ESTIMATION OF ADDITIVE ERROR IN MIXED SPECTRA
FOR STABLE PROCESSES
Rachid Sabre1
Laboratory LE2I UMR6306, CNRS, University of Bourgogne, Agrosup, Department DSIP, Dijon, France.
1. INTRODUCTION
In this paper, the class of symmetric alpha stable processes have been considered. It is a particular family of processes with infinite energy. Theory of these processes have been covered in numerous papers including Cambanis (1983), Cambanis and Mae-jima (1989), Marcus and Shen (1989), Masry and Cambanis (1984), Samorodnitsky and Taqqu (1994), Panki and Renming (2014), Zhen-Qing and Longmin (2016) to name a few. Symmetric alpha processes are considerably accurate models for many phenomenons in several fields such as: physics, biology, electronic and electric, hy-drology, economies, communications and radar applications, see Sousa (1992), Shao and Nikias (1993), Nikias and Shao (1995), Kogon and Manolakis (1996), Azzaouiet al.(2002), Montillet and Kegen (2015), Pereyra and Batalia (2012), Zhong and Premku-mar (2012), Ligang and Zidong (2015), Pankiet al.(2017), Briceet al. (2017).
In this work, a stationary symmetricαstable harmonizable process is precisely discussed,Z={Zn:n∈Z}. AlternativelyZhas the integral representation
Zn=
Zπ
−π
exp[i(nλ)]dξ(λ),
where 1< α <2 andξ is a complex valued symmetricα-stable random measure onR
with independent and isotropic increments. The measure defined bym(A) =|ξ(A)|αα (see Masry and Cambanis, 1984) is called "control" measure or spectral measure. Sup-pose that this measure is absolutely continuous with respect to Lebegue measure: md(x) = φ(x)d x. The functionφis called the spectral density. The spectral den-sity function was already estimated by Masry and Cambanis (1984), when the time of the process is continuous, by Sabre (1995) when the time of the process is discrete and by Sabre (2012) when the time of the process is p-adic.
In this work, we consider the case where can not be observed this process without an unknown constant error. The processXn=a+Znis observed instead of the process Z alone.
Our goal is to estimate this constanta, when the spectral measure is the sum of an absolutely continuous measure with respect to Lebesgue measure and a discrete measure
dµ(λ) =φ(x)d x+
q X
i=1
ciδw
i,
whereδ is a Dirac measure, φis nonnegative integrable and bounded function, ci is unknown positive real number and wi is unknown real number. Assume that wi 6= 0. We study the case where the spectral density is zero at origin, particularly atφ(λ) =sin2kαλ2g(λ)andφ(λ) =|λ|βg(λ). We show that the rate of convergence is improved in accordance with the value ofβ.
This paper is organized as follows. Section 2 gives some definitions and proprieties of symmetric stable processes and an estimator of the constantais defined. We show that this estimate converges in probability toa. Then we show that the estimate con-verges toainLp(p< α)which will replace the convergence in mean square because the second moment of the processes is infinite. In Section 3, the spectral density of Z is assumed vanishing at origin precisely:φ(λ) =|λ|βg(λ). We improve the rate of convergence in accordance with the values ofβ. Section 4 is reserved to numerical studies. Section 5, Appendix, is reserved to prove the elementary results.
2. THE ESTIMATE OF THE CONSTANTa
First, are introduced some basic notation and properties used throughout the paper. A random variableXis symmetricα-stable (SαS), 0< α <2, if its characteristic function is defined by
φX(θ) =e−σα|θ|α,
whereαis the characteristic exponent andσis the dispersion of the distribution. By lettingαtake the values 1 and 2, we obtain two important special cases of (SαS) distri-bution, namely Cauchy distribution and Gaussian distribution.
The random variablesX1, . . . ,Xd are jointly (SαS) if there is a single positive mea-sureΓX(d) onSd, unit sphere ofRd, where its characteristic function is of the form
φX(θ1, . . . ,θd) =exp §
−
Z
Sd
|θ1s1+. . .+θdsd|αdΓX(d)(s1, . . . ,sd)
ª
.
