Kernel deconvolution estimation for random fields

0.28 MB·English

by  Ahmed El Ghini

Kernel deconvolution estimation for random fields
Ahmed EL GHINI and Mohamed EL MACHKOURI
Abstract
Inthiswork,weestablishtheasymptoticnormalityofthedeconvolutionkernel
density estimator in the context of strongly mixing random fields. Only minimal
2 conditions on the bandwidth parameter are required and a simple criterion on
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the strong mixing coefficients is provided. Our approach is based on the Lin-
0
2
deberg’s method rather than on Bernstein’s technique and coupling arguments

n
a widely used in previous works on nonparametric estimation for spatial processes.
J
We deal also with nonmixing random fields which can be written as a (nonlinear)
2
functional of i.i.d. random fields by considering the physical dependence measure

]
T
coefficients introduced by Wu [20].
S
.
h
t AMS Subject Classifications (2000): 62G05, 62G07, 60G60.
a
m
Key words and phrases: Central limit theorem, deconvolution kernel density esti-
[
mator, strongly mixing random fields, nonmixing random fields, physical depen-

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v dence measure.
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Short title: Deconvolution kernel density estimator for random fields.
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4
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. 1 Introduction and main results
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2
1 Let X = (Xi)i∈Zd be a stationary real random field defined on the probability space
:
v (Ω, ,P). We observe the random field X on a region Λ , n N∗, but the observations
n
i F ∈
X
are contamined with noise such as measurement errors. In fact, we observe only the
r
a random field Y = (Y ) defined for any i in Zd by Y = X + θ where the error
i i∈Zd i i i
variables (θ ) are identically distributed and independent of X. We denote by
i i∈Zd
f , f and f the marginal density of Y, X and θ respectively and we have f =
Y X θ Y
f ⋆f . We observe a sample of Y and we want to estimate f using the deconvolution
X θ X
kernel approach introduced by Stefanski and Carroll [18]. Previous key results on
deconvolution kernel density estimators for time series are Fan [7], [8] and Masry [12],
[13]. For strongly mixing random fields indexed by the lattice Zd, Li [10] obtained
a central limit theorem for the deconvolution kernel density estimator using the so-
called Bernstein’s small and large blocks technique and coupling arguments initiated
by Tran [19]. Note that the extension of asymptotic result for time series to the spatial
setting is not trivial because of difficulties coming from spatial ordering. The purpose
of this work is to put on light a new approach for the asymptotic normality of kernel
density estimators. In fact, we are going to apply the Lindeberg’s method (see [11])
in order to improve the result by Li [10] in several directions. This new approach was
recently applied successfully in El Machkouri and Stoica [5] and El Machkouri [4] for
the Nadaraya-Watson estimator and the Parzen-Rosenblatt estimator respectively in
the setting of random fields.
For any finite subset B of Zd, denote B the number of elements in B and ∂B its
| |
boundary defined by ∂B = i B; j / B i j = 1 where s = max s for
1≤k≤d k
{ ∈ ∃ ∈ | − | } | | | |
any s = (s ,...,s ) in Zd. In the sequel, we assume that we observe (X ) on a
1 d i i∈Zd
sequence (Λ ) of finite subsets of Zd which satisfies
n n≥1
∂Λ
n
lim Λ = and lim | | = 0. (1)
n
n→∞| | ∞ n→∞ Λn
| |
Given two σ-algebras and of , we recall the α-mixing coefficient introduced by
U V F
Rosenblatt [16] defined by α( , ) = sup P(A B) P(A)P(B) , A , B .
U V {| ∩ − | ∈ U ∈ V}
For any τ in N∗ and any positive integer n, we consider the mixing coefficient
∪{∞}
α (n) defined by
1,τ
α (n) = sup α(σ(X ), ), k Zd, B τ, ρ(B, k ) n ,
1,τ k B
{ F ∈ | | ≤ { } ≥ }
where = σ(X ; i B) and the distance ρ is defined for any subsets B and B of
B i 1 2
F ∈
Zd by ρ(B ,B ) = min i j , i B , j B . We say that the random field (X )
1 2 1 2 i i∈Zd
{| − | ∈ ∈ }
is strongly mixing if lim α (n) = 0 for some τ in N∗ .
n→∞ 1,τ
∪{∞}
Let (b ) be a sequence of positive numbers going to zero as n goes to infinity. The
n n≥1
deconvolving kernel density estimator for f is defined for any x in R by
X
1 x Y
ˆ i
f (x) = g − (2)
n n
Λ b b
| n| n iX∈Λn (cid:18) n (cid:19)
where for any z in R,
1 φ (t)
g (z) = e−itz K dt.
n
2π φ (t/b )
R θ n
Z
The kernel density estimator fˆ defined by (2) can be written for any x in R as
n
1 φ (tb )
fˆ (x) = e−itxφˆ (t) K n dt (3)
n n
2π φ (t)
R θ
Z
where
1
φˆ (t) = eitYi.
n
Λ
n
| | iX∈Λn
We consider the following assumptions:
2
(A1) The marginal probability distribution of each Y is absolutely continuous with
k
continuous positive density function f .
Y
(A2) There exists κ > 0 such that sup f (y x) κwhere f is the conditional
(x,y)∈R2 i|j | ≤ i|j
density function of Y given Y for any i and j in Zd.
i j
(A3) There exists β > 0 and B > 0 such that t β φ (t) B.
θ
| | | | −−t→−−+−∞→
(A4) The characteristic function φ of the kernel K vanishes outside [ 1,1].
K
−
(A5) The bandwidth b converges to zero and Λ b goes to infinity.
n n n
| |
ˆ
The following result establishes the asymptotic bias of the estimator f .
n
Proposition 1 (Li [10], 2008) If φ is continuous then, for any real x,
K
E(fˆ (x)) f(x).
n
−−n−→−+−∞→
ˆ
Now, we investigate the asymptotic variance of the estimator f .
n
Proposition 2 Assume that m2d−1α (m) < . For any x in R, we have
m≥1 1,1 ∞
P f (x)
lim Λ b2β+1V(fˆ (x)) = Y t 2β φ (t) 2dt := σ2(x). (4)
n→∞| n| n n B2 R| | | K |
Z
Our main result is the following.
Theorem 1 Assume that Assumptions (A1),...,(A5) hold and
+∞
m2d−1α (m) < . (5)
1,∞
∞
m=1
X
Then for any positive integer k and any distinct points x ,...,x in R,
1 k
fˆ (x ) Efˆ (x )
n 1 n 1
−
(|Λn|bn2β+1)1/2 ...  −−n−→L−+−∞→ N (0,V) (6)
fˆ (x ) Efˆ (x )
 n k − n k 
 
