is sym- metric α-stable (S.α.S) if its characteristic function is of the form

R. S A B R E (Dijon)

SPECTRAL DENSITY ESTIMATION FOR STATIONARY STABLE RANDOM FIELDS

Abstract. We consider a stationary symmetric stable bidimensional pro- cess with discrete time, having the spectral representation (1.1). We consider a general case where the spectral measure is assumed to be the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. We estimate the density of the absolutely continuous measure and the density on the lines.

1. Introduction. A complex random variable X = X

+ iX

is sym- metric α-stable (S.α.S) if its characteristic function is of the form

exp n

− R

S2

|t

x

+ t

x

|

dΓ

(x

, x

) o

,

where t = t

+ it

and Γ

is a symmetric measure on the unit sphere S

of R

.

A stochastic process {X

, t ∈ T } is (S.α.S) if all linear combinations, a

X

+ . . . + a

X

, are (S.α.S). When X = X

+ iX

and Y = Y

+ iY

are jointly (S.α.S) and 1 < α ≤ 2, the covariation of X with Y is defined in [3] by

[X, Y ]

= R

S4

(x

+ ix

)(y

+ iy

)

dΓ

(x

, x

, y

, y

), where by convention, for b > 0 and Z a complex number, Z

= |Z|

Z

, Z

being the complex conjugate of Z. The quantity kXk

= [X, X]

is a norm [11] on the linear space of (S.α.S) random variables. If (ξ

)

is a complex (S.α.S) process with independent increments, then the measure µ

1991 Mathematics Subject Classification: 62G07, 60G35.

Key words and phrases: periodogram, Jackson kernel, double kernel method, (S.α.S) process.

[107]

defined by µ(]s, t]) = kξ

− ξ

k

is a Lebesgue–Stieltjes measure [3], called the spectral measure of ξ. When µ is absolutely continuous, its density is called the spectral density of ξ.

For stationary complex symmetric α-stable (S.α.S) processes having the spectral representation

X(t) =

∞

R

−∞

e

dξ(λ),

where ξ is a (S.α.S) process with independent isotropic increments, the spectral density function φ is estimated in [11].

Let us consider a bidimensional (random field), complex (S.α.S) process in discrete time, having the spectral representation

(1.1) X(n, m) =

π

R

−π π

R

−π

e

dξ(λ

, λ

),

where ξ is a (S.α.S) process with independent isotropic increments; that means ξ is an additive complex function defined on the Borel subsets of [−π, π]

such that:

• for any integer k and any Borel sets B

, . . . , B

, (ξ(B

), . . . , ξ(B

)) is (S.α.S);

• for any integer k and any disjoint Borel sets B

, . . . , B

, ξ(B

), . . . . . . , ξ(B

) are complex (S.α.S) independent random variables;

• for all Borel sets B, the distribution of the random variable e

ξ(B) is independent of θ.

For more details about the spectral representation see [8] and [1].

We define the spectral measure by [ξ(B), ξ(B)]

= R

B

dµ(λ

, λ

) for any Borel subset B of [−π, π]

, where [X, Y ]

denotes the covariation of X with Y [3] when X and Y are jointly (S.α.S).

To our knowledge the case we deal with is more general than those considered in [11], [6]: we suppose that the spectral measure of the process is the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines:

dµ(λ

, λ

) = φ(λ

, λ

)dλ

dλ

+

q

X

j=1

a

δ

(1.2)

+

q⁰

X

i=1

φ

(u

)δ

,

where φ and φ

for i = 1, . . . , q

are nonnegative continuous functions, a

∈ R

, a

, b

∈ R and w

, w

∈ [−π, π], j = 1, . . . , q, i = 1, . . . , q

.

