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Slepian’s inequality with respect to majorization
Linear Algebra and its Applications 434 (2011) 1107–1118
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Slepian’s inequality with respect to majorization Longxiang Fang a,b , Xinsheng Zhang a,∗ a b
Department of Statistics, Fudan University, Shanghai 200433, PR China Department of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, PR China
A R T I C L E
I N F O
A B S T R A C T
In this paper, we obtain some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 24 May 2010 Accepted 23 October 2010 Available online 18 November 2010 Submitted by H. Schneider AMS classification: 60E15 62G30 Keywords: Slepian’s inequality Gaussian random vectors Majorization Schur-convex
1. Introduction Let X = (X1 , . . . , Xn ) and X ∗ = (X1∗ , . . . , Xn∗ ) be two centered Gaussian random vectors with covariance matrices = (σij ) and ∗ = (σij∗ ), respectively. The well-known Slepian’s inequality [8] states that if σii = σii∗ and σij σij∗ for every i, j = 1, . . . , n, then for any x ∈ R, P min Xi x P min Xi∗ x , P max Xi x P max Xi∗ x . 1in
1in
1in
1in
Slepian’s inequality and its variations provide a very useful tool in the theory of Gaussian random variables and Gaussian processes. See, for examples, [1,5,7]. Majorization is a very interesting topic in various fields of mathematics and statistics, which is a partial ordering on vectors by sorting all components in nonincreasing order . If – and –∗ are two vectors ∗ Corresponding author. Tel.: +86 21 2501 1226; fax: +86 21 6564 2351. E-mail address: [email protected] (X. Zhang). 0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.10.019
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L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
m ∗ with all components in nonincreasing order such that m i=1 λi i=1 λi for m = 1, 2, . . . , n − 1 with ∗ equality at m = n, then – is said to be majorized by – and denote – m –∗ . The history of investigating on majorization should date back to Schur [9] and Hardy et al. [4]. In the reference of Marshall and Olkin [6], the reader can find that majorization has been connected with combinatorics, analytic inequalities, numerical analysis, matrix theory, probability and statistics. Recent research on majorization with respect to matrix inequalities and norm inequalities has been considered by Ando [2]. In this paper, we search some new conditions via majorization to obtain Slepian’s inequality. We consider two cases as follows. One case is two Gaussian random vectors have equal mean, sharing the same correlation coefficient for all components and the vector being constituted by the reciprocal of each component’s standard deviation satisfying majorization; the other case is two Gaussian random vectors with equal covariance matrix and the mean vector satisfying majorization. Our results cannot be derived from the well-known Slepian’s inequality. The explanation will be given in the later. 2. Main results and applications In order to state our main results, we first give the definitions of majorization and weak majorization. Definition 2.1. Let – = (λ1 , λ2 , . . . , λn ), –∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors. Let λ[1] λ[2] · · · λ[n] , λ∗[1] λ∗[2] · · · λ∗[n] , λ(1) λ(2) · · · λ(n) , λ∗(1) λ∗(2) · · · λ∗(n) be their ordered components. (1) –∗ is said to be majorized by –, in symbols – k i =1
λ[i]
k i =1
m –∗ , if
λ∗[i]
for k = 1, 2, . . . , n − 1 and ni=1 λi = ni=1 λ∗i . (2) –∗ is said to be weak lower majorized by –, in symbols – k i =1
λ[i]
k i =1
λ∗[i]
for k = 1, 2, . . . , n − 1 and ni=1 λi ni=1 λ∗i . ∗ (3) – is said to be weak upper majorized by –, in symbols – k i =1
λ(i)
k i =1
w –∗ , if
w –∗ , if
λ∗(i)
for k = 1, 2, . . . , n − 1 and ni=1 λi ni=1 λ∗i . The following lemma gives the relations of majorization and weak majorization. Lemma 2.1. Let – = (λ1 , λ2 , . . . , λn ), –∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors. Then – w –∗ if and only if there exists n-dimensional vector ν satisfying – m , and –∗ (i.e. νi λ∗i , i = 1, . . . , n). It is well-known that the conception of majorization (weak majorization) is link to the conception of doubly stochastic matrices(doubly substochastic matrices). We call that a n × n matrix A = (aij ) is doubly stochastic (substochastic) matrix if aij 0 and sum of all elements in any row and column equal one (is less or equal to one). Using the conception of doubly stochastic matrices, we have the following result. Lemma 2.2 [3]. Let – = (λ1 , λ2 , . . . , λn ), –∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors, then – m –∗ if there exists such a doubly stochastic matrix A such that –∗ = –A.
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
1109
Now, we state our main results. The proofs will be given in the next section.
∼ N (μ, σi2 ), cov(Xi , Xj ) = ρσi σj ; Xi∗ ∼ N (μ, σi∗ 2 ), cov(Xi∗ , Xj∗ ) = ρσi∗ σj∗
Theorem 2.1. Let Xi
< ρ < 1) , i = / j, i, j = 1, . . . , n. If
P
min Xi x P
1in
σ1 , σ2 , . . . , σn 1
1
1
min Xi∗ x ,
1in
m
σ1∗ , σ2∗ , . . . , σn∗ 1
1
1
, then for all x max Xi∗ x .
max Xi x P
P
1in
1 1 −n
∈ R,
1in
∼ N (μi , σ 2 ), cov(Xi , Xj ) = ρσ 2 ; Xi∗ ∼ N (μ∗i , σ 2 ), cov(Xi∗ , Xj∗ ) = ρσ 2 1−1 n < ρ < 1) , i = / j, i, j = 1, . . . , n, and let u = min(μ1 , μ2 , . . . , μn , μ∗1 , μ∗2 . . . , μ∗n ), v = max(μ1 , μ2 , . . . , μn , μ∗1 , μ∗2 , . . . , μ∗n ). Let f : [u, v] → R be a strictly monotone function satisfying (f (μ1 ), f (μ2 ), . . . , f (μn )) m (f (μ∗1 ), f (μ∗2 ), . . . , f (μ∗n )). Theorem 2.2. Let Xi
min Xi x P min Xi∗ x for all x 1 i n 1 i n [u, v], then P max Xi x P max Xi∗ x for all x
(1) if f (y)f (y) 0 for all y ∈ [u, v], then P
(2) if f (y)f (y) 0 for all y ∈
In Theorem 2.1, if we suppose ⎞ 2
⎛
σ1 ⎜ρσ σ ⎜ 1 2 ⎜ . ⎜ . ⎝ .
ρσ1 σn
ρσ1 σ2 σ22
··· ··· ..
ρσ2σn
ρσ1 σn ρσ2 σn .. .
.
···
1in (X1 , . . . , Xn )T
⎟ ⎟ ⎟ . We let A ⎟ ⎠
1in
∈ R; ∈ R.
∼ N (—, ), where — = (μ, . . . , μ)T , =
= (aij )n×n be a doubly stochastic matrix, and ff =
σn2
σ1 , σ2 , . . . , σn 1
1
ffA
1
=
, then ⎛ ⎞ n n 1 1 ⎝ ai1 , . . . , ain ⎠ . i =1
σi
σi
i =1
n Obviously, we have σ1 , σ1 , . . . , σ1 m ai1 σ1 , . . . , ni=1 ain σ1 . n i i 1 2 i =1 When n = 2, then σ1 , σ1 m σ1∗ , σ1∗ is equivalent to min(σ1 , σ2 ) min(σ1∗ , σ2∗ ), max(σ1 , σ2 ) 1 2 1 2 max(σ1∗ , σ2∗ ), and σ1 + σ1 = σ1∗ + σ1∗ .
1
2
So, if we let (X1∗ , . . . , Xn∗ )T ⎛ 2
1
∗ =
ρ n
i =1
ai1 σ1
i
i
n
2
∼ N (—, ∗ ), where — = (μ, . . . , μ)T ,
1 n 1 i=1 ai1 σ
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ρ n 1 1 ⎜ i=1 ai1 σi ⎜ ⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎝ 1
1 ai2 σ1
i =1
i
ρ n 1a 1 n 1a 1 i=1 i1 σi i=1 i2 σi 2 1 n 1 i=1 ai2 σ
···
ρ n
1 1 i=1 ain σ
ρ n
1 1 1 n 1 i=1 ain σi i=1 ai2 σi
i
1in
In Theorem 2.2, if we let f (x)
P
.
2
1 1 i=1 ain σ
n
i
max Xi x P
1in
.. .
···
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
1 1 n ai2 σ1 ain σ1 i = 1 i i
i =1
i
then, we have P min Xi x P min Xi∗ x , 1in
···
.. n
⎞
ρ n 1a 1 n 1a 1 i=1 i1 σi i=1 in σi
max Xi∗ x
1in
.
= x, we can easily get the following result.
Corollary 2.1. Suppose Xi and Xi∗ as in the setting of Theorem 2.2. If (μ1 , μ2 , . . . , μn ) m μ∗n ), then we have P min Xi x P min Xi∗ x , P max Xi x P max Xi∗ x . 1in
1in
1in
1in
(μ∗1 , μ2∗ , . . . ,
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L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
In Corollary 2.1, we let (X1 , . . . , Xn )T ∼ N (—, ), where — ⎛ 2 2 2 ⎞
σ
=
ρσ σ2
⎜ρσ 2 ⎜ ⎜ . ⎜ . ⎝ .
..