where V is a diagonal matrix with diagonal elements v = fY(xi) t 2β φ (t) 2dt.
ii B2 R| | | K |
R
Remark 1. Theorem 1 improves Theorem 4.1 and Theorem 4.2 in [10] in three direc-
tions: the regions Λ where the random field is observed are not reduced to rectangular
n
ones, the assumption (A5) on the bandwidth b is minimal and the mixing condition
n
3
(5) does not dependent on the bandwith parameter b .
n
Since the mixing property is often unverifiable and might be too restrictive, it is impor-
tant to provide limit theorems for nonmixing and possibly nonlinear spatial processes.
So, in the sequel, we consider that (X ) is a field of identically distributed real
i i∈Zd
random variables with a marginal density f such that
X = F ε ; s Zd , i Zd, (7)
i i−s
∈ ∈
(cid:0) (cid:1)
where (ε ) are i.i.d. random variables and F is a measurable function defined on
j j∈Zd
RZd. In the one-dimensional case (d = 1), the class (7) includes linear as well as
many widely used nonlinear time series models as special cases. More importantly,
it provides a very general framework for asymptotic theory for statistics of stationary
time series (see [20] and the review paper [21]). Let (ε′) be an i.i.d. copy of
j j∈Zd
(ε ) and consider for any positive integer n the coupled version X∗ of X defined
j j∈Zd i i
by X∗ = F ε∗ ; s Zd where ε∗ = ε 11 + ε′ 11 for any j in Zd. In other
i i−s ∈ j j {j6=0} 0 {j=0}
words, we obtain X∗ from X by just replacing ε by its copy ε′. Following Wu [20],
(cid:0) i (cid:1) i 0 0
we introduce appropriate dependence measures: let i in Zd and p > 0 be fixed. If X
i
belongs to L (that is, E X p is finite), we define the physical dependence measure
p i
| |
δ = X X∗ where . is the usual Lp-norm and we say that the random field
i,p k i − ikp k kp
(X ) is p-stable if δ < . For d 2, the reader should keep in mind the
i i∈Zd i∈Zd i,p ∞ ≥
following two examples already given in [6] :
P
Linear random fields: Let (ε ) be i.i.d random variables with ε in Lp, p 2. The
i i∈Zd i
≥
linear random field X defined for any i in Zd by
X = a ε
i s i−s
sX∈Zd
with (a ) in RZd such that a2 < is of the form (7) with a linear functional
s s∈Zd i∈Zd i ∞
g. For any i in Zd, δ = a ε ε′ . So, X is p-stable if a < . Clearly,
i,p k ikkP0 − 0kp i∈Zd | i| ∞
if H is a Lipschitz continuous function, under the above condition, the subordinated
P
process Y = H(X ) is also p-stable since δ = O( a ).
i i i,p i
| |
Volterra field : Another class of nonlinear random field is the Volterra process which
plays an important role in the nonlinear system theory (Casti [2], Rugh [17]): consider
the second order Volterra process
X = a ε ε ,
i s1,s2 i−s1 i−s2
s1,Xs2∈Zd
4
where a are real coefficients with a = 0 if s = s and (ε ) are i.i.d. random
s1,s2 s1,s2 1 2 i i∈Zd
variables with ε in Lp, p 2. Let
i
≥
A = (a2 +a2 ) and B = ( a p + a p).
i s1,i i,s2 i | s1,i| | i,s2|
s1,Xs2∈Zd s1,Xs2∈Zd
By the Rosenthal inequality, there exists a constant C > 0 such that
p
δ = X X∗ C A1/2 ε ε +C B1/p ε 2.
i,p k i − ikp ≤ p i k 0k2k 0kp p i k 0kp
Theorem 2 Let (X ) be defined by the relation (7) and assume that (A1),...,(A5)
i i∈Zd
hold. If (5) is replaced by the condition
5d
i 2 δi < (8)
| | ∞
i∈Zd
X
then the conclusion of Theorem 1 still holds.
2 Proofs
Throughout this section, the symbol κ will denote a generic positive constant which the
value is not important and for any i = (i ,...,i ) Zd, we denote i = max i .
1 d 1≤k≤d k
∈ | | | |
Recall also that for any finite subset B of Zd, we denote B the number of elements
| |
in B. Let τ N∗ be fixed and consider the sequence (m ) defined by
n,τ n≥1
∈ ∪{∞}
1
d
1
m = max v , i dα ( i ) +1 (9)
n,τ  n  1,τ  
b | | | | 
 n |iX|>vn 
 