WhenX1andX2are jointly (SαS), the covariation of(X1,X2)is defined in Camba-nis (1983) by
[X1,X2]α=
Z S2
s1.(s2)<α−1>dΓX
1,X2(s1,s2),
where s<β> = sign(s).|s|β. This covariation plays the same role as the covariance because the moment of second order is infinity.
From the definition of the covariation Schilder (1970) defined the following norm on the linear space of (SαS) random variables
The processξ = (ξt,t∈R)is symmetricα-stable if all linear combinationsPni=1λiξt
i
are (SαS) variables.
This paper considers a (SαS) process where its spectral representation is
ZN=
Zπ
−π
eiNλdξ(λ),
whereξ is a isotropic symmetricα-stable with independent increments.
The measure defined byµ(]s,t]) =|ξ(t)−ξ(s)|ααis Lebesgue-Stiel measure called the spectral measure(see Cambanis, 1983; Masry and Cambanis, 1984). Whenµis ab-solutely continuous dµ(x) = f(x)d x, the function f is calledthe spectral densityof the processZ.
As in Demesh (1988), Sabre (1994) and Sabre (1995), we give the definition of the Jack-son polynomial kernel.
Let Z1, . . . ,ZN observations of the process Z: Z(n)
0≤n≤N−1, where N satisfies:
N−1=2k(n−1) with n∈N k∈N∪ ¦1
2 ©
if k=1
2 then n=2n1−1,n1∈N.
The Jackson’s polynomial kernel is defined by: |HN(λ)|α= ANH
(N)(λ)
α where
H(N)(λ) = 1 qk,n
sin(n2λ)
sin(λ2)
!2k
with qk,n= 1 2π
Zπ
−π sinn2λ
sinλ2
!2k
dλ.
In addition, we have AN= (Bα,N)−α1 with B
α,N= Zπ
−π
H
(N)(λ)
α dλ.
We give the following lemmas which are used in the reminder of this paper. Their proof are given in Section 5.
LEMMA1. There is a non negative function hksuch as
H(N)(λ) = k(n−1)
X m=−k(n−1)
hkm n
cos(mλ).
LEMMA2. Let Bα0,N=
Zπ
−π
sinn2λ
sinλ2
2kα
dλ and JN,α=
Zπ
−π|
u|γ|HN(λ)|αdλ,
whereγ∈]0, 2].
Then, B0
α,N
≥2π
2 π
2kα
n2kα−1 if 0< α <2
≤ 4πkα 2kα−1n
2kα−1 if 1
and JN,α≤
πγ+2kα 22kα(γ−2kα+1)
1
n2kα−1 if
1 2k < α <
γ+1 2k , 2kαπγ+2kα
22kα(γ+1)(2kα−γ−1) 1 nγ if
γ+1
2k < α <2.
In this paper, we propose an estimate of the constant erroradefined by
ˆ
a= AN
HN(0)
k((n−1)
X n0=−k(n−1)
hk
n0 n
X n0+k(n−1)
. (1)
THEOREM3. Let p a real number such that0< p< α. Then
|aˆ−a|p=O
1
npα
.
PROOF. From the spectral representation of the process, the estimator proposed
becomes
ˆ
a= AN
HN(0)
k((n−1)
X n0=−k(n−1)
hk
n0 n
Zπ
−π e x p
i [n0+k(n−1)]λ
dξ(λ) +a
.
Using Cambanis (1983), the characteristic function of(aˆ−a)can be written as
Eexp[iℜe r(aˆ−a)] =exp
−Cα|r|α
Zπ
−π
AN HN(0)
k((n−1)
X n0=−k(n−1)
hk
n0 n
ei n0λ
α dξ(λ)
.
wherer=r1+i r2. It is easy to show that
Eexp[iℜe r(aˆ−a)] =exp(−Cα|r|αψN),
whereψN=ψN,1+ψN,2with
ψN,1= Zπ
−π
|HN(λ)|α
|HN(0)|αφ(λ)dλandψn,2=
q X
i=1
ci|HN(wi)| α
|HN(0)|α .