Using Jackson polynomial kernels and the double kernel method, we estimate the function φ(λ

, λ

) for every (λ

, λ

) in ]−π, π[

. Under appro- priate conditions on the function φ, we obtain rates of convergence. Let us also indicate that our methods allow us to estimate the positive num- bers a

(j = 1, . . . , q) and the functions φ

(i = 1, . . . , q

). For brevity, we consider here only the estimation of φ; for details, see [13].

2. The periodogram and its statistical properties. Given N M observations of the process X : (X(n, m))

0≤m≤M −1

, where N and M satisfy:

• N − 1 = 2k(n − 1) with n ∈ N, k ∈ N ∪ {1/2}; if k = 1/2 then n = 2n

− 1, n

∈ N.

• M − 1 = 2k(m − 1) with m ∈ N, k ∈ N ∪ {1/2}; if k = 1/2 then m = 2m

− 1, m

∈ N.

We consider the function H

(λ) =

k(n−1)

X

m⁰=−k(n−1)

h

(m

/n) cos(λm

),

where h

is defined in [7] such that H

(λ) = 1

q

 sin

sin



with q

= 1 2π

π

R

−π

 sin

sin



dλ.

The Jackson polynomial kernel is |H

(λ)|

= |A

H

(λ)|

, where A

= B

with B

= R

−π

|H

(λ)|

dλ. We define H

(λ) in the same way.

The following lemma, proved in [6], will be frequently used in the sequel.

Lemma 2.1 [6]. Let B

=

π

R

−π

sin

sin

2kα

dλ and J

=

π

R

−π

|u|

|H

(λ)|

dλ,

where γ ∈ ]0, 2]. Then

B



 



 



≥ 2π  2 π



n

if 0 < α < 2,

≤ 4πkα

2kα − 1 n

if 1

2k < α < 2,

and

J

≤



 



 



π

2

(γ − 2kα + 1) · 1

n

if 1

2k < α < γ + 1 2k , 2kαπ

2

(γ + 1)(2kα − γ − 1) · 1

n

if γ + 1

2k < α < 2.

The proof of the following lemma is the same as in the one-dimensional case in [3].

Lemma 2.2. If ξ is a (S.α.S ) process with independent and isotropic increments, then

E exp 

i Re h R

−π π

R

−π

f (u

, u

) dξ(u

, u

) i

= exp



−C

π

R

−π

|f (u

, u

)|

dµ(u

, u

)



for every f ∈ L

(µ), where C

= (2πα)

R

−π

|cos θ|

dθ.

Now we denote by A the set of points where there are no atoms:

A =



(λ

, λ

) ∈ ]−π, π[

: λ

6= w

, λ

6= w

and λ

− b

− a

λ

2π 6∈ Z for 1 ≤ i ≤ q

and 1 ≤ j ≤ q

 . In the sequel we choose large k such that kα > 1 and we consider the following periodogram:

d

(λ

, λ

) = A

A

Re



X

n⁰=−k(n−1)

k(m−1)

X

m⁰=−k(m−1)

h

 n

n

 h

 m

m



×e

X(n

+ k(n − 1), m

+ k(m − 1))

 . Following the study realized in the one-dimensional case by Masry and Cam- banis in [11], one can show easily that, for (λ

, λ

) ∈ A,

(2.1) lim

N →∞

E{exp[ird

(λ

, λ

)]} = exp[−C

|r|

φ(λ

, λ

)].

We modify this periodogram taking the power p, with 0 < p < α/2, and multiplying by a normalization constant:

I b

(λ

, λ

) = C

|d

(λ

, λ

)|

.