 
 
 
−1  
where v = b2d and [.] denotes the integer part function. The following technical
n n
lemma is a spatial version of a result by Bosq, Merlevède and Peligrad ([1], pages
(cid:2) (cid:3)
88-89).
Lemma 1 If τ N∗ and m2d−1α (m) < then
∈ ∪{∞} m≥1 1,τ ∞
P
1
m , md b 0 and i dα ( i ) 0.
n,τ → ∞ n,τ n → md b | | 1,τ | | →
n,τ n
|i|>Xmn,τ
5
Proof of Proposition 2. Let z be fixed in R. For any i in Zd, we denote
1 z Y z Y
Z (z) = g − i Eg − i .
i n n
b b − b
n (cid:18) (cid:18) n (cid:19) (cid:18) n (cid:19)(cid:19)
The proof of the following lemma is done in the appendix.
Lemma 2 For any z in R,
f (z)
b2β+1E(Z2(z)) σ2(z) := Y u 2β φ (u) 2du (10)
n 0 −−n−→−∞−→ B2 R| | | K |
Z
and sup E Z (s)Z (t) = O(b−2β) for any s and t in R.
i∈Zd\{0} | 0 i | n
Let x in R be fixed. We have
b2β+1
Λ b2β+1V(fˆ (x)) = b2β+1E Z2(x) + n Cov(Z (x),Z (x)). (11)
| n| n n n 0 Λ i j
n
(cid:0) (cid:1) | | i,Xij6=∈Λjn
Since (Z ) is stationary, we have
i i∈Zd
b2β+1
n Cov(Z (x),Z (x)) b2β+1 E(Z (x)Z (x))
Λ (cid:12) i j (cid:12) ≤ n | 0 i |
n (cid:12) (cid:12)
| | (cid:12)i,Xj∈Λn (cid:12) i∈ZXd\{0}
(cid:12) i6=j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) b2β+1 md sup E Z (x)Z (x) + E(Z (x)Z (x)) .
≤ n  n,1 | 0 i | | 0 i |
i∈Zd\{0}
|i|X>mn,1
 
