (i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then,
lim P Mn ≤ un
n→∞
= exp(−τ ),
where Mn = maxi∈In Xi . (ii) Let an =
√
2 log χn and bn = an −
lim P an Mn − bn ≤ x
n→∞
log log χn +log(4π) . 2an
(3)
Then,
= exp(−e−x ).
(4)
Pereira (2010) extended Theorem 1.1 to the non-stationary Gaussian random fields by verifying the coordinatewisemixing type condition D(un,i ) and a local dependence condition D′ (un,i ) from Pereira and Ferreira (2006). However, condition (1) was excluded from the hypothesis of Choi (2002). For proving the non-stationary case, Pereira (2010) found that a similar condition was needed. Thus, he inferred from the proofs of results for the non-stationary case that condition (1) was needed for the stationary case. It is worth mentioning that conditions (1) and (2) are extensions of the well-known Berman condition from sequences to fields and also not necessary for (3) and (4). However, Choi (2010) showed that (3) and (4) still held under conditions of another type. Theorem 1.2. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field. Assume that the covariance functions rm = EXj Xj+m satisfy lim ran = 0 for any a ∈ {−1, 1}2
n→∞
(5)
and lim
n→∞
1
χn
|rk | log χk exp γ |rk | log χk = 0 for some γ > 4.
(6)
k∈In
(i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then, (3) holds. (ii) Let an and bn be defined as in Theorem 1.1. Then (4) holds. The almost sure limit versions of Theorems 1.1 and 1.2 are dealt with by Tan and Wang (in press) and Choi (2010), respectively. For further results concerning extremes in Gaussian random fields we refer the reader to Berman (1992), Piterbarg (1996), Adler and Taylor (2007), and Pereira (2010). For other related results on random fields, we refer the reader to Pereira (2009, 2012), Ferreira and Pereira (2008, 2012). In this work, we concentrate on the limit theorem on extremes of Gaussian random fields and have two goals. The first one is to extend Theorem 1.1 to the strongly dependent case. The second one is to extend Theorem 1.2 to the non-stationary case. As a by-product, we find that (3) and (4) still holds under some conditions which are weaker than condition (1). 2. The main results Now, we state our main results. The first one is the Cox limit theorem for the stationary Gaussian random fields while the second one is the Gumbel type extreme limit theorem for the non-stationary Gaussian random fields. Theorem 2.1. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field. Assume that the covariance functions rn = EXj Xj +n satisfy r(n1 ,0) log n1
and
r(0,n2 ) log n2
are bounded,
(7)
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Z. Tan / Statistics and Probability Letters 83 (2013) 436–444
for n ∈ Z2 , lim rn log χn = r ∈ [0, ∞),
(8)
n→∞
and supn∈N2 −{0} |rn | < 1. (i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then,
lim P Mn ≤ un
+∞
exp −τ exp(−r +
=
n→∞
√
2rx) φ(x)dx,
(9)
−∞
where Mn = maxi∈In Xi . √ (ii) Let an = 2 log χn and bn = an −
log log χn +log(4π) . 2an
lim P an Mn − bn ≤ x
+∞
exp − exp(−x − r +
=
n→∞
Then,
√
2rz ) φ(z )dz .
(10)
−∞
Remark 2.1. (i) Let r = 0; we obtain Theorem 1.1, but the condition (7) is weaker than (1). Thus, Theorem 2.1 not only extended Theorem 1.1 to the strongly dependent case, but also weakened the condition. (ii) Theorem 2.1 extended Theorem 6.5.1 of Leadbetter et al. (1983) to the random fields. (iii) We conclude that the assertions (9) and (10) still hold for non-stationary strongly dependent Gaussian fields which satisfy some conditions similar to (7) and (8). Theorem 2.2. Let X = {Xn }n≥1 be a non-stationary standardized Gaussian random field. Assume that the covariance functions rij = EXi Xj satisfy
σ = sup |rij | < 1
(11)
i̸=j
and the Cesàro convergence, i.e., for some θ > lim
n→∞
1
χn2
i,j∈In ,i≤j,i̸=j
4(1+σ ) , 1−σ
|rij | log χj−i exp θ|rij | log χj−i = 0.