The functionφbeing bounded on[−π,π]and|HN(.)|αbeing a kernel, it can be shown
that
Zπ
−π|
HN(λ)|αφ(λ)dλis converging toφ(0). On the other hand, from Lemma 2,
we have
1 |HN(0)|α =
B0 α,N
n2kα =O
1
n
. (2)
ThereforeψN,1converges to 0.
ψn,2≤ q X
i=1
ci B0
α,N
1
sin
1 2wi
2kα B0
α,N
Therefore
ψN,2=O 1
n2kα
.
Thus
ψN=O 1
n
.
Consequently, the characteristic function of ˆa−aconverges to 1 whenNapproaches infinity. Hence we have the convergence in probability of ˆatoa.
We study now the convergence of ˆatoainLpwhere 0< p< α, which replaces the convergence in mean square, because the second order moment ofX is infinity.
Let
Dp=ℜe
Z∞
−∞
Z π4
−π4
1−ei rcosθ |r|1+p d r dθ.
Lett=r0eiθ0 andx=%eiτ0.
Assuming nowr="r0, τ=τ0−τ
0
Dp=ℜe
Z∞
−∞
Z π4+τ0
−π4+τ0
1−ei"r0cos(τ0−τ 0)
|"|1+p|r0|1+p "d r
0dθ0
Dp|x|p=ℜe
Z∞ 0
Z π4+τ0
−π4+τ0
1−eiℜe(t x¯)
|t|1+p d|t|dθ
0− ℜeZ0 −∞
Z π4+τ0
−π4+τ0
1−eiℜe(¯t x)
|t|1+p d|t|dθ
0.
Replacing in this formulaxby ˆa−a, from the previous result we have
DpE|x|p =
Z∞
−∞
Z π4+τ0
−π4+τ0
1−e−Cα|t|αψN
|t|1+p d|t|dθ
0
= π 2
Z∞
−∞
1−e−Cα|t|αψN
|t|1+p d t.
Letu=t[ψN]1α and using (2), we obtain
2
πCp,αE|aˆ−a|p= (ψN)
p
α =O
1
np/α
, (3)
where
Cp,α=RpFp−,α1(Cα)−αp
with
Rp=
Z 1
−cos(u)
|u|1+p d uandFp,α=
Z 1
−e−|u|α |u|1+pα
3. IMPROVEMENT OF THE RATE OF CONVERGENCE
In this section, we study the cases where the spectral density is zero at the origin. We prove that the convergence rate of the estimator ˆawill be improved.
THEOREM4. Assume that the spectral density is satisfying
φ(λ) =|λ|βg(λ),
whereβ∈]0, 2kα−1[,λ∈[−π,π]and g(λ)is a bounded function on[−π,π], contin-uous in neighborhood of0and g(0)6=0. Then
24k pL≤ lim
N→∞n
p(β+1)
α E|aˆ−a|p≤π4k pL,
where L is the following constant
L= π
2Cp,α
g(0)
Z∞
−∞
sinu2
2kα
|u|2kα−β d u
p α
.
PROOF. From (2), the functionψNcan be written as
ψN=n−2kα Zπ −π
sinn2λ
sinλ2
2kα
|λ|βg(λ)dλ+n−2kα
q X
i=1
ci
sinnwi 2
sinwi
2
2kα .
Using the following inequality
sinx 2 ≥ x
π 0≤x≤π, (4)
we maximizeψNas follows
ψN≤π4kαn−2kα Zπ −π sin nλ 2
2kα
|λ|2kα−β g(λ)dλ+n−
2kαXq i=1
ci 1 sinwi
2
2kα .
Puttingnλ=u, we have
ψN≤π4kαn−1−β Z∞ −∞ sin u 2
2kα
|u|2kα−β g
u
n
d u+n−
2kα+1+β
π4kα
q X
i=1
ci 1 sinwi
2
2kα .