ByRio’scovarianceinequality(cf. [15],Theorem1.1),weknowthat E(Z (x)Z (x))
0 i
| | ≤
κ Z (x) 2 α ( i ). Since Z (x) κb−β−1 and τ N∗ , we obtain
k 0 k∞ 1,1 | | k 0 k∞ ≤ n ∈ ∪{∞}
b2β+1 1
n Cov(Z (x),Z (x)) κ md b2β+1 sup E Z (x)Z (x) + i dα ( i ) .
Λ (cid:12) i j (cid:12) ≤  n,1 n | 0 i | md b | | 1,1 | | 
| n| (cid:12)(cid:12)i,Xj∈Λn (cid:12)(cid:12) i∈Zd\{0} n,1 n |i|X>mn,1
(cid:12) i6=j (cid:12)
 
(cid:12) (cid:12)
(cid:12) (cid:12)
Apply(cid:12)ing Lemma 1 and the s(cid:12)econd part of Lemma 2, we derive
b2β+1
lim n Cov(Z (x),Z (x)) = 0. (12)
(cid:12) i j (cid:12)
n→∞ Λn (cid:12) (cid:12)
| | (cid:12)i,Xj∈Λn (cid:12)
(cid:12) i6=j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Combining (10), (11) and (12), we(cid:12)obtain (4). The proof (cid:12)of Proposition 2 is complete.
6
Proof of Theorem 1. Without loss of generality, we consider only the case k = 2
and we refer to x and x as x and y (x = y). Let λ and λ be two constants such
1 2 1 2
6
that λ2 +λ2 = 1 and denote
1 2
β+1
b 2∆
S = λ ( Λ b2β+1)1/2(fˆ (x) Efˆ(x))+λ ( Λ b2β+1)1/2(fˆ (y) Efˆ (y)) = n i
n 1 | n| n n − n 2 | n| n n − n Λ 1/2
n
iX∈Λn | |
where ∆ = λ Z (x)+λ Z (y) and for any z in R,
i 1 i 2 i
1 z Y z Y
Z (z) = g − i Eg − i .
i n n
b b − b
n (cid:18) (cid:18) n (cid:19) (cid:18) n (cid:19)(cid:19)
We consider the notation
λ2f (x)+λ2f (y)
η = 1 Y 2 Y t 2β φ (t) 2dt. (13)
B2 | | | K |
R
Z
The proof of the following technical result is postponed to the annex.
Lemma 3 b2β+1E(∆2) converges to η as n goes to infinity and sup E ∆ ∆ =
n 0 i∈Zd\{0} | 0 i|
O(b−2β).
n
In order to prove the convergence in distribution of S to √ητ where τ (0,1),
n 0 0
∼ N
we follow the Lindeberg’s method used in the proof of the central limit theorem for
stationary random fields by Dedecker [3]. Let ϕ be a one to one map from [1,κ] N∗
∩
to a finite subset of Zd and (ξ ) a real random field. For all integers k in [1,κ], we
i i∈Zd
denote
k κ
S (ξ) = ξ and Sc (ξ) = ξ
ϕ(k) ϕ(i) ϕ(k) ϕ(i)
i=1 i=k
X X
with the convention S (ξ) = Sc (ξ) = 0. To describe the set Λ , we define the
ϕ(0) ϕ(κ+1) n
one to one map ϕ from [1, Λ ] N∗ to Λ by: ϕ is the unique function such that
n n
| | ∩
ϕ(k) <lex ϕ(l) for 1 k < l Λn . From now on, we consider a field (τi)i∈Zd of i.i.d.
≤ ≤ | |
random variables independent of (∆ ) such that τ has the standard normal law
i i∈Zd 0
(0,1). We introduce the fields Γ and γ defined for any i in Zd by
N
β+1
b 2∆ τ √η
n i i
Γ = and γ =
i Λ 1/2 i Λ 1/2
n n
| | | |
whereη isdefined by(13). Lethbeany functionfromRto R. For0 k l Λ +1,
n
≤ ≤ ≤ | |