The normalization constant C

is given by C

= D

F

C

, where

D

=

∞

R

−∞

1 − cos u

|u|

du and F

=

∞

R

−∞

1 − e

|u|

du, 0 < p < α 2 . We show that

E b I

(λ

, λ

) = [ψ

(λ

, λ

)]

, (2.2)

Var[ b I

(λ

, λ

)] = V

[ψ

(λ

, λ

)]

, (2.3)

where V

= C

C

− 1 and

ψ

(λ

, λ

) = I

(λ

, λ

) + J

(λ

, λ

) + K

(λ

, λ

) with

I

(λ

, λ

) =

π

R

−π π

R

−π

|H

(λ

− u

)|

|H

(λ

− u

)|

φ(u

, u

) du

du

,

J

(λ

, λ

) =

q

X

j=1

a

|H

(λ

− w

)|

|H

(λ

− w

)|

,

K

(λ

, λ

) =

q⁰

X

i=1 π

R

−π

|H

(λ

− v

)|

|H

(λ

− a

v

− b

)|

φ

(v

) dv

. As in [11] one can show that for (λ

, λ

) ∈ A, the function ψ

(λ

, λ

) converges to φ(λ

, λ

). Therefore b I

(λ

, λ

) is an asymptotically unbiased estimator of [φ(λ

, λ

)]

but it is not consistent because the variance is proportional to [φ(λ

, λ

)]

.

3. The smoothed periodogram. In this section, using two spectral windows, we smooth the periodogram b I

and we obtain consistent esti- mators of [φ(λ

, λ

)]

at the points (λ

, λ

) where there are no atoms. Let W be a nonnegative, even, continuous function vanishing for |λ| > 1, with R

−1

W (λ) dλ = 1. The spectral windows W

, W

are defined by W

(λ) = M

W (M

λ) and W

(λ) = L

W (L

λ) where M

and L

satisfy

N →∞

lim M

= ∞, lim

N →∞

M

N = 0,

M →∞

lim L

= ∞, lim

M →∞

L

M = 0.

We consider the following estimator:

f

(λ

, λ

) =

π

R

−π π

R

−π

W

(λ

− u

)W

(λ

− u

) b I

(u

, u

) du

du

. As in [6], for giving the best rate of convergence of this estimator we intro- duce on φ two hypotheses (h

) and (h

) called regularity hypotheses:

(h

) φ(λ

− u

, λ

− u

) = φ(λ

, λ

) + R

(λ

, λ

, u

, u

)

with |R

(λ

, λ

, u

, u

)| ≤ C

k(u

, u

)k

, where 0 < γ ≤ 1, (h

) φ(λ

− u

, λ

− u

) = φ(λ

, λ

) + ∂φ

∂x (λ

, λ

)u

+ ∂φ

∂y (λ

, λ

)u

+ R

(λ

, λ

, u

, u

)

with |R

(λ

, λ

, u

, u

)| ≤ C

k(u

, u

)k

, where 1 ≤ γ < 2, C

and C

being nonnegative constants.

Theorem 3.1. Let λ

, λ

∈ A. Then:

(i) f

(λ

, λ

) is an asymptotically unbiased estimator of the quantity [φ(λ

, λ

)]

.

(ii) Choosing k so large that γ + 1 < 2kα, we have E[f

(λ

, λ

)] − [φ(λ

, λ

)]

=



 



 

 O

 1

M

+ 1 (L

)



if φ satisfies (h

), O

 1 M

+ 1 L



if φ satisfies (h

).

P r o o f. Using (2.2), we have E[f

(λ

, λ

)] =

MN(λ1+π)

R

MN(λ1−π)

LM(λ2+π)

R

LM(λ2−π)

W (u)W (v)

× [ψ

(λ

− u/M

, λ

− v/L

)]

du dv.

Since W is vanishing for |λ| > 1 and p/α < 1, for N and M large enough we obtain

|E[f

(λ

, λ

)] − [φ(λ

, λ

)]

|

≤

1

R

−1 1

R

−1

W (u)W (v)

× |ψ

(λ

− u/M

, λ

− v/L

) − φ(λ

, λ

)|

du dv.

Using the facts that H

and H

are 2π-periodic, |H

|

and |H

|

are two kernels, and φ is uniformly continuous on [−π, π]

, we find that

(3.1) lim

N →∞

M →∞

I

(λ

− u/M

, λ

− v/L

) − φ(λ

, λ

) = 0.