On the other hand, using Lebesgue’s dominated convergence theorem, we show that
lim
N→∞
Z∞
−∞
sinu2
2kα
|u|2kα−β g
u
n
d u+n−
2kα+1+β
π4kα
q X
i=1
ci 1 sinwi
2
2kα =g(0)
Z∞
−∞
sinu2
2kα
Lemma 2 gives
lim
N→∞n
p(β+1)
α (ψ
N)
p
α ≤π4k p g(0)
Z+∞
−∞
sinu2
2kα
|u|2kα−β d u
!pα
.
ThusψN converges to zero. Using the following inequality
|sinx| ≤ |x| ∀x∈[−π,π], (6)
we obtain
ψN≥24kαn−2kα
Zπ −π sin nλ 2
2kα
|λ|2kα−β g(λ)dλ+n−
2kαXq i=1
ci
sinnwi 2
sinwi 2
2kα ,
ψN≥24kαn−β−1
Zπ
−π
sinu2
2kα
|u|2kα−β g( u
n)d u+Rn
,
whereRn=n−22kα+β+4kα 1P
q i=1ci
sin[nwi2 ] sin[wi2]
2kα
. SinceRnconverges to zero, the equality (5)
gives
lim
N→∞n
p(β+1)
α (ψ
N)
p
α ≥24k p g(0)
Z+∞
−∞
sinu2
2kα
|u|2kα−β d u
!αp
.
The first equality of (3) reaches the result of this theorem.
THEOREM5. Assuming that the spectral density satisfies
φ(λ) =sin2kα
λ
2
g(λ),
where the function g is integrable on[−π,π]and g(0)6=0. Then
c t e π 2Cp,α
Zπ
−π g(λ)dλ
αp
≤ lim
N→∞n
2pkE|aˆ−a|p≤
π
2Cp,α
Zπ
−π
g(λ)dλ+
q X
i=1
ci 1 sinwi
2
2kα!αp
.
PROOF. From the definition ofψNand (2), we have
ψN=n−2kα Zπ −π
sinnλ 2
2kα
g(λ)dλ+n−2kα
q X
i=1
ci
sinnwi 2
sinwi 2
2kα .
As 1≤kαand the sinus function is smaller than 1, the next expression is connected to
ψN≤n−2kα Zπ −π sin nλ 2 2
g(λ)dλ+n−2kα
q X
i=1
ci
sinnwi 2
sinwi
2
ψN≤n−2kα
Zπ
−π
g(λ)dλ+
q X
i=1
ci
sinnwi 2
sinwi 2
2kα .
So, from Lemma 2, we have
lim
N→∞ψNn
2kα≤Zπ −π
g(λ)dλ+
q X
i=1
ci 1 sinwi
2
2kα .
Using the fact that the sinus function is between−1 and 1 and thatkα <[kα] +1 where[kα]represents the integer part ofkα, we obtain
ψN≥n−2kα Zπ
−π
sinnλ 2
2[kα]+1
g(λ)dλ+n−2kαXq i=1
ci
sinnwi
2
sinwi 2
2kα
ψN≥n−2kα
n−1
2B0 α,N
Zπ
−π
1−cosnλ
2
[kα]+1
g(λ)dλ+n−2kαXq i=1
ci
sinnwi
2
sinwi 2
2kα .
The binomial formula gives
2[kα]+1ψN ≥ n−2kα
Zπ
−π g(λ)dλ
+ n−2kα
[kα]+1 X
r=1
(−1)rC[rkα]+1
Zπ
−π
cosr(nλ)g(λ)dλ
+ n−2kα
q X
i=1
ci
sinnwi 2
sinwi 2
2kα .
We know that
Zπ
−π
cosr(nλ)g(λ)dλ is converging to a constant. Indeed, using the
binomial formula, we obtain
cosr(nλ)=
ei nλ+e−i nλ 2 r = 1 2r r X
j=0
Crjei j nλe−i(r−j)nλ.
Hence
Zπ
−π
cosr(nλ)g(λ)dλ = 1 2r
r X
j=0
Crj
Zπ
−π
ei(2j−r)nλ
= 1 2r
r X
j=0
Crj
Zπ
−π
cos((2j−r)nλ)g(λ)dλ.