we introduce h (Γ) = h(S (Γ)+Sc (γ)). With the above convention we have that
k,l ϕ(k) ϕ(l)
h (Γ) = h(S (Γ)) and also h (Γ) = h(Sc (γ)). In the sequel, we will often
k,|Λn|+1 ϕ(k) 0,l ϕ(l)
7
write h instead of h (Γ). We denote by B4(R) the unit ball of C4(R): h belongs to
k,l k,l 1 b
B4(R) if and only if it belongs to C4(R) and satisfies max h(i) 1. It suffices
1 0≤i≤4k k∞ ≤
to prove that for all h in B4(R),
1
E h S (Γ) E(h(τ √η)).
ϕ(|Λn|) −−n−→−+−∞→ 0
(cid:0) (cid:0) (cid:1)(cid:1)
We use Lindeberg’s decomposition:
|Λn|
E h S (Γ) h(τ √η) = E(h h ).
ϕ(|Λn|) − 0 k,k+1− k−1,k
k=1
(cid:0) (cid:0) (cid:1) (cid:1) X
Now,
h h = h h +h h .
k,k+1 k−1,k k,k+1 k−1,k+1 k−1,k+1 k−1,k
− − −
Applying Taylor’s formula we get that:
1
h h = Γ h′ + Γ2 h′′ +R
k,k+1− k−1,k+1 ϕ(k) k−1,k+1 2 ϕ(k) k−1,k+1 k
and
1
h h = γ h′ γ2 h′′ +r
k−1,k+1− k−1,k − ϕ(k) k−1,k+1− 2 ϕ(k) k−1,k+1 k
where R Γ2 (1 Γ ) and r γ2 (1 γ ). Since (Γ,τ ) is inde-
| k| ≤ ϕ(k) ∧| ϕ(k)| | k| ≤ ϕ(k) ∧| ϕ(k)| i i6=ϕ(k)
pendent of τ , it follows that
ϕ(k)
η
E γ h′ = 0 and E γ2 h′′ = E h′′
ϕ(k) k−1,k+1 ϕ(k) k−1,k+1 Λ k−1,k+1
(cid:16) (cid:17) (cid:16) (cid:17) (cid:18)| n| (cid:19)
Hence, we obtain
|Λn|
E h(S (Γ)) h(τ √η) = E(Γ h′ )
ϕ(|Λn|) − 0 ϕ(k) k−1,k+1
k=1
(cid:0) (cid:1) X
|Λn| η h′′
+ E Γ2 k−1,k+1
ϕ(k) − Λ 2
k=1 (cid:18) | n|(cid:19) !
X
|Λn|
+ E(R +r ).
k k
k=1
X
Let 1 k Λ be fixed. Noting that ∆ is bounded by κb−β−1 and applying the
≤ ≤ | n| | 0| n
first part of Lemma 3, we derive
b3β+23E ∆ 3 1
E R n | 0| = O
k
| | ≤ Λ 3/2 ( Λ 3b )1/2
| n| (cid:18) | n| n (cid:19)
8
and
E γ 3 η3/2E τ 3 1
E r | 0| | 0| = O .
| k| ≤ Λ 3/2 ≤ Λ 3/2 Λ 3/2
| n| | n| (cid:18)| n| (cid:19)
Consequently, we obtain
|Λn| 1 1
E( R + r ) = O + = o(1).
| k| | k| ( Λ b )1/2 Λ 1/2
k=1 (cid:18) | n| n | n| (cid:19)
X
Now, it is sufficient to show
|Λn| η h′′
lim E(Γ h′ )+E Γ2 k−1,k+1 = 0. (14)
n→+∞k=1 ϕ(k) k−1,k+1 (cid:18) ϕ(k) − |Λn|(cid:19) 2 !!
X
First, wefocuson |Λn|E Γ h′ . OnthelatticeZd wedefine thelexicographic
k=1 ϕ(k) k−1,k+1
order as follows: if i = (i ,...,i ) and j = (j ,...,j ) are distinct elements of Zd, the
P (cid:0)1 d (cid:1) 1 d
notation i <lex j means that either i1 < j1 or for some p in 2,3,...,d , ip < jp and
{ }
i = j for 1 q < p. Let the sets VM ; i Zd, M N∗ be defined as follows:
q q ≤ { i ∈ ∈ }
V1 = j Zd; j <lex i and for M 2, VM = V1 j Zd; i j M .
i { ∈ } ≥ i i ∩{ ∈ | − | ≥ }
For any subset L of Zd define = σ(∆ ; i L) and set
L i
F ∈
E (∆ ) = E(∆ ), M N∗.
M i i|FViM ∈
For all M in N∗ and all integer k in [1, Λ ], we define
n
| |
EM = ϕ([1,k] N∗) VM and SM (Γ) = Γ .