ρσ 2
ρσ 2
∗
—
··· ···
ρσ ρσ 2 .. .
.
···
⎟ ⎟ ⎟ , and A ⎟ ⎠
= (aij ) be a doubly stochastic matrix. Let
σ2
⎛ ⎞ n n ⎝ ai1 μi , . . . , ain μi ⎠
= —T A =
= (μ1 , . . . , μn )T ,
i =1
i =1
(μ∗1 , μ∗2 . . . , μ∗n ).
Then, we have (μ1 , μ2 , . . . , μn ) m (μ∗1 , μ∗2 , . . . , μ∗n). Suppose that (X1∗ , . . . , Xn∗ )T ∼ N (—∗T , ), where n n ∗ ∗ ∗ — = (μ1 , . . . , μn ) = i=1 ai1 μi , . . . , i=1 ain μi , then we have P min Xi x P min Xi∗ x , P max Xi x P max Xi∗ x . 1in
Moreover, let —
=
1in
1in
n −k i =1
1in
μi , μn , μn−1 , . . . , μn−k+1 , 0, . . . , 0 , k = 0, 1, . . . , n − 1, where μi 0, i = 1, . . . , n. There are n − k − 1 zero components among the n components of the vector —k . According k
to Definition 2.1, we have 0
—
m —1 m —2 · · · m —n−1 .
Let (X1k , . . . , Xnk )T ⎛ 2 2
σ
⎜ρσ 2 ⎜ ⎜ . ⎜ . ⎝ .
∼ N ((—k )T , ), where —k = ⎞ · · · ρσ 2 · · · ρσ 2 ⎟ ⎟ .. ⎟ .. ⎟ . Then, we obtain . . ⎠
ρσ σ2
ρσ 2 ρσ 2
···
n −k i =1
μi , μn , μn−1 , . . . , μn−k+1 , 0, . . . , 0 and =
σ n −1 0 x , P min Xi x P min Xi1 x · · · P min Xi 1in 1in 1in n −1 x . P max Xi0 x P max Xi1 x · · · P max Xi 1in
2
1in
1in
We can extend the results of Theorem 2.2 to weak majorization case. Corollary 2.2. Suppose Xi , Xi∗ and u, v as in the setting of Theorem 2.2. Let f : [u, v] → R be a strictly monotone function, and denote —f = (f (μ1 ), f (μ2 ), . . . , f (μn )), —f∗ = (f (μ∗1 ), f (μ∗2 ), . . . , f (μn∗ )),
(1) if f (y) > 0, f (y) 0 for all y ∈ [u, v], and —f P min Xi x P min Xi∗ x 1in
1in
for all x ∈ R; (2) if f (y) < 0, f (y) 0 for all y ∈ [u, v], and —f P min Xi x P min Xi∗ x 1in
for all x
∈ R;
w —∗f , then
1in
for all x ∈ R; (3) if f (y) > 0, f (y) 0 for all y ∈ [u, v], and —f P max Xi x P max Xi∗ x 1in
w —∗f , then
1in
w —∗f , then
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
w —∗f , then
(4) if f (y) < 0, f (y) 0 for all y ∈ [u, v], and —f P max Xi x P max Xi∗ x . 1in
for all x
1111
1in
∈ R.
From Corollary 2.2, we get two interesting results as follows. Corollary 2.3. In Corollary 2.2, there exists a real number a ∗ aμn ),
∗
> 0 satisfying (aμ1 , . . . , aμn ) w (aμ1 , . . . ,
min Xi x P min Xi∗ x for all x ∈ R; n 1 i 1 i n 1, then P max Xi x P max Xi∗ x for all x ∈ R.
(1) if a
> 1, then P
(2) if 0
1in
1in
Corollary 2.4. In Corollary 2.2, we suppose μ∗τ n ), where τ is a nonzero real number,
μi > 0, μ∗i > 0. Let —τ = (μτ1 , . . . , μτn ), —∗τ = (μ∗τ 1 ,...,
min Xi x P min Xi∗ x for all x ∈ R; n 1 i 1 i n ∗ τ ∗τ (2) if 0 < τ 1, and — w — , then P max Xi x P max Xi x for all x ∈ R; 1in 1in (3) if τ < 0, and μτ w μ∗τ , then P max Xi x P max Xi∗ x for all x ∈ R. (1) if τ 1, and —τ
w —∗τ , then P
1in
1in
Finally, we present a concrete example. Let ⎛⎛ ⎞ ⎛ ⎞⎞ ⎛ ⎛ ⎞ ⎞ 1 1 1 X1 1 X1∗ 15 9 ⎜ ⎜ ⎟ ⎟ ∗ 1 1 1 ⎝X2 ⎠ ∼ N ⎝⎝1 ⎠ , ⎝ ⎠⎠ , ⎝X2 ⎠ 15 25 45 1 1 1 1 X3 X3∗ 9
45
9
⎛⎛
∼
⎞ ⎛1 1 ⎜⎝ ⎠ ⎜ 19 N⎝ 1 , ⎝ 18 1 1
16
1 18 1 4 1 24
1 36 1 24 1 16
⎞⎞ ⎟⎟ ⎠⎠ .
Obviously, we have correlation coefficient ρ = and 1 1 1 1 1 1 , , = (1, 5, 3) m (3, 2, 4) = , , ∗ ∗ ∗ . 1 3
σ1 σ2 σ3
σ1 σ2 σ3
From Theorem 2.1, we get P min Xi x P min Xi∗ x , 1i3
1i3
P
max Xi x P max Xi∗ x ,
1i3
1i3
for all x
∈ R.
From this example, we can see that the elements of the covariance matrices do not satisfy the conditions of the well-known Slepian’s inequality. But the Slepian’s inequality still holds.
3. Preliminaries and the proofs of two theorems Before we prove our main results, we need the following well-known definition 3.1 and some lemmas. Definition 3.1. A real-valued function φ(–):Rn function if for all – m —, then φ(–) ()φ(—).
→ R is said to be a Schur-concave (Schur-convex)
Lemma 3.1 [6]. A permutation-symmetric differentiable function φ(X ) is Schur-concave (Schur-convex) if and only if
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L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
∂φ(X ) ∂φ(X ) 0 ( 0) (Xi − Xj ) − ∂ Xi ∂ Xj = / j.
for all i
The following lemma can be proved easily, so we omit its proof here. 2 x2 +∞ − t2 (1) Let g (x) = e 2 x e dt, for all x ∈ R, then g (x) is a nonincreasing function. x2 t2 x − (2) Let h(x) = e 2 −∞ e 2 dt, for all x ∈ R, then h(x) is a nondecreasing function. Now we prove the first main result as stated in Theorem 2.1.
Lemma 3.2
Proof of Theorem 2.1. Let λ1
. . . , λ∗n ).
F
min Xi (x)= P
1in
=
σ1 , . . . , λn 1
1in
=
1 ∗ σn ; λ1
=
1 ∗ σ1∗ , . . . , λn
min Xi x
1in
+∞ x
...
1 n 2
(2π ) | |
x
e− 2 (t −—) 1
1 2
−1 (t −—)
dt1 . . . dtn
ρσ1 σ2 σ22
··· ··· ..
ρσ1 σn ρσ2 σn .. .
.
⎟ ⎟ ⎟, then −1 ⎟ ⎠
=
= (μ, . . . , μ)T . 1
(1−ρ)[(n−1)ρ+1]
σn2 ρσ1 σn ρσ2 σn · · · 2 [(n − 2)ρ + 1]λ1 −ρλ1 λ2 ··· −ρλ1 λn ⎜ −ρλ1 λ2 [(n − 2)ρ + 1]λ22 · · · −ρλ2 λ1 ⎜ ·⎜ .. .. .. ⎜ ⎝ . . . −ρλ2 λn · · · [(n − 2)ρ + 1]λ2n −ρλ1 λn (n−2)ρ+1 −ρ Denote (1−ρ)[(n−1)ρ+1] a, (1−ρ)[(n−1)ρ+1] b, then for all x ∈ R, min Xi (x)
1in
∂λi
+∞
=
x
...
+∞ x
· dt1 . . . dtn − +b 1
Since
∂| −1 | 2 ∂λi
∂F
=
∂λi =
−
k= /i
∂F
1 n 2
(2π ) | |
1 2
·
+∞
⎞
x
...
+∞
1
aλi (ti
x
λk (tk − μ)(ti − μ)⎠ e− 2 (t −—)
−1 (t −—)
min Xi (x)
1
(2π) 2 | | 2
x
∂λj ...
+∞ +∞ +∞ 1 x
⎟ ⎟ ⎟ ⎟ ⎠
(Vij )n×n .
1
1in
+∞
1 n
⎞
∂| −1 | 2 − 1 (t −—) −1 (t −—) e 2 n (2π ) 2 ∂λi 1
1 − 21 , we can get λi | |
min Xi (x)
1in
1
= P (X1 x, . . . , Xn x)
+∞
⎛
∂F
σn∗ , then (λ1 , . . . , λn )
1in
is Schur-concave function with respect to (λ1 , λ2 , . . . , λn ), where — ⎛ ⎞ 2
σ1 ⎜ρσ σ ⎜ 1 2 Since = ⎜ ⎜ .. ⎝ .