The right side of the last equality converges to 1 2r
Zπ
−π
g(λ)dλwhen r is even, and
converges to 0 whenris odd. Consequently,
lim
N→∞
[kα]+1 X r=1
(−1)rC[rkα]+1
Zπ
−π
[kα]+1 2 X
p=1
(−1)2pC[2kpα]+1 1 22p
Zπ
−π
g(λ)dλ if [kα] +1 is even
[kα]
2 X
p=1
(−1)2pC[2kpα]+1 1 22p
Zπ
−π
g(λ)dλ else.
Since limN→∞n−2kαPq i=1ci
sin[nwi2 ] sin[wi2]
2kα
= 0. The similar arguments are used for showing that
lim
N→∞(ψN)
p
αn2k p≥c t e
Zπ
−π g(λ)dλ
pα
. (7)
THEOREM6. Assume that spectral density satisfies
φ(λ) =|λ|βg(λ),
whereβ >2kα−1and g is measurable positive function bounded on[−π,π]. Then
c t e×R≤ lim
N→∞n
2k pE|aˆ−a|p≤R,
where R= π
2Cp,α
Zπ
−π |λ|β
sin
λ
2
2kαg(λ)dλ
p α
.
PROOF. Define the continuous functionlas follows
l(λ) =
π2kα if |λ|> π
|λ|2kα
sin
λ
2
2kα if 0<|λ| ≤π
22kα ifλ=0. The functionψNcan be written as
ψN=n−2kα Zπ
−π
sinn2λ
sinλ2
2kα
sin2kα
λ
2
h(λ)dλ+n−2kα
q X
i=1
ci
sinnwi 2
sinwi 2
2kα ,
whereh(λ) =l(λ)|λ|β−2kαg(λ). We know that the functionhis integrable on[−π,π]. Indeed, since the function g is bounded, we get
Zπ
−π
h(λ)dλ≤sup(g)
Zπ
−π
l(λ)|λ|β−2kαdλ.
Using the inequality (6), we obtainl(λ)≤π2kα. Thus
Zπ
−π
h(λ)dλ≤2π2kαsup(g)
Zπ
0
(λ)β−2kαdλ.
THEOREM7. Assume that the spectral density has the following form:φ(λ) =|λ|βg(λ) whereβ <2kα−1and g is a continuous positive function on[−π,π]. Then
Q×22k p≤ lim
N→∞n
(β+1)p
α |E|aˆ−a|p≤π2k p×Q,
where Q=
g(0)
Z∞ −∞
sinu1 2
2kα
|u1|β1−2kαd u 1
pα
.
PROOF. From the definition of the kernel|HN(λ)|α and the inequality (6), we obtain
ψN≤π
2kαn−1
B0 α,N
Zπ −π
sinnλ 2
2kα
|λ|β−2kαg(λ)dλ+n−2kα
q X
i=1
ci
sinnwi 2
sinwi 2
2kα .
Puttingnλ=u, we obtain
ψN≤π
2kαn2kα−β1−2
B0 α,N
Z∞ −∞ sinu 2
2kα
|u|β−2kαg(u
n)d u+n −2kαXq
i=1
ci 1 sinwi
2
2kα .
whenN approaches infinity, we obtain
lim
N→∞n β+1ψ
N≤π2kα Z∞ −∞ sin u 2
2kα
|u|β−2kαg(0)d u.
Similarly, from the inequality (6) we minimizeψNas follow
nψN≥ 2
2kα
B0 α,N
Zπ −π sin nλ 2
2kα
|λ|β−2kαdλ+Qn,
whereQn=n−2kα+1Pq i=1ci
sin[nwi2 ] sin[wi2]
2kα
. SinceQnconverges to zero and the function
λ−→g(λ)is continuous, Lemma 2 gives
lim
N→∞n β+1ψ
N≥22kα Z∞ −∞
sinu1 2
2kα
|u1|β−2kαg(0)d u.
Thus from (3) we obtain the result of this theorem.