(12)
(i) Let the constants {un }n≥1 be such that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞]. Then, (3) holds. (ii) Let an and bn be defined as in Theorem 1.1. Then, (4) holds. Remark 2.2. Theorem 2.2 extended Theorem 1 of Xie (1984) to the random fields. 3. Auxiliary results In this section, we state and prove several lemmas which will be used in the proofs of our main results. As usual, an ≪ bn means an = O(bn ). Let C denote positive constants whose values may vary from place to place. The first lemma is the so called Normal Comparison lemma which can be found in Leadbetter et al. (1983). A simple special form of this theorem is given here. Lemma 3.1. Let X = {Xn }n≥1 and Y = {Yn }n≥1 be standardized Gaussian random fields with covariance functions Λ1ij and Λ2ij ,
respectively. Let maxi̸=j| γij | = γ < 1, where γij = max{Λ1ij , Λ2ij }. Then, for constants {un }n≥1 , we have
u2n { X i ≤ un } − P {Yi ≤ un } ≤ K |Λ1ij − Λ2ij | exp − , P i∈I 1 + |γij | i∈I i,j∈I ,i≤j,i̸=j n
n
n
where K is some constant, depending only on γ . Lemma 3.2. Let X = {Xn }n≥1 be a stationary standardized Gaussian random field with covariance functions rn satisfying (7), (8) and supn∈N2 −{0} |rn | < 1. Let the constants {un }n≥1 be such that χn (1 − Φ (un )) is bounded. Put ρn = r / log χn . Then, we have
χn
k∈Jn ,k̸=0
|rk − ρn | exp −
u2n 1 + ωk
as n → ∞, where ωk = max{ρn , |rk |}.
→0
(13)
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439
Proof. We prove this result for when χn (1 − Φ (un )) actually converges to a finite limit, from which it follows as stated. If χn (1 − Φ (un )) < K for some K > 0 and vn is defined to satisfy χn (1 − Φ (vn )) = K then (13) follows with vn for un . But since vn ≤ un , it follows at once that (13) holds for un itself. Suppose that χn (1 − Φ (un )) converges to a finite limit. Since 1 − Φ (x) ∼ φ(x)/x as x → ∞, we have that
exp −
u2n
v
2
Kun
χn
,
un v
2 log χn
(14)
for large n. Let δ = supk∈N2 −{0} ωk . Clearly, δ < 1. Let α be a positive constant satisfying α < 2(11−δ . +δ) Split the sum on the left hand side of (13) into three parts, the first for k > 0, the second for k1 = 0 and k2 > 0, and the third for k2 = 0 and k1 > 0, and denote them by Tn,i , i = 1, 2, 3, respectively. First, we consider the term Tn,1 . Split Tn,1 into two parts:
T n ,1 = χ n
k∈Jn ,k>0
=: Tn(1,1) + Tn(2,1) .
+χn
k∈Jn ,k>0
χk ≤χnα
χk >χnα
For the first term, using the facts (14), we have (1) T n ,1
≤ δχn
u2n
exp −
k∈Jn ,k>0
χk ≤χnα
1+δ
u2 ≤ δχn χn2α exp − n 1+δ 2 1+δ un ≪ δχn χn2α χn 2 1+2α− 1+δ
≪ δχn
1
(log χn ) 1+δ . (1)
−σ1
2 Since α < 2(11−δ and 0 < δ < 1, we have 1 + 2α − 1+δ < 0. Hence, there exists a constant σ1 > 0 such that Tn,1 ≪ χn +δ) (2) (1) For the term Tn,1 , letting δp = supχk >p ωk , where p = χnα , we have
(2)
Tn,1 = χn
|rk − ρn | exp −
k∈Jn ,k>0
χk >χnα
≤ χn exp −
u2n
.
1 + ωk
u2n
(1)
1 + δp
|rk − ρn |
k∈Jn ,k>0
χk >χnα
χn2 u2n = exp − (1) log χn 1 + δp
log χn
χn
k∈Jn ,k>0
|rk − ρn |.