On the other hand,

J

(λ

− u/M

, λ

− v/L

)

≤

q

X

j=1

a

B

B

× 1

 sin 

2

(λ

− u/M

− w

)  

2kα

  sin 

2

(λ

− v/L

− w

)  

2kα

. Since λ

6= w

and λ

6= w

for j ∈ {1, . . . , q}, using Lemma 2.1 we obtain (3.2) J

(λ

− u/M

, λ

− v/L

) = O

 1

n

m

 .

Considering all possible cases, and partitioning the integrals, we show easily (see [13]) that

(3.3) K

(λ

− u/M

, λ

− v/L

)

= O

 1

m

+ 1

n

+ 1

n

m

 . Using the inequality

|x

− y

| ≤ r

2 (x

+ y

)|x − y|, x, y ∈ R

, r ∈ [0, 1] ∪ [2, ∞[, and the equalities (3.2), (3.3) we can show that

[ψ

(λ

− u/M

, λ

− v/L

)]

+ [φ(λ

, λ

)]

is bounded by a constant, for N and M large enough. We get

|E[f

(λ

, λ

)] − [φ(λ

, λ

)]

| ≤ const

1

R

−1 1

R

−1

W (u)W (v)

×|ψ

(λ

− u/M

, λ

− v/L

) − φ(λ

, λ

)| du dv.

1) If φ satisfies the hypothesis (h

) then we have

|ψ

(λ

− u/M

, λ

− v/L

) − φ(λ

, λ

)|

≤ C

λ1−u/MN+π

R

λ1−u/MN−π

λ2−v/LM+π

R

λ2−v/LM−π

|H

(t)H

(t

)|

× (|u/M

+ t| + |v/L

+ t

|)

dt dt

+ J

(λ

− u/M

, λ

− v/L

) + K

(λ

− u/M

, λ

− v/L

).

Using twice the inequality

(3.4) |x + y|

≤  |x|

+ |y|

if 0 < r ≤ 1, 2

(|x|

+ |y|

) if r ≥ 1, we show that

1

const |E[f

(λ

,λ

)][φ(λ

, λ

)]

|

≤ C

M

1

R

−1

W (u)|u|

du

+ C

R

−1

W (u)

λ1−u/MN+π

R

λ1−u/MN−π

|H

(t)|

|t|

dt du

+ C

L

1

R

−1

W (v)|v|

dv

+ C

R

−1

W (v)

λ2−v/LM+π

R

λ2−v/LM−π

|H

(t

)|

|t

|

dt

dv

+

1

R

−1 1

R

−1

W (u)W (v)

× [J

(λ

− u/M

, λ

− v/L

) + K

(λ

− u/M

, λ

− v/L

)] du dv.

On the other hand, since H

is an even function, we have

λ1−u/MN+π

R

λ1−u/MN−π

|H

(t)|

|t|

dt

≤

π

R

−π

|H

(t)|

|t|

dt + (2π)

|λ1|+|u/MN|+π

R

π

|H

(t)|

|t|

dt.

Using Lemma 2.1, for N large enough we have (3.5)

|λ1|+|u/MN|+π

R

π

|H

(t)|

|t|

dt = O(T

(λ

)), where

T

(λ

) =



 

  1

n

if λ

6= 0, 1

M

n

if λ

= 0.

We gather the results (3.2), (3.3) and (3.5) to obtain

|E[f

(λ

, λ

)] − [φ(λ

,λ

)]

|

= O



T

(λ

) + T

(λ

) + 1

M

+ 1 L

+ 1

n

+ 1 m

+ 1

n

+ 1

m

+ 1

n

m

 . Since 0 < γ ≤ 1 and γ + 1 < 2kα, it is clear that

|E[f

(λ

, λ

)] − [φ(λ

, λ

)]

| = O

 1

M

+ 1 L

 .