4. NUMERICAL STUDIES
their arrival times are modeled by Poisson distributions independent to the direction (isotropic). From the series representation theory shown by Janicki and Weron (1993), the sum of the arrival times can be modeled by a stable harmonizable processZn as the process defined by (8). It is supposed that these signals are observed in an aquatic environment causing a constant disturbance. The signal arrival delay model in this case can be considered to be affected by a constant error (Xn=Zn+a).
Throughout this section, we give the simulation of the studied process
Zn=
Zπ
−π
ei nλdξ(λ), (8)
where 1< α <2 andξ is a complex symmetricα-stable measure onRwith
indepen-dent and isotropic increments and with control measuremsuch thatmd x=φ(x)d x. For that, we use the series representations given by Janick and Weron (1993). Therein the authors have shown that the processZdefined by (8) can be expressed as follows
Zn=Cα
Z
φ(x)d x
1αX∞
k=1 "kΓ
−1α
k e
i nVkeiθk, (9)
where
• "kis a sequence of i.i.d. random variables such asP["k=0] =P["k=1] = 1 2,
• Γk is a sequence of arrival times of Poisson process,
• Vk is a sequence of i.i.d. random variables independent of"k and ofΓk having the same distribution of control measurem, which has probability densityφ,
• θkis a sequence of i.i.d. random variables that have the uniform distribution on [−π,π], independent of"k,Γk andVk.
To generateNvalues(N=101, 501, 1001)of the processZn, we use the following steps:
• generate 2000 values of"k,
• generate 2000 values ofΓk,
• generate 2000 values ofVk,
• generate 2000 values ofθk.
Then we calculate for all 0≤n≤N
Zn=Cα
Z
φ(x)d x
1α2000 X k=1
"kΓ
−1α
k e
i nVkeiθk, (10)
We calculate the estimator ˆagiven in (1) for different sizes of sampleN=101, 501, 1001. The result is given in Table 1.
Comparing ˆato 30 (value of the constanta), we find that the estimator ˆa increas-ingly approaching the constant errorawhen the sample size is large. And the approx-imation is better when the parameterβof the spectral density is greater than 2kα−1 compared to the case whereβis smaller than 2kα−1.
TABLE 1
Values of the estimator for two values ofβone is smaller than2kα−1other is bigger than
2kα−1.
2kα−1=12.6 β=0.21 β=38 N=101 aˆ=20.30 aˆ=36.45 N=501 aˆ=35.10 aˆ=31.03 N=1001 aˆ=28.89 aˆ=30.28
5. APPENDIX
PROOF OF LEMMA1. For all nonnegative integer,n, we have
n−1 X s=0
ei sλ=sin(
nλ
2)
sin(λ2)e
i(n−1)λ
2 .
Therefore we can writeH(N)(λ) = 1 qk,n
n−1 X s=0
ei sλ
2k
e−i k(n−1)λ
HenceH(N)(λ) =e−
i k(n−1)λ
qk,n
X t∈S2k
exp
i(t1+t2+· · ·+t2k)λ
, where
S2k=
§
t= (t1,t2,· · ·,t2k)∈Z2k; ∀j∈ {1, 2,· · ·, 2k} 0≤tj≤n−1 ª
.
Let τ=t1+t2+· · ·+t2k, we haveH(N)(λ) =e
−i k(n−1)λ
qk,n
2k(n−1)
X
τ=0
exp
iτλ
QN(k)(τ),
where QN(k)(τ) =X
t∈S2k
χ0(t1+t2+· · ·+t2k−τ).
QN(k)(τ)is the number of the terms (t1,t2,· · ·,t2k) which satisfy: ∀j∈ {1, 2,· · ·, 2k} tj∈Z, 0≤tj≤n−1 et t1+t2+· · ·+t2k=τ.
Ifk=1
2 andτ∈Z then Q
(k)
N (τ) =1. On the other hand, putting
m=τ−k(n−1), we get
H(N)(λ) = 1 qk,n
k(n−1)
X m=−k(n−1)
exp[iλm]QN(k)(m+k[n−1]).
SinceH(N)(.)is even function, by using H(N)(λ) = H(
N)(λ) +H(N)(−λ)
2 we obtain
H(N)(λ) = 1 qk,n
k(n−1)
X m=−k(n−1)
exp
iλm
+exp
−iλm
2 Q
(k)
H(N)(λ) 1 qk,n
k(n−1)
X m=−k(n−1)
cos(λm)QN(k)(m+k[n−1]).