=
min Xi x P min Xi∗ x . It is sufficient to prove
We first prove that P
=
x
x
λi
−
1
λj
− μ)2
dt1 . . . dtn .
m (λ∗1 ,
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
⎡ a
· exp ⎣− λ2i (ti − μ)2 − b 2
−b +b
k= / i,k= /j
k= /i
a
λk λi (tk − μ)(ti − μ) − λ2j (tj − μ)2 2
⎤
λk λj (tk − μ)(tj − μ)⎦ dti dtj − ⎡
⎞
+∞ +∞ x
⎡
2
b2 2a
⎛
⎝
k= /i
⎞2
a k= /i
a
2
k= / i,k= /j
λk (tk − μ)⎠ − λ2j (tj − μ)2 − b
· λj (tk − μ)(tj − μ) · dti dtj +
+∞ +∞ x
aλj (tj
x
⎡
⎞
⎡ a ⎢ b λk (tk − μ)(tj − μ)⎠ exp ⎣− ⎣λj (tj 2 k= /j
+
⎞2
⎛
2a
− μ)2
b λk (tk − μ)(ti − μ)⎠ · exp ⎣− ⎣λi (ti − μ) + λk
k= /i
b2
aλi (ti
x
a
· (tk − μ)]2 +
+
1113
⎝
k= /j
− μ)2
a
2
k= / i,k= /j
· λi (tk − μ)(ti − μ)] dti dtj e− 2 k=/ i,k=/ j · dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn . 1
b
− μ) +
λk (tk − μ)⎠ − λ2i (ti − μ)2 − b
k = / i,k = /j
λk
a k= /j
⎤2
λk (tk − μ)⎦
λk
Vkk (tk −μ)(tk −μ)
Let +∞ +∞ 1
I=
x
a
−
2
λi
x
λ2j (tj
x
⎡
x
a
−
2
x
·
⎛ ⎝aλi (ti
⎡
x
⎛ ⎝aλj (tj
⎡ ⎢ a⎣ exp ⎣− λj (tj 2
−
a 2
⎡
2
k= /i
− μ)2 + b
k= /i
b a k= /i
k= / i,k= /j
− μ) + b b a k= /j
k= / i,k= /j
(3.1)
⎞
λk (tk − μ)(ti − μ)⎠ ⎤2
λk · (tk − μ)⎦ +
⎛
b2
2a ⎤
⎝
k= /i
⎞2
λk (tk − μ)⎠
λk · λj (tk − μ)(tj − μ)⎦ dti dtj ,
2
⎤
λk λi (tk − μ)(ti − μ)
λk λj (tk − μ)(tj − μ)⎦ dti dtj ,
k= / i,k= /j
− μ) +
λ2i (ti − μ)2 − b
a
· exp ⎣− λ2i (ti − μ)2 − b
− μ) +
λ2j (tj − μ)2 − b
+∞ +∞
III =
2
⎡ ⎢ a⎣ exp ⎣− λi (ti 2
·
λj
− μ) − b
+∞ +∞
II =
−
1
k= /j
⎞
λk (tk − μ)(tj − μ)⎠ ⎤2
λk (tk − μ)⎦ +
b2 2a ⎤
⎛
⎝
k= /j
λk λi (tk − μ)(ti − μ)⎦ dti dtj .
⎞2
λk (tk − μ)⎠
1114
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
Now, we calculate II . ⎧ ⎛ +∞ ⎨ +∞ ⎝aλi (ti II = ⎩ x x b
+
−b
a
1 2
k= / i,k= /j
λi λj
k= /i
⎞
a
λk (tk − μ)(ti − μ)⎠ exp − [λi (ti − μ) 2
⎞2
a
− μ)⎠ − λ2j (tj − μ)2 2
⎤
1
=
⎡ ⎛ ⎤2 ⎤ ⎫ ⎪ ⎬ b2 ⎢ ⎥ · (tk − μ)⎦ ⎦ dti ⎪ exp ⎣ ⎝ λk (tk 2a k= ⎭ /i
λk
a k= /i
− μ) + b 2
λk λj (tk − μ)(tj − μ)⎦ dtj
(x − μ)
+∞ √ % a
λj (x−μ)+ ba λi (x−μ)+ ba
k= / i,k= /j
λk (tk −μ)
&
e−
u2 2
du
⎡
⎞2 ⎛ 2 b2 − a2 2 b2 − ab ⎢b ⎝ exp ⎣ λk (tk − μ)⎠ + λi (x − μ)2 + λi (x − μ) 2a k= 2a a / i,k= /j ⎡ ⎤ +∞ +∞ 1 a λk (tk − μ)⎦ + exp ⎣− λ2i (ti − μ)2 − b λk λi (tk − μ) λi x 2 x k= / i,k= /j k= /i
·
·
a
· (ti − μ) −
2
λ2j (tj
− μ) − b
2
k= / i,k= /j
⎤
λk λj (tk − μ)(tj − μ)⎦ dti dtj .
(3.2)
Similarly ⎧ ⎛ +∞ ⎨ +∞ ⎝aλj (tj ⎩ x x
III =
+
b a k= /j
−b
k= / i,k= /j
1
=
a
1 2
λk
λi λj
− μ)2 + b
k= /j
⎞
λk (tk − μ)(tj − μ)⎠ exp −
⎡ ⎤2 ⎤ ⎫ ⎛ ⎪ ⎬ b2 ⎢ ⎥ · (tk − μ)⎦ ⎦ dtj exp ⎣ ⎝ λk (tk ⎪ 2a k= ⎭ /j
⎞2
a' 2
λj (tj − μ)
a
− μ)⎠ − λ2i (ti − μ)2 2
⎤
λk λj (tk − μ)(tj − μ)⎦ dtj
(x − μ)
+∞ √ % a
λi (x−μ)+ ba λj (x−μ)+ ba
k= / i,k= /j
λk (tk −μ)
&
e−
u2 2
du
⎡
·
·
⎞2 ⎛ 2 b2 − a 2 2 b2 − ab ⎢b ⎝ exp ⎣ λk (tk − μ)⎠ + λj (x − μ)2 + λj (x − μ) 2a k= 2a a / i,k= /j ⎡ ⎤ +∞ +∞ 1 a ⎦ λk (tk − μ) + exp ⎣− λ2j (tj − μ)2 − b λk λj (tk − μ) λj x 2 x k= / i,k= /j k= /j
· (tj − μ) −
a 2
λ2i (ti
− μ) − b 2
k= / i,k= /j
⎤
λk λi (tk − μ)(ti − μ)⎦ dti dtj .
(3.3)
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
1115
From (3.1)–(3.3), we can obtain I − II + III 1
=
a
1 2
(x − μ)
λi λj
+∞ √ % a
λi (x−μ)+ ba λj (x−μ)+ ba
k= / i,k= /j
λk (tk −μ)
&
u2 2
e−
du
⎡
·
·
⎛ ⎞2 2 b 2 − a2 2 b2 − ab ⎢b ⎝ exp ⎣ λk · (tk − μ)⎠ + λj (x − μ)2 + λj (x − μ) 2a k= 2a a / i,k= /j ⎤ +∞ 2 −u & e 2 du λk (tk − μ)⎦ − (x − μ) √ % b b a
k= / i,k= /j
⎡
⎞2
⎛ 2 ⎢b ⎝ · exp ⎣ λk · (tk 2a k= / i,k= /j ⎤⎫ ⎬ λk (tk − μ)⎦⎭ · k= / i,k= /j
1
=
1
a 2 λi λj
·
⎧ ⎪ ⎨
λj (x−μ)+ a λi (x−μ)+ a
− μ)⎠ +
2a
√
a
b2
− ab a
λi (x − μ)
⎡ b b ⎢a ⎣ [( x − μ) exp ⎣ λ λk (tk i (x − μ) + λj (x − μ) + ⎪ 2 a a k= ⎩ / i,k= /j
λi (x−μ)+ ba λj (x−μ)+ ba
· (x − μ) +
b
k= / i,k= / j λk (tk −μ)
a k= / i,k= /j
a
a
· exp ⎣− λ2j (x − μ)2 − b 2
k= / i,k= /j
e−
&
u2 2
du − (x − μ) exp
⎤2 ⎤ ⎥ +∞ λk (tk − μ)⎦ ⎦ √ %
⎡
−b
λ2i (x − μ)2 +
λk (tk −μ)
⎡
+∞ %
− a2
b2
k= / i,k= /j
k= / i,k= /j
λj (x−μ)+ ba λi (x−μ)+ ba
a
2
⎤2 ⎤ ⎥ − μ)⎦ ⎦
a
k= / i,k= /j
λk (tk −μ)
a
λk λj (tk − μ)(x − μ) − λ2i (x − μ)2 2
⎤
λk λi (tk − μ)(x − μ) − bλj λi (x − μ)
2⎦
.
By Lemma 3.2(1), we get
(λi − λj )(I − II + III) 0. That is ⎛
(λi − λj ) ⎜ ⎝
∂F
min Xi (x)
1in
∂λi
−
∂F
1
1in
∂λj
· (λi − λj )(I − II + III)e− 2 Therefore P
min Xi (x)
k= / i,k= /j
1in
⎟ ⎠
=
/j k = / i,k =
min Xi x P min Xi∗ x .
1in
⎞ 1 n
1
(2π) 2 | | 2
·
Vkk (tk −μ)(tk −μ)
+∞ x
...
b
λj (x − μ) + λi
+∞ x
dt1 . . . dtn 0.