2) If φ satisfies the hypothesis (h

), using the facts that H

, H

are 2π-periodic kernels, and (3.5), (3.4) we get

1

const |E[f

(λ

, λ

)] − [φ(λ

, λ

)]

|

≤    

∂φ

∂x (λ

, λ

)    

1

R

−1

|u/M

|W (u) du

+    

∂φ

∂y (λ

, λ

)    

T

(λ

)

+    

∂φ

∂x (λ

, λ

)    

T

(λ

) +    

∂φ

∂y (λ

, λ

)    

1

R

−1

|v/L

|W (v) dv

+ 2

C

M

1

R

−1

|u|

W (u) du + 2

C

L

1

R

−1

|v|

W (v) dv

+ 2

C

λ1−u/MN+π

R

λ1−u/MN−π

|H

(t)|

|t|

dt

+ 2

C

λ2−v/LM+π

R

λ2−v/LM−π

|H

(t

)|

|t

|

dt

+

1

R

−1 1

R

−1

W (u)W (v)

× [J

(λ

− u/M

, λ

− v/L

)

+ K

(λ

− u/M

, λ

− v/L

)] du dv.

We use again the results (3.2), (3.3) and (3.5) to obtain

|E[f

(λ

, λ

)] − [φ(λ

, λ

)]

|

= O

 1 M

+ 1 L

+ 1 L

+ 1

M

+ T

(λ

) + T

(λ

) + 1

n

+ 1

m

+ 1

n

+ 1

m

+ 1

n

m

 . Since 1 < γ ≤ 2, we have 1/k < (γ + 1)/(2k) < α. Thus, we obtain

|E[f

(λ

, λ

)] − [φ(λ

, λ

)]

| = O

 1 M

+ 1 L

 . Now we show the following lemma which will be used in the sequel.

Lemma 3.2. Let (λ

, λ

) and (λ

, λ

) belong to A. Write Q

(λ

, λ

, λ

, λ

)

=

π

R

−π π

R

−π

|H

(λ

− v

)H

(λ

− v

)

× H

(λ

− v

)H

(λ

− v

)|

dµ(v

, v

).

Then

Q

(λ

, λ

, λ

, λ

)

≤ O

 1

n

+ 1

m

+ 1

n

m



+  sup(φ) π

 π 2



1 (nm)



<(δ, λ

, λ

)<(δ

, λ

, λ

), where the function < is defined by

<(x, y, z) = π

sin

+ 2x

inf 

sin

, sin

 , with two real numbers δ, δ

satisfying

0 < δ < inf[π − λ

; π + λ

; |λ

− λ

|/2], 0 < δ

< inf[π − λ

; π + λ

; |λ

− λ

|/2].

P r o o f. First using the expression of dµ in (1.1) and the inequality x

y

≤

(x

+ y

), we obtain

Q

(λ

, λ

, x

, x

) ≤ B +

J

(λ

, λ

)

J

(λ

, λ

)

+

K

(λ

, λ

) +

K

(λ

, λ

),

where

B =

π

R

−π π

R

−π

|H

(λ

− v

)H

(λ

− v

)

× H

(λ

− v

)H

(λ

− v

)|

φ(v

, v

) dv

dv

. Since φ is bounded on [−π, π]

, we have

B ≤ sup(φ)

 R

−π

|H

(λ

− v

)H

(λ

− v

)|

dv



×  R

−π

|H

(λ

− v

)H

(λ

− v

)|

dv

 .

We split the first integral into five integrals: over a neighbourhood of λ

, a neighbourhood of λ

and the remainders:

π

R

−π

|H

(λ

− v

)H

(λ

− v

)|

dv

= 1

B



λ1,1−δ

R

−π

+

λ1,1+δ

R

λ1,1−δ

+

λ1,2−δ

R

λ1,1+δ

+

λ1,2+δ

R

λ1,2−δ

+

π

R

λ1,2+δ

    

sin 

2

(λ

− w) 

sin

· sin 

2

(λ

− w)  sin

kα

dw.