It follows from the definition ofqk,n, we obtain
qk,n = 1 2π
Zπ
−π
sin(n2λ)
sin(λ2)
!2k
dλ
qk,n = 1 2π
Zπ
−π
k(n−1)
X
τ=−k(n−1)
cos(λτ)QN(k)(τ+k[n−1])
dλ
qk,n = 1 2π
k(n−1)
X
τ=−k(n−1)
Zπ
−π
cos(λτ)dλ
QN(k)(τ+k[n−1]).
Since τ∈Z, we have
Zπ
−π
cos(λτ)dλ=
§ 0 if τ
6
=0
2π if τ=0
We haveqk,n=QN(k)(k[n−1]). By substituting this in the expression forH(N)(λ), we
obtainH(N)(λ) =
k(n−1)
X m=−k(n−1)
cos(λm)
QN(k)(m+k[n−1])
QN(k)(k[n−1])
. Therefore the function
hkis defined by
hk(u) =
( Q(k)
N(nu+k[n−1])
QN(k)(k[n−1])
ifnu∈Z
0 ifnu∈/Z
We note that ifk=1
2 then qk,n=hk
m n
=1.
PROOF OFLEMMA2. We first state some inequalities which we are using in this proof
|sin(λ)| ≤ |λ| ∀λ∈R (11)
|sin(λ 2)| ≥
|λ|
π ∀λ∈[−π,π] (12)
|sin(nλ)| ≤n|sin(λ)| ∀λ∈R, (13)
It follows from (11) and (12) that
Bα0,N=
Zπ
−π
sinn2λ
sinλ2
2kα
dλ≥2
Z πn
0
nλ/π λ/2
2kα
dλ=2π
2 π
2kα
n2kα−1. (14)
Bα0,N =
Zπ
−π
sinn2λ
sinλ2
2kα
dλ≤2
Z πn
0
n2kαdλ+
Zπ
π n
1
λ
π
2kαdλ
Thus we obtain
Bα0,N≤
4πkα
2kα−1
n2kα−1. (15)
Using (12) and (11) we get
JN,α = 2
B0 α,N
Zπ
0
(λ)γ
sinn2λ
sinλ2
2kα
dλ≤ 2
B0 α,N
Z πn
0
λγn2kαdλ+Zπ
π n
λγπ λ
2kα dλ
≤ 2
B0 α,N
πγ+1 γ+1n
2kα−γ−1+ π2kα γ−2kα+1
πγ−2kα+1−π
n
γ−2kα+1
≤ 2
B0 α,N
πγ+1 γ+1n
2kα−γ−1+ πγ+1 γ−2kα+1−
πγ+1 γ−2kα+1
1 nγ−2kα+1
.
We distinguish two cases according to the sign ofγ−2kα+1
JN,α≤ 2 B0
α,N
πγ+1 γ+1−2kα−
2kαπγ+1
(γ+1)(γ−2kα+1) 1
nγ−2kα+1 if γ−2kα+1>0
2kαπγ+1
(γ+1)(2kα−γ−1) 1 nγ−2kα+1−
πγ+1
2kα−γ−1 if γ−2kα+1<0.
From (13) we get
JN,α≤
2πγ+1 γ−2kα+1
1
2ππ22kαn2kα−1
if 1
2k < α <
γ+1 2k
4kαπγ+1
(γ+1)(2kα−γ−1)
n2kα−γ−1
2π2
π
2kα n2kα−1
if 2> α >γ+1 2k .
This inequality implies the results of the lemma.
CONCLUSION
ACKNOWLEDGEMENTS
I would like to thank the referees for their interest in this paper and their valuable comments and suggestions.
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SUMMARY
Consider a symmetricαstable process having a spectral representation with an additive con-stant error. An estimator of that error and its rate of convergence are given. We study the rate of convergence when the spectral density have some behaviors at origin. Few long memory processes are taken here as example.