&
e
2 − u2
⎫ ⎪ ⎬ du
⎪ ⎭
1116
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
Similarly, we can show that F max Xi (x)= P max Xi
> x = 1 − P (X1 x, . . . , Xn x)
1in
1in
=1 −
x
...
−∞
x
1
−∞ (2π )
n 2
1
| |
e− 2 (t −—)
1 2
−1 (t −—)
dt1 . . . dtn
(λ1 , λ2 , . . . , λn ) for all x ∈ R. So P
is Schur-convex function with respect to P max Xi∗ x . 1in
max Xi x
1in
We have completed the proof of Theorem 2.1.
Proof of Theorem 2.2. We first let λ1 = f (μ1 ), . . . , λn = f (μn ), λ∗1 = f (μ∗1 ), . . . , λ∗n = f (μ∗n ), then μ1 = f −1 (λ1 ), . . . , μn = f −1 (λn ), μ∗1 = f −1 (λ∗1 ), . . . , μ∗n = f −1 (λn∗ ), so we get (λ1 , . . . , λn ) m (λ∗1 , . . . , λ∗n ). (1) Now we prove that P min Xi x P min Xi∗ x by proving that 1in 1in F min Xi (x) = P min Xi x = P (X1 x, . . . , Xn x) 1in
1in
+∞
=
x
...
+∞
1
(2π ) | |
1 2
· e− 2 (t −—) 1
n 2
x
−1 (t −—)
dt1 . . . dtn
is Schur-concave function with respect to (λ1 , . . . , λn ), where — = (μ, . . . , μ)T . ⎛ ρ ··· ρ ⎞ 1 ⎜ρ 1 ··· ρ ⎟ ⎜ ⎟ Since = σ 2 ⎜ . , then −1 = (1−ρ)σ 2 [(1n−1)ρ+1] .. ⎟ .. ⎝ .. . . ⎠ ρ ρ ··· 1 ⎛ ⎞ (n − 2)ρ + 1 −ρ ··· −ρ ⎜ ⎟ −ρ −ρ (n − 2)ρ + 1 · · · ⎜ ⎟ ·⎜ ⎟ (Vij )n×n . . . . . . . ⎝ ⎠ . . . −ρ −ρ · · · (n − 2)ρ + 1 Denote Vii
=
(n−2)ρ+1 (1−ρ)[(n−1)ρ+1]σ 2
Therefore, for all x
∂F
∈ R,
d, i = / j, i, j = 1, . . . , n.
min Xi (x)
+∞
1 n
1
(2π) 2 | | 2 f (μi )
· e− 2 (t −—) 1
=
−ρ (1−ρ)[(n−1)ρ+1]σ 2
1in
∂λi =
c, Vij =
− 1 ( t − —)
1 n
1
(2π) 2 | | 2 f (μi ) ·e
x
1 n 2
1
x
⎛ ⎝Vii (ti
− μi ) +
dt1 . . . dtn ⎡ ⎛ +∞ +∞ +∞ ⎣ ⎝Vii (ti ...
− 12 (t −—) −1 (t −—)
=−
...
+∞
x
(
x
x
k= /i
⎞ Vik (tk
− μi ) +
− μk )⎠
k= /i
⎞ Vik (tk
− μk )⎠
dti dt1 . . . dti−1 dti+1 . . . dtn ( +∞ +∞ +∞ 1 −1 ... d(e− 2 (t −—) (t −—) ) dt1 . . . dti−1
(2π) | | 2 f (μi ) · dti+1 . . . dtn
x
x
x
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
+∞
1
=
n 2
1 2
...
+∞
⎡ exp ⎣−
1
[c (x − μi )2 + 2d(x − μi )
2 x (2π) | | f (μi ) x 2 (tk − μk )(tj − μj ) + + c (tj − μj ) + 2d k= / i,k= /j
· (tk − μk )]] dt1 . . . dti−1 dti+1 . . . dtn +∞
1
=
...
1117
+∞ ) +∞
k = / i,k = /j
1
Vkk (tk
k= /i
(tk − μk )
− μk )
− c (tj − μj ) +
d
((x − μi ) c ⎫ ⎤ ⎡ ⎞⎤2 ⎛ ⎞2 ⎪ 2 ⎬ ⎥ ⎢ d ⎝ ⎠ ⎦ (tk − μk ) (x − μi ) + (tk − μk )⎠ + ⎦ dtj exp ⎣− ⎪ c ⎭ k= / i,k= /j k= / i,k= /j 2 + 2d (tk − μk )(x − μi ) + c (x − μi ) + Vkk (tk − μk ) n 2
(2π) | |
1 2
f (μ
i)
x
x
exp
2
x
k= / i,k= / j k = / i,k = /j
k= / i,k= /j
· (tk − μk )] dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn 1
=
n 2
(2π) | | d
1 2
+∞
√
cf (μi )
x
⎤ 2⎦
...
+∞ )
exp
x
d
c 2
[(x − μj ) + (x − μi ) c
+∞
⎫ ⎬
2
− u2
(tk − μk )] e du √ ⎭ c k= c [(x−μj )+ dc (x−μi )+ dc k= / i,k= / j (tk −μk )] / i,k= /j ⎡ ⎤ c c 2 2 · exp ⎣− (x − μi ) − (x − μj ) − d (tk − μk )[(x − μi ) + (x − μj )⎦ 2 2 k= / i,k= /j ⎤ − d(x − μi )(x − μj ) + Vkk (tk − μk )(tk − μk )⎦ +
k= / i,k= / j k = / i,k = /j
· dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn .
(3.4)
So from (3.4), we have
∂F
min Xi (x)
1in
∂λi =
∂F
−
min Xi (x)
1in
∂λj
+∞
1
1√
n 2
cf (μi )f (μj )
(2π) | | 2 d
+ (x − μi ) + c
⎡
−
f (μi ) exp ⎣
c
d c
·
k= / i,k= /j
x
...
+∞ ) x
⎤ 2⎦
(tk − μk )] d
f (μj ) exp
c 2
[(x − μj )
+∞ √
d
k= / i,k= / j (tk −μk )]
e−
u2 2
du
⎤ 2⎦
(tk − μk )] c k= / i,k= /j * +∞ 2 c c − u2 · √ e du · exp − (x − μj )2 − (x − μi )2 d d 2 2 c [(x−μi )+ c (x−μj )+ c k= / i,k= / j (tk −μk )] −d (tk − μk )[(x − μi ) + (x − μj )] − d(x − μi )(x − μj ) + k= / i,k= /j
k = / i,k = /j
Vkk (tk
2
[(x − μi ) + (x − μj ) +
c [(x−μj )+ dc (x−μi )+ dc
c
⎤
k= / i,k= /j
− μk )(tk − μk )⎦ dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn .
(3.5)
1118
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
Thus, if f (y)f
(y) 0 for all y ∈ [u, v], we can obtain the following result from Lemma 3.2(1), ⎧ ⎨
⎡
d
d
2
c
c k= / i,k= /j
(λi − λj ) ⎩f (μj ) exp ⎣ [(x − μj ) + (x − μi ) + ·
+∞
+
√
c [(x−μj )+ dc (x−μi )+ dc
d
c k= / i,k= /j
c
⎤
(tk − μk )]2 ⎦
k= / i,k= / j (tk −μk )]
e−
u2 2
⎤
2⎦
(tk − μk )]
du − f (μi ) exp
c 2
+∞ √
c [(x−μi )+ dc (x−μj )+ dc
k= / i,k= / j (tk −μk )]
d
[(x − μi ) + (x − μj )
e−
u2 2
⎫ ⎬
c
du 0. ⎭
(3.6)
Moreover, from (3.5) and (3.6), we get ⎛ ⎞ ∂ F min Xi (x) ∂ F min Xi (x) ⎜ ⎟ (λi − λj ) ⎝ 1 i n − 1 i n ⎠ 0.
∂λi
Therefore P
∂λj
min Xi x P min Xi∗ x .
1in
1in
(2) Similarly, if f (y)f (y) 0 for all y ∈ [u, v], we can show that F max Xi (x) = P max Xi > x = 1 − P (X1 x, . . . , Xn x) 1in
1in
=1−
x
−∞
...
x
−∞ (2π )
1 n 2
| |
e− 2 (t −—) 1
1 2
− 1 ( t − —)
dt1 . . . dtn
is Schur-convex function with respect to (λ1 , . . . , λn ) for all x ∈ R through Lemma 3.2(2). ∗ So P max Xi x P max Xi x . Finally, We have completed the proof of Theorem 2.2. 1in
1in
Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 10671037), the University Natural Science Research Project of Anhui Province of China (No. KJ2010B347), and the Grant for Youth of Anhui Normal University. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R.J. Adler, J.E. Taylor, Random Fields and Geometry, Springer, New York, 2007. T. Ando, Majorization and inequalities in matrix theory, Linear Algebra Appl. 199 (1994) 17–67. R. Bhatia, Matrix Analysis, Springer, New York, 1997. G.H. Hardy, J.E. Littlewood, G. Pólya, Some simple inequalities satisfied by convex functions, Messenger Math. 58 (1929) 145–152. M. Ledoux, M. Talagrand, Probability in Banach Spaces, Springer, 1991. A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. A. Maurer, Transfer bounds for linear feature learning, Mach. Learn. 75 (3) (2009) 327–350. D. Slepian, The one-sided barrier problem for Gaussian processes, Bell System Tech. J. 41 (1962) 463–501. I. Schur, Über eine klasse von mittelbildungen mit anwendungen auf die determinantentheorie,Sitzungsber. Berliner Math. Ges. 22 (1923) 9–20.