(15)
χk >χnα
(1)
Since rn log χn → r, there is a constant C such that rn log χn ≤ C , n ≥ 1. Hence also δp log p ≤ C , so by (14) we have
χn2 u2n χn2 u2n exp − ≤ exp − (1) log χn log χn 1 + C / log χnα 1 + δp 2/(1+C / log χnα ) χn2 un ≪ log χn χn α
α
≤ (χn )(2C / log χn )/(1+C / log χn ) = O(1)
(16) α
as n → ∞. Moreover, adding and subtracting ρn log χn / log χk = r / log χk and using the fact that log χk ≥ log χn for k > p gives log χn
χn
k∈Jn ,k>0
χk >χnα
|rk − ρn | ≤
1
αχn
k∈Jn ,k>0
χk >χnα
|rk log χk − r | + r
1 − log χn . χn k∈Jn ,k>0 log χk 1
χk >χnα
(17)
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Here the first term on the right tends to 0 by (8). Furthermore, estimating the second sum by an integral, we obtain
1 1 − log χn ≤ χn k∈Jn ,k>0 log χk α log χn
log χk 1 χ χ
1
χk >χnα
≪
χk >χnα
1
1
α log χn
n
k∈Jn ,k>0
n
1
| log x1 x2 |dx1 dx2 . 0
0
Hence the left hand side of (17) tends to 0. Since by (16), the first factor on the right of (15) is bounded, we thus conclude (2) that Tn,1 = o(1) as n → ∞. This completes the proof of Tn,1 . Second, we consider the term Tn,2 . We will discuss it for two cases, the first for χnα > n2 , and the second for χnα ≤ n2 . For the first case χnα > n2 , we have u2n
Tn,2 ≤ δχn
exp −
1+δ
k∈Jn ,k1 =0
u2n
≤ δχn n2 exp − ≪ δχn n2
un
1+δ
2 1+δ
χn
2 1+α− 1+δ
≤ δχn
1
(log χn ) 1+δ . −σ2
.
−σ3
.
2 Since α < 2(11−δ and 0 < δ < 1, we have 1 + α − 1+δ < 0. Hence, there exists a constant σ2 > 0 such that Tn,2 ≪ χn +δ) α For the second case χn ≤ n2 , split Tn,2 into two parts:
Tn,2 = χn
k∈Jn ,k1 =0 α k2 ≤χn
=: Tn(1,2) + Tn(2,2) .
+χn
k∈Jn ,k1 =0 α k2 >χn
For the first term, using the facts (14), we have (1)
Tn,2 ≤ δχn
u2n
exp −
k∈Jn ,k1 =0 α k2 ≤χn
1+δ
u2 ≤ δχn χnα exp − n 1+δ 2 1+δ un ≪ δχn χnα χn 2 1+α− 1+δ
≤ δχn
1
(log χn ) 1+δ . (1)
2 Since α < 2(11−δ and 0 < δ < 1, we have 1 + α − 1+δ < 0. Hence, there exists a constant σ3 > 0 such that Tn,2 ≪ χn +δ) (2) (2) For the term Tn,2 , letting δp = supk2 >p ω(0,k2 ) = supk2 >p max{ρn , |r(0,k2 ) |}, where p = χnα , we have
(2)
Tn,2 = χn
|rk − ρn | exp −
k∈Jn ,k1 =0 α k2 >χn
≤ χn exp −
u2n
1 + ωk
u2n
(2)
1 + δp
|rk − ρn |
k∈Jn ,k1 =0 α k2 >χn
u2n log χn χn2 = exp − (2) log χn χn 1 + δp
|rk − ρn |.
(18)
k∈Jn ,k1 =0 α k2 >χn
(2)
Since r(0,n2 ) log n2 is bounded, there is a constant C such that r(0,n2 ) log n2 ≤ C , n2 ≥ 1. Hence also δp log p ≤ C , so by (14), like in the proof of (16), we conclude that
χn2 u2n exp − = O(1) (2) log χn 1 + δp
(19)
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441
as n → ∞. Since r(0,n2 ) log n2 ≤ C , we have log χn
χn
log χn
|rk − ρn | ≤
χn
k∈Jn ,k1 =0 α k2 >χn
log χn
≤
χn
χnα
log χn
≪
(|r(0,k2 ) | + ρn )
k∈Jn ,k1 =0 α k2 >χn
(C / log k2 + r / log χn )
n2
χn log χn
= o(1),
(20)
∞. Substituting (19) and (20) into (18), we obtain Tn(2,2)
as n → that Tn,2 tends to 0 as n → ∞. Likewise we can bound the third term Tn,3 .