k ∩ ∩ ϕ(k) ϕ(k) i
iX∈EMk
For any function Ψ from R to R, we define ΨM = Ψ(SM (Γ) + Sc (γ)). We are
k−1,l ϕ(k) ϕ(l)
going to apply this notation to the successive derivatives of the function h. Our aim is
to show that
|Λn|
lim E Γ h′ Γ S (Γ) Smn,∞(Γ) h′′ = 0.
n→+∞ ϕ(k) k−1,k+1 − ϕ(k) ϕ(k−1) − ϕ(k) k−1,k+1
Xk=1 (cid:16) (cid:16) (cid:17) (cid:17)
where (m ) is the sequence defined by (9). First, we use the decomposition
n,∞ n≥1
Γ h′ = Γ h′mn,∞ +Γ h′ h′mn,∞ .
ϕ(k) k−1,k+1 ϕ(k) k−1,k+1 ϕ(k) k−1,k+1 − k−1,k+1
(cid:16) (cid:17)
9
We consider a one to one map ψ from [1, Emn,∞ ] N∗ to Emn,∞ and such that ψ(i)
| k | ∩ k | −
ϕ(k) ψ(i 1) ϕ(k) . This choice of ψ ensures that S (Γ) and S (Γ) are
ψ(i) ψ(i−1)
| ≤ | − − |
-measurable. The fact that γ is independent of Γ imply that
FV|ψ(i)−ϕ(k)|
ϕ(k)
E Γ h′ Sc (γ) = 0.
ϕ(k) ϕ(k+1)
(cid:16) (cid:17)
(cid:0) (cid:1)
Therefore
|Emn,∞|
k
E Γ h′mn,∞ = E Γ (Θ Θ ) (15)
ϕ(k) k−1,k+1 (cid:12) ϕ(k) i − i−1 (cid:12)
(cid:12) (cid:16) (cid:17)(cid:12) (cid:12)(cid:12) Xi=1 (cid:0) (cid:1)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where Θ = h′ S (Γ)+Sc (γ) . (cid:12)Since S (Γ) and S (Γ(cid:12)) are -
i ψ(i) ϕ(k+1) (cid:12) ψ(i) ψ(i−1) (cid:12) FV|ψ(i)−ϕ(k)|
ϕ(k)
measurable, we(cid:16)cantake the condition(cid:17)alexpectation of Γ withrespect to
ϕ(k) FV|ψ(i)−ϕ(k)|
ϕ(k)
in the right hand side of (15). On the other hand the function h′ is 1-Lipschitz, hence
Θ Θ Γ . Consequently,
i i−1 ψ(i)
| − | ≤ | |
E Γ (Θ Θ ) E Γ E Γ
ϕ(k) i i−1 ψ(i) |ψ(i)−ϕ(k)| ϕ(k)
− ≤ | |
(cid:12) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
and (cid:12) (cid:12)
|Emn,∞|
k
E Γ h′mn,∞ E Γ E (Γ ) .
ϕ(k) k−1,k+1 ≤ | ψ(i) |ψ(i)−ϕ(k)| ϕ(k) |
(cid:12) (cid:16) (cid:17)(cid:12) Xi=1
(cid:12) (cid:12)
Hence,
(cid:12) (cid:12)
|Λn| b2β+1 |Λn||Emkn,∞|
E Γ h′mn,∞ n E ∆ E (∆ )
(cid:12) ϕ(k) k−1,k+1 (cid:12) ≤ Λ | ψ(i) |ψ(i)−ϕ(k)| ϕ(k) |
(cid:12)(cid:12)Xk=1 (cid:16) (cid:17)(cid:12)(cid:12) | n| Xk=1 Xi=1
(cid:12)(cid:12) (cid:12)(cid:12) b2β+1 ∆ E (∆ ) .
(cid:12) (cid:12) ≤ n k j |j| 0 k1
|j|≥Xmn,∞
For any j in Zd, we have
k∆jE|j|(∆0)k1 = Cov |∆j| 11E|j|(∆0)≥0 − 11E|j|(∆0)<0 ,∆0 .
(cid:16) (cid:16) (cid:17) (cid:17)
So, applying Rio’s covariance inequality (cf. [15], Theorem 1.1), we obtain
α1,∞(|j|)
∆ E (∆ ) 4 Q2 (u)du
k j |j| 0 k1 ≤ ∆0
Z0
where Q is defined by Q (u) = inf t 0; P( ∆ > t) u for any u in [0,1].
∆0 ∆0 { ≥ | 0| ≤ }
Since ∆ is bounded by κb−β−1, we have
| 0| n
Q (u) κb−β−1 and ∆ E (∆ ) κb−2β−2α ( j ).
∆0 ≤ n k j |j| 0 k1 ≤ n 1,∞ | |
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