Contents lists available at ScienceDirect
Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa
Slepian’s inequality with respect to majorization Longxiang Fang a,b , Xinsheng Zhang a,∗ a b
Department of Statistics, Fudan University, Shanghai 200433, PR China Department of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, PR China
A R T I C L E
I N F O
A B S T R A C T
In this paper, we obtain some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 24 May 2010 Accepted 23 October 2010 Available online 18 November 2010 Submitted by H. Schneider AMS classification: 60E15 62G30 Keywords: Slepian’s inequality Gaussian random vectors Majorization Schur-convex
1. Introduction Let X = (X1 , . . . , Xn ) and X ∗ = (X1∗ , . . . , Xn∗ ) be two centered Gaussian random vectors with covariance matrices = (σij ) and ∗ = (σij∗ ), respectively. The well-known Slepian’s inequality [8] states that if σii = σii∗ and σij σij∗ for every i, j = 1, . . . , n, then for any x ∈ R, P min Xi x P min Xi∗ x , P max Xi x P max Xi∗ x . 1in
1in
1in
1in
Slepian’s inequality and its variations provide a very useful tool in the theory of Gaussian random variables and Gaussian processes. See, for examples, [1,5,7]. Majorization is a very interesting topic in various fields of mathematics and statistics, which is a partial ordering on vectors by sorting all components in nonincreasing order . If – and –∗ are two vectors ∗ Corresponding author. Tel.: +86 21 2501 1226; fax: +86 21 6564 2351. E-mail address: [email protected] (X. Zhang). 0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.10.019
1108
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
m ∗ with all components in nonincreasing order such that m i=1 λi i=1 λi for m = 1, 2, . . . , n − 1 with ∗ equality at m = n, then – is said to be majorized by – and denote – m –∗ . The history of investigating on majorization should date back to Schur [9] and Hardy et al. [4]. In the reference of Marshall and Olkin [6], the reader can find that majorization has been connected with combinatorics, analytic inequalities, numerical analysis, matrix theory, probability and statistics. Recent research on majorization with respect to matrix inequalities and norm inequalities has been considered by Ando [2]. In this paper, we search some new conditions via majorization to obtain Slepian’s inequality. We consider two cases as follows. One case is two Gaussian random vectors have equal mean, sharing the same correlation coefficient for all components and the vector being constituted by the reciprocal of each component’s standard deviation satisfying majorization; the other case is two Gaussian random vectors with equal covariance matrix and the mean vector satisfying majorization. Our results cannot be derived from the well-known Slepian’s inequality. The explanation will be given in the later. 2. Main results and applications In order to state our main results, we first give the definitions of majorization and weak majorization. Definition 2.1. Let – = (λ1 , λ2 , . . . , λn ), –∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors. Let λ[1] λ[2] · · · λ[n] , λ∗[1] λ∗[2] · · · λ∗[n] , λ(1) λ(2) · · · λ(n) , λ∗(1) λ∗(2) · · · λ∗(n) be their ordered components. (1) –∗ is said to be majorized by –, in symbols – k i =1
λ[i]
k i =1
m –∗ , if
λ∗[i]
for k = 1, 2, . . . , n − 1 and ni=1 λi = ni=1 λ∗i . (2) –∗ is said to be weak lower majorized by –, in symbols – k i =1
λ[i]
k i =1
λ∗[i]
for k = 1, 2, . . . , n − 1 and ni=1 λi ni=1 λ∗i . ∗ (3) – is said to be weak upper majorized by –, in symbols – k i =1
λ(i)
k i =1
w –∗ , if
w –∗ , if
λ∗(i)
for k = 1, 2, . . . , n − 1 and ni=1 λi ni=1 λ∗i . The following lemma gives the relations of majorization and weak majorization. Lemma 2.1. Let – = (λ1 , λ2 , . . . , λn ), –∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors. Then – w –∗ if and only if there exists n-dimensional vector ν satisfying – m , and –∗ (i.e. νi λ∗i , i = 1, . . . , n). It is well-known that the conception of majorization (weak majorization) is link to the conception of doubly stochastic matrices(doubly substochastic matrices). We call that a n × n matrix A = (aij ) is doubly stochastic (substochastic) matrix if aij 0 and sum of all elements in any row and column equal one (is less or equal to one). Using the conception of doubly stochastic matrices, we have the following result. Lemma 2.2 [3]. Let – = (λ1 , λ2 , . . . , λn ), –∗ = (λ∗1 , λ∗2 , . . . , λ∗n ) denote two n-dimensional real vectors, then – m –∗ if there exists such a doubly stochastic matrix A such that –∗ = –A.
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
1109
Now, we state our main results. The proofs will be given in the next section.
∼ N (μ, σi2 ), cov(Xi , Xj ) = ρσi σj ; Xi∗ ∼ N (μ, σi∗ 2 ), cov(Xi∗ , Xj∗ ) = ρσi∗ σj∗
Theorem 2.1. Let Xi
< ρ < 1) , i = / j, i, j = 1, . . . , n. If
P
min Xi x P
1in
σ1 , σ2 , . . . , σn 1
1
1
min Xi∗ x ,
1in
m
σ1∗ , σ2∗ , . . . , σn∗ 1
1
1
, then for all x max Xi∗ x .
max Xi x P
P
1in
1 1 −n
∈ R,
1in
∼ N (μi , σ 2 ), cov(Xi , Xj ) = ρσ 2 ; Xi∗ ∼ N (μ∗i , σ 2 ), cov(Xi∗ , Xj∗ ) = ρσ 2 1−1 n < ρ < 1) , i = / j, i, j = 1, . . . , n, and let u = min(μ1 , μ2 , . . . , μn , μ∗1 , μ∗2 . . . , μ∗n ), v = max(μ1 , μ2 , . . . , μn , μ∗1 , μ∗2 , . . . , μ∗n ). Let f : [u, v] → R be a strictly monotone function satisfying (f (μ1 ), f (μ2 ), . . . , f (μn )) m (f (μ∗1 ), f (μ∗2 ), . . . , f (μ∗n )). Theorem 2.2. Let Xi
min Xi x P min Xi∗ x for all x 1 i n 1 i n [u, v], then P max Xi x P max Xi∗ x for all x
(1) if f (y)f (y) 0 for all y ∈ [u, v], then P
(2) if f (y)f (y) 0 for all y ∈
In Theorem 2.1, if we suppose ⎞ 2
⎛
σ1 ⎜ρσ σ ⎜ 1 2 ⎜ . ⎜ . ⎝ .
ρσ1 σn
ρσ1 σ2 σ22
··· ··· ..
ρσ2σn
ρσ1 σn ρσ2 σn .. .
.
···
1in (X1 , . . . , Xn )T
⎟ ⎟ ⎟ . We let A ⎟ ⎠
1in
∈ R; ∈ R.
∼ N (—, ), where — = (μ, . . . , μ)T , =
= (aij )n×n be a doubly stochastic matrix, and ff =
σn2
σ1 , σ2 , . . . , σn 1
1
ffA
1
=
, then ⎛ ⎞ n n 1 1 ⎝ ai1 , . . . , ain ⎠ . i =1
σi
σi
i =1
n Obviously, we have σ1 , σ1 , . . . , σ1 m ai1 σ1 , . . . , ni=1 ain σ1 . n i i 1 2 i =1 When n = 2, then σ1 , σ1 m σ1∗ , σ1∗ is equivalent to min(σ1 , σ2 ) min(σ1∗ , σ2∗ ), max(σ1 , σ2 ) 1 2 1 2 max(σ1∗ , σ2∗ ), and σ1 + σ1 = σ1∗ + σ1∗ .
1
2
So, if we let (X1∗ , . . . , Xn∗ )T ⎛ 2
1
∗ =
ρ n
i =1
ai1 σ1
i
i
n
2
∼ N (—, ∗ ), where — = (μ, . . . , μ)T ,
1 n 1 i=1 ai1 σ
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ρ n 1 1 ⎜ i=1 ai1 σi ⎜ ⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎝ 1
1 ai2 σ1
i =1
i
ρ n 1a 1 n 1a 1 i=1 i1 σi i=1 i2 σi 2 1 n 1 i=1 ai2 σ
···
ρ n
1 1 i=1 ain σ
ρ n
1 1 1 n 1 i=1 ain σi i=1 ai2 σi
i
1in
In Theorem 2.2, if we let f (x)
P
.
2
1 1 i=1 ain σ
n
i
max Xi x P
1in
.. .
···
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
1 1 n ai2 σ1 ain σ1 i = 1 i i
i =1
i
then, we have P min Xi x P min Xi∗ x , 1in
···
.. n
⎞
ρ n 1a 1 n 1a 1 i=1 i1 σi i=1 in σi
max Xi∗ x
1in
.
= x, we can easily get the following result.