= o(1) as n → ∞. This completes the proof of the assertion
Lemma 3.3. Let X = {Xn }n≥1 be a non-stationary standardized Gaussian random field with covariance functions rij satisfying (11) and (12). Let the constants {un }n≥1 be such that χn (1 − Φ (un )) is bounded. Then, we have u2n
|rij | exp −
i,j∈In ,i≤j,i̸=j
→0
1 + |rij |
(21)
as n → ∞. Proof. Like in the proof of Lemma 3.2, we suppose that χn (1 − Φ (un )) converges to a finite limit. Let β = θ2 < 2(11−σ . Split +σ ) the sum into two parts:
+
i,j∈In ,i≤j,i̸=j
=: Sn,1 + Sn,2 .
i,j∈In ,i≤j,i̸=j
β χ|i−j| ≤χn
β χ|i−j| >χn
For the first term, using the facts (14), we have
|rij | exp −
Sn,1 =
i,j∈In ,i≤j,i̸=j
u2n
1 + |rij |
β χ|i−j| ≤χn
u2n
≤ σ
exp −
i,j∈In ,i≤j,i̸=j
1+σ
β χ|i−j| ≤χn
1+2β n
≤ σχ
1+2β n
≤ σχ
≪ σ χn1+2β
u2n
exp −
1+σ
exp −
un
u2n
2 1+σ
2
2 1+σ
χn
2 1− 1+σ +2β
≪ σ χn
1
(log χn ) 1+σ . −σ3
2 Since β = θ2 < 2(11−σ , we have 1 − 1+σ + 2β < 0. Hence, there exists a constant σ3 > 0 such that Sn,1 ≪ χn +σ )
For the second term, using the facts (14) again, we have Sn,2 =
|rij | exp −
i,j∈In ,i≤j,i̸=j
u2n
1 + |rij |
β χ|i−j| >χn
≪
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
|rij |
log χn
χn
1+|2r
ij |
.
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≤
|rij |
log χn
χ
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
ij |
2 n
2|rij |
≤ χn−2
1+|1r
|rij |χn
log χn
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
|rij |(χ|i−j| )θ|rij | log(χ|i−j| )
≤ β −1 χn−2
i,j∈In ,i≤j,i̸=j
β χ|i−j| >χn
≪
≪
1
χ
2 n i,j∈In ,i≤j,i̸=j β χ|i−j| >χn
1
χ
|rij | exp θ|rij | log χj−i log χj−i
|rij | exp θ|rij | log χj−i log χj−i .
2 n i,j∈In ,i≤j,i̸=j
Hence Sn,2 = o(1) as n → ∞, by condition (12). 4. Proof of the main results In this section, we give the proofs of our main results. The proof of Theorem 2.1. For case (i), we split the discussion into two cases. (1) Case τ ∈ [0, ∞). Let Z = {Zn }n≥1 be an independent standardized Gaussian random field. Define
ξi =
1 − ρn Zi +
√ ρn V ,
i ∈ In ,
where V is a standard Gaussian random variable independent of Z = {Zn }n≥1 . Note that the covariance function of
ξ = {ξi }i∈In is equal to ρn . Let Mn (ξ ) = maxi∈In ξi . First, we show that
|P (Mn ≤ un ) − P (Mn (ξ ) ≤ un )| → 0
(22)
as n → ∞. By Lemmas 3.1 and 3.2, we have
|P (Mn ≤ un ) − P (Mn (ξ ) ≤ un )| ≤ K χn
|rk − ρn | exp −
k∈Jn ,k̸=0
u2n
1 + ωk
= o(1) as n → ∞. Second, in view of the definition of ξ = {ξi }i∈In , we have
P Mn (ξ ) ≤ un
= P max ξi ≤ un i∈In √ =P 1 − ρn max Zi + ρn V ≤ un i∈In +∞ √ = P 1 − ρn max Zi + ρn x ≤ un φ(x)dx i∈In
−∞
+∞
= −∞
√
ρn x P max Zi ≤ √ φ(x)dx. i∈In 1 − ρn un −
It is easy to check that
√ ρn x 1 √ = (un − ρn x) 1 + ρn + o(ρn ) 2 1 − ρn √ r − 2rx 1 + o(u− = un + n ).