The first, third and last integrals are respectively bounded by 1

B

· λ

− δ + π

sin

, 1

B

· λ

− λ

− 2δ sin

and

1

B

· π − λ

− δ sin

. Using the inequality sin(nx) ≤ n sin x, we obtain

1 B

λ1,1+δ

R

λ1,1−δ

sin 

2

(λ

− w)  sin

sin 

2

(λ

− w)  sin

kα

dw

≤ 1

B

· 2δn

inf 

sin

, sin

 .

Similarly, the remaining integral is bounded by the same quantity. Thus

using Lemma 2.1, we have

π

R

−π

|H

(λ

− v

)H

(λ

− v

)|

dv

= 1 2π

 π 2



1 n

×

 2π

sin

+ 4δn

inf 

sin

, sin



 . Just as before, we have

π

R

−π

|H

(λ

− v

)H

(λ

− v

)|

dv

≤ 1 π

 π 2



1

m

<(δ

, λ

, λ

).

Equalities (3.2) and (3.3) give the result of this lemma.

Theorem 3.3. Let (λ

, λ

) ∈ A be such that φ(λ

, λ

) 6= 0. Then:

(i) Var[f

(λ

, λ

)] converges to zero.

(ii) If φ satisfies (h

) or (h

), and M

= n

and L

= m

, where c and c

are two real numbers satisfying

inf  2k

α

+ 1

6α

k

, kα + 2 3(kα + 1)



< c, c

< 1 2 , then

Var[f

(λ

, λ

)] = O

 1

n

+ 1 m

 .

P r o o f. When φ(λ

, λ

) = 0 we do not need to smooth b I

, since its variance tends to zero. We have

Var[f

(λ

, λ

)]

=

π

R

−π π

R

−π π

R

−π π

R

−π

W

(λ

− u

)W

(λ

− u

)W

(λ

− u

)W

(λ

− u

)

× Cov[b I

(u

, u

), b I

(u

, u

)] du

du

du

du

. Putting

x

= M

(λ

− u

), x

= M

(λ

− u

),

x

= L

(λ

− u

), x

= L

(λ

− u

).

and using the fact that W is vanishing for |λ| > 1, for N and M large enough

we have

Var[f

(λ

, λ

)]

=

1

R

−1 1

R

−1 1

R

−1 1

R

−1

W (x

)W (x

)W (x

)W (x

)

× Cov[b I

(λ

− x

/M

, λ

− x

/L

),

I b

(λ

− x

/M

, λ

− x

/L

)] dx

dx

dx

dx

. We define

L

= {(x

, x

) ∈ [−1, 1]

: |x

− x

| > σ

}, L

= {(x

, x

) ∈ [−1, 1]

: |x

− x

| > σ

},

L

= {(x

, x

, x

, x

) ∈ [−1, 1]

: |x

− x

| ≤ σ

or |x

− x

| ≤ σ

}, where σ

, σ

are two nonnegative real sequences converging to 0. We split the last integral into the integral over L

and the integral over L

and L

:

Var[f

(λ

, λ

)] = R R R R

L3

+ R R

L1

R R

L2

=: J

+ J

.

Using (2.3) and the uniform (in x

, x

) convergence of ψ

(λ

− x

/M

, λ

− x

/L

) to φ(λ

, λ

), we obtain

J

≤ const h R R

|x2−x⁰₂|≤σ⁰_M

W (x

)W (x

) dx

dx

+ R R

|x1−x⁰₁|≤σN

W (x

)W (x

) dx

dx

i .

Thus, J

≤ const[sup(W )]

[σ

+ σ

]. Hence J

tends to zero.