Corollary 2.1. Suppose Xi and Xi∗ as in the setting of Theorem 2.2. If (μ1 , μ2 , . . . , μn ) m μ∗n ), then we have P min Xi x P min Xi∗ x , P max Xi x P max Xi∗ x . 1in
1in
1in
1in
(μ∗1 , μ2∗ , . . . ,
1110
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
In Corollary 2.1, we let (X1 , . . . , Xn )T ∼ N (—, ), where — ⎛ 2 2 2 ⎞
σ
=
ρσ σ2
⎜ρσ 2 ⎜ ⎜ . ⎜ . ⎝ .
..
ρσ 2
ρσ 2
∗
—
··· ···
ρσ ρσ 2 .. .
.
···
⎟ ⎟ ⎟ , and A ⎟ ⎠
= (aij ) be a doubly stochastic matrix. Let
σ2
⎛ ⎞ n n ⎝ ai1 μi , . . . , ain μi ⎠
= —T A =
= (μ1 , . . . , μn )T ,
i =1
i =1
(μ∗1 , μ∗2 . . . , μ∗n ).
Then, we have (μ1 , μ2 , . . . , μn ) m (μ∗1 , μ∗2 , . . . , μ∗n). Suppose that (X1∗ , . . . , Xn∗ )T ∼ N (—∗T , ), where n n ∗ ∗ ∗ — = (μ1 , . . . , μn ) = i=1 ai1 μi , . . . , i=1 ain μi , then we have P min Xi x P min Xi∗ x , P max Xi x P max Xi∗ x . 1in
Moreover, let —
=
1in
1in
n −k i =1
1in
μi , μn , μn−1 , . . . , μn−k+1 , 0, . . . , 0 , k = 0, 1, . . . , n − 1, where μi 0, i = 1, . . . , n. There are n − k − 1 zero components among the n components of the vector —k . According k
to Definition 2.1, we have 0
—
m —1 m —2 · · · m —n−1 .
Let (X1k , . . . , Xnk )T ⎛ 2 2
σ
⎜ρσ 2 ⎜ ⎜ . ⎜ . ⎝ .
∼ N ((—k )T , ), where —k = ⎞ · · · ρσ 2 · · · ρσ 2 ⎟ ⎟ .. ⎟ .. ⎟ . Then, we obtain . . ⎠
ρσ σ2
ρσ 2 ρσ 2
···
n −k i =1
μi , μn , μn−1 , . . . , μn−k+1 , 0, . . . , 0 and =
σ n −1 0 x , P min Xi x P min Xi1 x · · · P min Xi 1in 1in 1in n −1 x . P max Xi0 x P max Xi1 x · · · P max Xi 1in
2
1in
1in
We can extend the results of Theorem 2.2 to weak majorization case. Corollary 2.2. Suppose Xi , Xi∗ and u, v as in the setting of Theorem 2.2. Let f : [u, v] → R be a strictly monotone function, and denote —f = (f (μ1 ), f (μ2 ), . . . , f (μn )), —f∗ = (f (μ∗1 ), f (μ∗2 ), . . . , f (μn∗ )),
(1) if f (y) > 0, f (y) 0 for all y ∈ [u, v], and —f P min Xi x P min Xi∗ x 1in
1in
for all x ∈ R; (2) if f (y) < 0, f (y) 0 for all y ∈ [u, v], and —f P min Xi x P min Xi∗ x 1in
for all x
∈ R;
w —∗f , then
1in
for all x ∈ R; (3) if f (y) > 0, f (y) 0 for all y ∈ [u, v], and —f P max Xi x P max Xi∗ x 1in
w —∗f , then
1in
w —∗f , then
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
w —∗f , then
(4) if f (y) < 0, f (y) 0 for all y ∈ [u, v], and —f P max Xi x P max Xi∗ x . 1in
for all x
1111
1in
∈ R.
From Corollary 2.2, we get two interesting results as follows. Corollary 2.3. In Corollary 2.2, there exists a real number a ∗ aμn ),
∗
> 0 satisfying (aμ1 , . . . , aμn ) w (aμ1 , . . . ,
min Xi x P min Xi∗ x for all x ∈ R; n 1 i 1 i n 1, then P max Xi x P max Xi∗ x for all x ∈ R.
(1) if a
> 1, then P
(2) if 0
1in
1in
Corollary 2.4. In Corollary 2.2, we suppose μ∗τ n ), where τ is a nonzero real number,
μi > 0, μ∗i > 0. Let —τ = (μτ1 , . . . , μτn ), —∗τ = (μ∗τ 1 ,...,
min Xi x P min Xi∗ x for all x ∈ R; n 1 i 1 i n ∗ τ ∗τ (2) if 0 < τ 1, and — w — , then P max Xi x P max Xi x for all x ∈ R; 1in 1in (3) if τ < 0, and μτ w μ∗τ , then P max Xi x P max Xi∗ x for all x ∈ R. (1) if τ 1, and —τ
w —∗τ , then P
1in
1in
Finally, we present a concrete example. Let ⎛⎛ ⎞ ⎛ ⎞⎞ ⎛ ⎛ ⎞ ⎞ 1 1 1 X1 1 X1∗ 15 9 ⎜ ⎜ ⎟ ⎟ ∗ 1 1 1 ⎝X2 ⎠ ∼ N ⎝⎝1 ⎠ , ⎝ ⎠⎠ , ⎝X2 ⎠ 15 25 45 1 1 1 1 X3 X3∗ 9
45
9
⎛⎛
∼
⎞ ⎛1 1 ⎜⎝ ⎠ ⎜ 19 N⎝ 1 , ⎝ 18 1 1
16
1 18 1 4 1 24
1 36 1 24 1 16
⎞⎞ ⎟⎟ ⎠⎠ .
Obviously, we have correlation coefficient ρ = and 1 1 1 1 1 1 , , = (1, 5, 3) m (3, 2, 4) = , , ∗ ∗ ∗ . 1 3
σ1 σ2 σ3
σ1 σ2 σ3
From Theorem 2.1, we get P min Xi x P min Xi∗ x , 1i3
1i3
P
max Xi x P max Xi∗ x ,
1i3
1i3
for all x
∈ R.
From this example, we can see that the elements of the covariance matrices do not satisfy the conditions of the well-known Slepian’s inequality. But the Slepian’s inequality still holds.
3. Preliminaries and the proofs of two theorems Before we prove our main results, we need the following well-known definition 3.1 and some lemmas. Definition 3.1. A real-valued function φ(–):Rn function if for all – m —, then φ(–) ()φ(—).
→ R is said to be a Schur-concave (Schur-convex)
Lemma 3.1 [6]. A permutation-symmetric differentiable function φ(X ) is Schur-concave (Schur-convex) if and only if
1112
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
∂φ(X ) ∂φ(X ) 0 ( 0) (Xi − Xj ) − ∂ Xi ∂ Xj = / j.
for all i
The following lemma can be proved easily, so we omit its proof here. 2 x2 +∞ − t2 (1) Let g (x) = e 2 x e dt, for all x ∈ R, then g (x) is a nonincreasing function. x2 t2 x − (2) Let h(x) = e 2 −∞ e 2 dt, for all x ∈ R, then h(x) is a nondecreasing function. Now we prove the first main result as stated in Theorem 2.1.
Lemma 3.2
Proof of Theorem 2.1. Let λ1
. . . , λ∗n ).
F
min Xi (x)= P
1in
=
σ1 , . . . , λn 1
1in
=
1 ∗ σn ; λ1
=
1 ∗ σ1∗ , . . . , λn
min Xi x
1in
+∞ x
...
1 n 2
(2π ) | |
x
e− 2 (t −—) 1
1 2
−1 (t −—)
dt1 . . . dtn
ρσ1 σ2 σ22
··· ··· ..
ρσ1 σn ρσ2 σn .. .
.
⎟ ⎟ ⎟, then −1 ⎟ ⎠
=
= (μ, . . . , μ)T . 1
(1−ρ)[(n−1)ρ+1]
σn2 ρσ1 σn ρσ2 σn · · · 2 [(n − 2)ρ + 1]λ1 −ρλ1 λ2 ··· −ρλ1 λn ⎜ −ρλ1 λ2 [(n − 2)ρ + 1]λ22 · · · −ρλ2 λ1 ⎜ ·⎜ .. .. .. ⎜ ⎝ . . . −ρλ2 λn · · · [(n − 2)ρ + 1]λ2n −ρλ1 λn (n−2)ρ+1 −ρ Denote (1−ρ)[(n−1)ρ+1] a, (1−ρ)[(n−1)ρ+1] b, then for all x ∈ R, min Xi (x)
1in
∂λi
+∞
=
x
...
+∞ x
· dt1 . . . dtn − +b 1
Since
∂| −1 | 2 ∂λi
∂F
=
∂λi =
−
k= /i
∂F
1 n 2
(2π ) | |
1 2
·
+∞
⎞
x
...
+∞
1
aλi (ti
x
λk (tk − μ)(ti − μ)⎠ e− 2 (t −—)
−1 (t −—)
min Xi (x)
1
(2π) 2 | | 2
x
∂λj ...
+∞ +∞ +∞ 1 x
⎟ ⎟ ⎟ ⎟ ⎠
(Vij )n×n .
1
1in
+∞
1 n
⎞
∂| −1 | 2 − 1 (t −—) −1 (t −—) e 2 n (2π ) 2 ∂λi 1
1 − 21 , we can get λi | |
min Xi (x)
1in
1
= P (X1 x, . . . , Xn x)
+∞
⎛
∂F
σn∗ , then (λ1 , . . . , λn )
1in
is Schur-concave function with respect to (λ1 , λ2 , . . . , λn ), where — ⎛ ⎞ 2
σ1 ⎜ρσ σ ⎜ 1 2 Since = ⎜ ⎜ .. ⎝ .