u(nx) :=
un −
√
un
(23)
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443
Thus, it follows from limn→∞ χn (1 − Φ (un )) = τ ∈ (0, ∞) that lim χn (1 − Φ (un(x) )) = τ e−r +
√
n→∞
2rx
.
Now, noting that Z = {Zn }n≥1 is an independent standardized Gaussian random field, we have
(x)
P max Zi ≤ un
=
i∈In
(x)
P Z i ≤ un
i∈In
= χn (1 − Φ (u(nx) )) = τ e −r +
√
2rx
(1 + o(1)).
Combining the last result with (22), (23) and applying the dominated convergence theorem, we obtain the desired result. (2) Case τ = ∞. Since Φ (·) is continuous, we know that for arbitrarily large τ ′ < ∞, there exists a real sequence vn such that limn→∞ χn (1 − Φ (vn )) = τ ′ . Clearly, for n sufficiently large, un ≤ vn , and hence
P Mn ≤ un
√ ≤ Mn ≤ vn → E exp −τ ′ exp(−r + 2rV ) ,
n → ∞.
Since this holds for arbitrarily large τ ′ < ∞, on letting τ ′ → ∞ we see that
lim
n→∞
Mn ≤ un
= 0,
which completes the proof.
For case (ii), we proceed as follows. Note that in this case, un = un (x) = x/an + bn . It is easy to check that
χn (1 − Φ (un )) = χn (1 − Φ (x/an + bn )) → e−x as n → ∞. Thus, case (ii) follows from case (i) with τ = e−x . The proof of Theorem 2.2. For case (i), we also split the discussion into two cases. (1) Case τ ∈ [0, ∞). Let Y = {Yn }n≥1 be an independent standardized Gaussian random field. It is easy to see that
P (Yi ≤ un ) − exp(−τ ) . |P (Mn ≤ un ) − exp(−τ )| ≤ P (Mn ≤ un ) − P (Yi ≤ un ) + i∈I i∈I n
(24)
n
By Lemmas 3.1 and 3.3, we have
u2n |rij | exp − P Y i ≤ un ≤ K P Mn ≤ un − 1 + |rij | i,j∈I ,i≤j,i̸=j i∈I n
n
= o(1). Now, note that limn→∞ χn (1 − Φ (un )) = τ ∈ [0, ∞). It is easy to check that the second term on the right hand side of (24) is also o(1). The proof is complete. (2) Case τ = ∞. The proof is similar to that for case τ = ∞ of Theorem 2.1. (ii) Case (ii) follows from case (i) with τ = e−x . Acknowledgments The author would like to thank the two referees for several corrections and important suggestions which led to significant improvement of this work. References Adler, R., Taylor, J.E., 2007. Random Fields and Geometry. Springer, New York. Berman, S., 1992. Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Boston. Choi, H., 2002. Central limit theory and extremes of random fields. Ph.D. Dissertation in Univ. of North Carolina at Chapel Hill. Choi, H., 2010. Almost sure limit theorem for stationary Gaussian random fields. Journal of the Korean Statistical Society 39, 449–454. Ferreira, H., Pereira, L., 2008. How to compute the extremal index of stationary random fields. Statistics and Probability Letters 78, 1301–1304. Ferreira, H., Pereira, L., 2012. Point processes of exceedances by random fields. Journal of Statistical Planning and Inference 142, 773–779. Leadbetter, M.R., Lindgren, G., Rootzén, H., 1983. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York. Pereira, L., 2009. The asymptotic location of the maximum of a stationary random field. Statistics and Probability Letters 79, 2166–2169. Pereira, L., 2010. On the extremal behavior of a nonstationary normal random field. Journal of Statistical Planning and Inference 140, 3567–3576.
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