It remains to show that J

tends to zero. First we define for simplicity λ

= λ

− x

/M

, λ

= λ

− x

/M

,

λ

= λ

− x

/L

, λ

= λ

− x

/L

, and

C(λ

, λ

)

= Cov[ b I

(λ

− x

/M

, λ

− x

/L

), b I

(λ

− x

/M

, λ

− x

/L

)].

We use the equality

|x|

= D

∞

R

−∞

1 − cos(xu)

|u|

du = D

Re

∞

R

−∞

1 − e

|u|

du, for all real x and p ∈ ]0, 2[. We have

(3.6) I b

(λ

, λ

) = 1 F

C

Re

∞

R

−∞

1 − e

|u|

du.

From (2.1) we obtain

(3.7) E b I

(λ

, λ

) = 1 F

C

∞

R

−∞

1 − exp{−C

|u|

ψ

(λ

, λ

)}

|u|

du.

By using (3.6) and (3.7) we show easily that

C(λ

, λ

) = F

C

∞

R

−∞

∞

R

−∞

 E

h Y

k=1

cos(u

d

(λ

, λ

)) i

− exp n

−C

X

k=1

|u

|

ψ

(λ

, λ

)

o du

du

|u

u

|

. From the equality 2 cos x cos y = cos(x + y) + cos(x − y), we have

E h Y

k=1

u

d

(λ

, λ

) i

= 1 2 exp h

−C

R 

2

X

k=1

u

H

(λ

− v

)H

(λ

− v

)   

α

dµ(v

, v

) i

+ 1 2 exp h

−C

R 

2

X

k=1

(−1)

u

H

(λ

− v

)

×H

(λ

− v

)   

α

dµ(v

, v

) i . Substituting this expression in C(λ

, λ

) and changing the variable u

to

−u

in the second term, we obtain C(λ

, λ

) = F

C

∞

R

−∞

∞

R

−∞

(e

− e

) du

du

|u

u

|

, where

K = C

R

−π π

R

−π

2

X

k=1

u

H

(λ

− v

)H

(λ

− v

)   

α

dµ(v

, v

),

K

= C

X

k=1

|u

|

π

R

−π π

R

−π

|H

(λ

− v

)H

(λ

− v

)|

dµ(v

, v

)

= C

X

k=1

|u

|

ψ

(λ

, λ

).

Since K, K

> 0, we have

|e

− e

| ≤ |K − K

|e

. Using the inequality

kx + y|

− |x|

− |y|

| ≤ 2|xy|

, x, y ∈ R and 1 ≤ α ≤ 2, we obtain

|K − K

| ≤ 2C

|u

u

|

Q

(λ

, λ

, λ

, λ

).

Therefore, C(λ

, λ

)

≤ F

C

2C

Q

(λ

, λ

, λ

, λ

)

∞

R

−∞

∞

R

−∞

e

|u

u

|

du

du

. Now we have

(3.8) |K − K

| − K

≤ −C

2

X

k=1

|u

|

[ψ

(λ

, λ

) − Q

(λ

, λ

, λ

, λ

)].

Let δ(N ), δ

(M ) be two real numbers depending respectively on N and M such that

0 < δ(N ) < inf[|λ

− λ

|/2; π − λ

; π + λ

], 0 < δ

(M ) < inf[|λ

− λ

|/2; π − λ

; π + λ

].

Moreover, we suppose that

(3.9) lim

N →∞

δ(N )M

σ

= 0, lim

M →∞

δ

(M )L

σ

= 0.

Using Lemma 3.2 and the inequality

(3.10) sin(x/2) ≥ x/π, 0 ≤ x ≤ π, we obtain

π

R

−π

|H

(λ

− v)H

(λ

− v)|

dv

≤ 1 π

 π 2





π

n

δ(N )

+ 2δ(N )(2π)

n

(σ

/M

)



.

We choose δ(N ) = n

, δ

(M ) = m

with β > 0 and β

> 0. In order to

have