=
min Xi x P min Xi∗ x . It is sufficient to prove
We first prove that P
=
x
x
λi
−
1
λj
− μ)2
dt1 . . . dtn .
m (λ∗1 ,
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
⎡ a
· exp ⎣− λ2i (ti − μ)2 − b 2
−b +b
k= / i,k= /j
k= /i
a
λk λi (tk − μ)(ti − μ) − λ2j (tj − μ)2 2
⎤
λk λj (tk − μ)(tj − μ)⎦ dti dtj − ⎡
⎞
+∞ +∞ x
⎡
2
b2 2a
⎛
⎝
k= /i
⎞2
a k= /i
a
2
k= / i,k= /j
λk (tk − μ)⎠ − λ2j (tj − μ)2 − b
· λj (tk − μ)(tj − μ) · dti dtj +
+∞ +∞ x
aλj (tj
x
⎡
⎞
⎡ a ⎢ b λk (tk − μ)(tj − μ)⎠ exp ⎣− ⎣λj (tj 2 k= /j
+
⎞2
⎛
2a
− μ)2
b λk (tk − μ)(ti − μ)⎠ · exp ⎣− ⎣λi (ti − μ) + λk
k= /i
b2
aλi (ti
x
a
· (tk − μ)]2 +
+
1113
⎝
k= /j
− μ)2
a
2
k= / i,k= /j
· λi (tk − μ)(ti − μ)] dti dtj e− 2 k=/ i,k=/ j · dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn . 1
b
− μ) +
λk (tk − μ)⎠ − λ2i (ti − μ)2 − b
k = / i,k = /j
λk
a k= /j
⎤2
λk (tk − μ)⎦
λk
Vkk (tk −μ)(tk −μ)
Let +∞ +∞ 1
I=
x
a
−
2
λi
x
λ2j (tj
x
⎡
x
a
−
2
x
·
⎛ ⎝aλi (ti
⎡
x
⎛ ⎝aλj (tj
⎡ ⎢ a⎣ exp ⎣− λj (tj 2
−
a 2
⎡
2
k= /i
− μ)2 + b
k= /i
b a k= /i
k= / i,k= /j
− μ) + b b a k= /j
k= / i,k= /j
(3.1)
⎞
λk (tk − μ)(ti − μ)⎠ ⎤2
λk · (tk − μ)⎦ +
⎛
b2
2a ⎤
⎝
k= /i
⎞2
λk (tk − μ)⎠
λk · λj (tk − μ)(tj − μ)⎦ dti dtj ,
2
⎤
λk λi (tk − μ)(ti − μ)
λk λj (tk − μ)(tj − μ)⎦ dti dtj ,
k= / i,k= /j
− μ) +
λ2i (ti − μ)2 − b
a
· exp ⎣− λ2i (ti − μ)2 − b
− μ) +
λ2j (tj − μ)2 − b
+∞ +∞
III =
2
⎡ ⎢ a⎣ exp ⎣− λi (ti 2
·
λj
− μ) − b
+∞ +∞
II =
−
1
k= /j
⎞
λk (tk − μ)(tj − μ)⎠ ⎤2
λk (tk − μ)⎦ +
b2 2a ⎤
⎛
⎝
k= /j
λk λi (tk − μ)(ti − μ)⎦ dti dtj .
⎞2
λk (tk − μ)⎠
1114
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
Now, we calculate II . ⎧ ⎛ +∞ ⎨ +∞ ⎝aλi (ti II = ⎩ x x b
+
−b
a
1 2
k= / i,k= /j
λi λj
k= /i
⎞
a
λk (tk − μ)(ti − μ)⎠ exp − [λi (ti − μ) 2
⎞2
a
− μ)⎠ − λ2j (tj − μ)2 2
⎤
1
=
⎡ ⎛ ⎤2 ⎤ ⎫ ⎪ ⎬ b2 ⎢ ⎥ · (tk − μ)⎦ ⎦ dti ⎪ exp ⎣ ⎝ λk (tk 2a k= ⎭ /i
λk
a k= /i
− μ) + b 2
λk λj (tk − μ)(tj − μ)⎦ dtj
(x − μ)
+∞ √ % a
λj (x−μ)+ ba λi (x−μ)+ ba
k= / i,k= /j
λk (tk −μ)
&
e−
u2 2
du
⎡
⎞2 ⎛ 2 b2 − a2 2 b2 − ab ⎢b ⎝ exp ⎣ λk (tk − μ)⎠ + λi (x − μ)2 + λi (x − μ) 2a k= 2a a / i,k= /j ⎡ ⎤ +∞ +∞ 1 a λk (tk − μ)⎦ + exp ⎣− λ2i (ti − μ)2 − b λk λi (tk − μ) λi x 2 x k= / i,k= /j k= /i
·
·
a
· (ti − μ) −
2
λ2j (tj
− μ) − b
2
k= / i,k= /j
⎤
λk λj (tk − μ)(tj − μ)⎦ dti dtj .
(3.2)
Similarly ⎧ ⎛ +∞ ⎨ +∞ ⎝aλj (tj ⎩ x x
III =
+
b a k= /j
−b
k= / i,k= /j
1
=
a
1 2
λk
λi λj
− μ)2 + b
k= /j
⎞
λk (tk − μ)(tj − μ)⎠ exp −
⎡ ⎤2 ⎤ ⎫ ⎛ ⎪ ⎬ b2 ⎢ ⎥ · (tk − μ)⎦ ⎦ dtj exp ⎣ ⎝ λk (tk ⎪ 2a k= ⎭ /j
⎞2
a' 2
λj (tj − μ)
a
− μ)⎠ − λ2i (ti − μ)2 2
⎤
λk λj (tk − μ)(tj − μ)⎦ dtj
(x − μ)
+∞ √ % a
λi (x−μ)+ ba λj (x−μ)+ ba
k= / i,k= /j
λk (tk −μ)
&
e−
u2 2
du
⎡
·
·
⎞2 ⎛ 2 b2 − a 2 2 b2 − ab ⎢b ⎝ exp ⎣ λk (tk − μ)⎠ + λj (x − μ)2 + λj (x − μ) 2a k= 2a a / i,k= /j ⎡ ⎤ +∞ +∞ 1 a ⎦ λk (tk − μ) + exp ⎣− λ2j (tj − μ)2 − b λk λj (tk − μ) λj x 2 x k= / i,k= /j k= /j
· (tj − μ) −
a 2
λ2i (ti
− μ) − b 2
k= / i,k= /j
⎤
λk λi (tk − μ)(ti − μ)⎦ dti dtj .
(3.3)
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
1115
From (3.1)–(3.3), we can obtain I − II + III 1
=
a
1 2
(x − μ)
λi λj
+∞ √ % a
λi (x−μ)+ ba λj (x−μ)+ ba
k= / i,k= /j
λk (tk −μ)
&
u2 2
e−
du
⎡
·
·
⎛ ⎞2 2 b 2 − a2 2 b2 − ab ⎢b ⎝ exp ⎣ λk · (tk − μ)⎠ + λj (x − μ)2 + λj (x − μ) 2a k= 2a a / i,k= /j ⎤ +∞ 2 −u & e 2 du λk (tk − μ)⎦ − (x − μ) √ % b b a
k= / i,k= /j
⎡
⎞2
⎛ 2 ⎢b ⎝ · exp ⎣ λk · (tk 2a k= / i,k= /j ⎤⎫ ⎬ λk (tk − μ)⎦⎭ · k= / i,k= /j
1
=
1
a 2 λi λj
·
⎧ ⎪ ⎨
λj (x−μ)+ a λi (x−μ)+ a
− μ)⎠ +
2a
√
a
b2
− ab a
λi (x − μ)
⎡ b b ⎢a ⎣ [( x − μ) exp ⎣ λ λk (tk i (x − μ) + λj (x − μ) + ⎪ 2 a a k= ⎩ / i,k= /j
λi (x−μ)+ ba λj (x−μ)+ ba
· (x − μ) +
b
k= / i,k= / j λk (tk −μ)
a k= / i,k= /j
a
a
· exp ⎣− λ2j (x − μ)2 − b 2
k= / i,k= /j
e−
&
u2 2
du − (x − μ) exp
⎤2 ⎤ ⎥ +∞ λk (tk − μ)⎦ ⎦ √ %
⎡
−b
λ2i (x − μ)2 +
λk (tk −μ)
⎡
+∞ %
− a2
b2
k= / i,k= /j
k= / i,k= /j
λj (x−μ)+ ba λi (x−μ)+ ba
a
2
⎤2 ⎤ ⎥ − μ)⎦ ⎦
a
k= / i,k= /j
λk (tk −μ)
a
λk λj (tk − μ)(x − μ) − λ2i (x − μ)2 2
⎤
λk λi (tk − μ)(x − μ) − bλj λi (x − μ)
2⎦
.
By Lemma 3.2(1), we get
(λi − λj )(I − II + III) 0. That is ⎛
(λi − λj ) ⎜ ⎝
∂F
min Xi (x)
1in
∂λi
−
∂F
1
1in
∂λj
· (λi − λj )(I − II + III)e− 2 Therefore P
min Xi (x)
k= / i,k= /j
1in
⎟ ⎠
=
/j k = / i,k =
min Xi x P min Xi∗ x .
1in
⎞ 1 n
1
(2π) 2 | | 2
·
Vkk (tk −μ)(tk −μ)
+∞ x
...
b
λj (x − μ) + λi
+∞ x
dt1 . . . dtn 0.
&
e
2 − u2
⎫ ⎪ ⎬ du
⎪ ⎭
1116
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
Similarly, we can show that F max Xi (x)= P max Xi
> x = 1 − P (X1 x, . . . , Xn x)
1in
1in
=1 −
x
...
−∞
x
1
−∞ (2π )
n 2
1
| |
e− 2 (t −—)
1 2
−1 (t −—)
dt1 . . . dtn
(λ1 , λ2 , . . . , λn ) for all x ∈ R. So P
is Schur-convex function with respect to P max Xi∗ x . 1in
max Xi x
1in
We have completed the proof of Theorem 2.1.
Proof of Theorem 2.2. We first let λ1 = f (μ1 ), . . . , λn = f (μn ), λ∗1 = f (μ∗1 ), . . . , λ∗n = f (μ∗n ), then μ1 = f −1 (λ1 ), . . . , μn = f −1 (λn ), μ∗1 = f −1 (λ∗1 ), . . . , μ∗n = f −1 (λn∗ ), so we get (λ1 , . . . , λn ) m (λ∗1 , . . . , λ∗n ). (1) Now we prove that P min Xi x P min Xi∗ x by proving that 1in 1in F min Xi (x) = P min Xi x = P (X1 x, . . . , Xn x) 1in
1in
+∞
=
x
...
+∞
1
(2π ) | |
1 2
· e− 2 (t −—) 1
n 2
x
−1 (t −—)
dt1 . . . dtn
is Schur-concave function with respect to (λ1 , . . . , λn ), where — = (μ, . . . , μ)T . ⎛ ρ ··· ρ ⎞ 1 ⎜ρ 1 ··· ρ ⎟ ⎜ ⎟ Since = σ 2 ⎜ . , then −1 = (1−ρ)σ 2 [(1n−1)ρ+1] .. ⎟ .. ⎝ .. . . ⎠ ρ ρ ··· 1 ⎛ ⎞ (n − 2)ρ + 1 −ρ ··· −ρ ⎜ ⎟ −ρ −ρ (n − 2)ρ + 1 · · · ⎜ ⎟ ·⎜ ⎟ (Vij )n×n . . . . . . . ⎝ ⎠ . . . −ρ −ρ · · · (n − 2)ρ + 1 Denote Vii
=
(n−2)ρ+1 (1−ρ)[(n−1)ρ+1]σ 2
Therefore, for all x
∂F
∈ R,
d, i = / j, i, j = 1, . . . , n.
min Xi (x)
+∞
1 n
1
(2π) 2 | | 2 f (μi )
· e− 2 (t −—) 1
=
−ρ (1−ρ)[(n−1)ρ+1]σ 2
1in
∂λi =
c, Vij =
− 1 ( t − —)
1 n
1
(2π) 2 | | 2 f (μi ) ·e
x
1 n 2
1
x
⎛ ⎝Vii (ti
− μi ) +
dt1 . . . dtn ⎡ ⎛ +∞ +∞ +∞ ⎣ ⎝Vii (ti ...
− 12 (t −—) −1 (t −—)
=−
...
+∞
x
(
x
x
k= /i
⎞ Vik (tk
− μi ) +
− μk )⎠
k= /i
⎞ Vik (tk
− μk )⎠
dti dt1 . . . dti−1 dti+1 . . . dtn ( +∞ +∞ +∞ 1 −1 ... d(e− 2 (t −—) (t −—) ) dt1 . . . dti−1
(2π) | | 2 f (μi ) · dti+1 . . . dtn
x
x
x
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
+∞
1
=
n 2
1 2
...
+∞
⎡ exp ⎣−
1
[c (x − μi )2 + 2d(x − μi )
2 x (2π) | | f (μi ) x 2 (tk − μk )(tj − μj ) + + c (tj − μj ) + 2d k= / i,k= /j
· (tk − μk )]] dt1 . . . dti−1 dti+1 . . . dtn +∞
1
=
...
1117
+∞ ) +∞
k = / i,k = /j
1
Vkk (tk
k= /i
(tk − μk )
− μk )
− c (tj − μj ) +
d
((x − μi ) c ⎫ ⎤ ⎡ ⎞⎤2 ⎛ ⎞2 ⎪ 2 ⎬ ⎥ ⎢ d ⎝ ⎠ ⎦ (tk − μk ) (x − μi ) + (tk − μk )⎠ + ⎦ dtj exp ⎣− ⎪ c ⎭ k= / i,k= /j k= / i,k= /j 2 + 2d (tk − μk )(x − μi ) + c (x − μi ) + Vkk (tk − μk ) n 2
(2π) | |
1 2
f (μ
i)
x
x
exp
2
x
k= / i,k= / j k = / i,k = /j
k= / i,k= /j
· (tk − μk )] dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn 1
=
n 2
(2π) | | d
1 2
+∞
√
cf (μi )
x
⎤ 2⎦
...
+∞ )
exp
x
d
c 2
[(x − μj ) + (x − μi ) c
+∞
⎫ ⎬
2
− u2
(tk − μk )] e du √ ⎭ c k= c [(x−μj )+ dc (x−μi )+ dc k= / i,k= / j (tk −μk )] / i,k= /j ⎡ ⎤ c c 2 2 · exp ⎣− (x − μi ) − (x − μj ) − d (tk − μk )[(x − μi ) + (x − μj )⎦ 2 2 k= / i,k= /j ⎤ − d(x − μi )(x − μj ) + Vkk (tk − μk )(tk − μk )⎦ +
k= / i,k= / j k = / i,k = /j
· dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn .
(3.4)
So from (3.4), we have
∂F
min Xi (x)
1in
∂λi =
∂F
−
min Xi (x)
1in
∂λj
+∞
1
1√
n 2
cf (μi )f (μj )
(2π) | | 2 d
+ (x − μi ) + c
⎡
−
f (μi ) exp ⎣
c
d c
·
k= / i,k= /j
x
...
+∞ ) x
⎤ 2⎦
(tk − μk )] d
f (μj ) exp
c 2
[(x − μj )
+∞ √
d
k= / i,k= / j (tk −μk )]
e−
u2 2
du
⎤ 2⎦
(tk − μk )] c k= / i,k= /j * +∞ 2 c c − u2 · √ e du · exp − (x − μj )2 − (x − μi )2 d d 2 2 c [(x−μi )+ c (x−μj )+ c k= / i,k= / j (tk −μk )] −d (tk − μk )[(x − μi ) + (x − μj )] − d(x − μi )(x − μj ) + k= / i,k= /j
k = / i,k = /j
Vkk (tk
2
[(x − μi ) + (x − μj ) +
c [(x−μj )+ dc (x−μi )+ dc
c
⎤
k= / i,k= /j
− μk )(tk − μk )⎦ dt1 . . . dti−1 dti+1 . . . dtj−1 dtj+1 . . . dtn .
(3.5)
1118
L. Fang, X. Zhang / Linear Algebra and its Applications 434 (2011) 1107–1118
Thus, if f (y)f
(y) 0 for all y ∈ [u, v], we can obtain the following result from Lemma 3.2(1), ⎧ ⎨
⎡
d
d
2
c
c k= / i,k= /j
(λi − λj ) ⎩f (μj ) exp ⎣ [(x − μj ) + (x − μi ) + ·
+∞
+
√
c [(x−μj )+ dc (x−μi )+ dc
d
c k= / i,k= /j
c
⎤
(tk − μk )]2 ⎦
k= / i,k= / j (tk −μk )]
e−
u2 2
⎤
2⎦
(tk − μk )]
du − f (μi ) exp
c 2
+∞ √
c [(x−μi )+ dc (x−μj )+ dc
k= / i,k= / j (tk −μk )]
d
[(x − μi ) + (x − μj )
e−
u2 2
⎫ ⎬
c
du 0. ⎭
(3.6)
Moreover, from (3.5) and (3.6), we get ⎛ ⎞ ∂ F min Xi (x) ∂ F min Xi (x) ⎜ ⎟ (λi − λj ) ⎝ 1 i n − 1 i n ⎠ 0.
∂λi
Therefore P
∂λj
min Xi x P min Xi∗ x .
1in
1in
(2) Similarly, if f (y)f (y) 0 for all y ∈ [u, v], we can show that F max Xi (x) = P max Xi > x = 1 − P (X1 x, . . . , Xn x) 1in
1in
=1−
x
−∞
...
x
−∞ (2π )
1 n 2
| |
e− 2 (t −—) 1
1 2
− 1 ( t − —)
dt1 . . . dtn
is Schur-convex function with respect to (λ1 , . . . , λn ) for all x ∈ R through Lemma 3.2(2). ∗ So P max Xi x P max Xi x . Finally, We have completed the proof of Theorem 2.2. 1in
1in
Acknowledgements This research is supported by the National Natural Science Foundation of China (No. 10671037), the University Natural Science Research Project of Anhui Province of China (No. KJ2010B347), and the Grant for Youth of Anhui Normal University. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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