Construction of some special classes of stable processes that - - PowerPoint PPT Presentation
Motivation and application examples Overview on -stable variables and processes Construction of some special classes of stable processes that generalises spatial or temporal Gaussian processes. Nourddine Azzaoui works with G.W. Peters, L.
Motivation and application examples Overview on α-stable variables and processes
Nourddine Azzaoui works with G.W. Peters, L. Clavier, M. Egan and A. Guillin STM 2016: International Workshop on Spatial and Temporal Modeling from Statistical, Machine Learning and Engineering perspectives 20 July 2016
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
Let {s1, . . . , sn} measuring sensors placed in a spatial field D. n Zt(s), s 2 D ⇢ Rd, d = 1, 2, 3
varying spatial random variable. The question is to find Zt(s) at any time t or/and any location s in the field D. Parametric solution: reduce the knowledge of the process to some sufficient functionals: = ) Gaussian Processes (GP): reduce the processes to some sufficient functionals (covariances and mean functionals) = ) why not heavy tailed models (stables processes) with covariation, spectral densities... Estimates these functionals from observations of the process. Other concurrent solutions exist: non parametric techniques, Bayesian methods...
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
Consider a channel or a filter, s(t) = e ⇤ h(t) = Z e(t τ)h(τ)dτ Or equivalently by Fourier transform, = ) H(ω) = Z eιωth(t)dt h(t) =
N
X
k=1
akδtτk eiθk , But real world is random and (h(t), t 0) is considered as a stochastic process = ) A harmonizable process H(ω) = Z eιωtdξ(t) (ξt) (resp dξ(.) ) is heavy tailed process (resp. random measure)
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
Theoretical interest It is an extension of gaussian distributions and processes (case α = 2) The convolution stability: a combination of i.i.d stable variables is a stable one The central limit theorem: α-stable distributions are the only possible limit distribution for normalized sum of random variables. It is a parametric family having only 4 parameters (tail index α, scale, location and skewness parameters) Practical modelings Heavier tail with the decrease of α. α-stables take into account extreme values usually seen as outliers for Gaussians. α-stable are better models the high variability phenomena (infinite variance, impulsive signals...).
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
Let us consider a stochastic integral: Xt = Z f (t, λ)dξ(λ), where ξ is an α-stable stochastic process, We focus here on the symmetric case (SαS process) How to characterize this process with a spectral bi-measure? = ) spectral representation We Focus on the particular case of harmonisable processes (f (t, λ) = eιtλ)
In this case how the bimeasure is linked to the dependance structure of the process. Given observations how to estimate this bi-measure. What is the physical interpretation of the spectral measure in this case.
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
1
A random variable X is said stable (or have a stable distribution) if and only if for any positive real A and B their exist a unique positive C and real D s.t: AX1 + BX2 =d CX + D X1 and X2 i.i.d copies of X (in the symmetric case D=0)
2
It was shown that in this cas there exist a unique 0 α 2 such that C is given by Aα + Bα = C α Hence the prefix α
3
the characteristic function of Symmetric α-stable variables (SαS) is given by: φX (θ) = I E[eıθX ] = eσα|θ|α where 0 < α 2 and σ > 0.
4
Unfortunately the form of the characteristic function suggest that the density function of these distribution is impossible to calculate except for three special cases (α = 1, 2, or 1
2 )
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Motivation and application examples Overview on α-stable variables and processes
1
A random vector X = (X1, . . . , Xd) is α-stable SαS if for every A and B positives, their exist C > 0 such that: AX (1) + BX (2)
d
= CX, where X (1) and X (2) are i.i.d. copies of X and Aα + Bα = C α
2
equivalently we can show that the vector X is symmetric α-stable if and only if every linear combinaison Y =
d
X
k=1
bkXk is a an SαS univariate variable.
3
The Characteristic function of an SαS real vector X
(d) = (X1, . . . , Xd) is given by:
φX (θ1, . . . , θd ) = exp{ Z
Sd
|θ1s1 + · · · + θd sd |αdΓ
X(d) (s1, . . . , sd )}
where Γ
X(d) is a unique positive finite measure on the unit sphere of Rd 4
Complexe random variables and vectors: X = X1 + ı.X2 est α-stable if and only iff the vector (X1, X2) is α-stable on R2. More generally a vector (X1, . . . , Xd) with Xj = X 1
j + ı.X 2 j , is α-stable if and only if
(X 1
1 , X 2 1 , . . . , X 1 d , X 2 d ) is α-stable vector on R2d.
5
A complexe SαS, X = X1 + ı.X2 is said isotropic (rotationally invariant) if for any φ 2 [0, 2π[ X
d
= eıφ.X.
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
1
A stochastic process ξ = (ξt, t 2 R) is symmetric if and only if its finite dimensional distributions are SαS vectors.
2
An SαS random measure is a random set function dξ : B(R) 7 ! R( or C) such that, for any Borel sets A1, . . . , An, the vector (dξ(A1), . . . , dξ(An)) is an SαS random vector
3
A random measure dξ is said independently scattered if for any disjoint Borel sets A1, . . . , An the variables dξ(A1), . . . , dξ(An) are independents.
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
Let X = (X1, X2) jointly SαS vector with corresponding measure on the sphere Γ, the covariation of X1
[X1, X2]α = Z
S2
s1.(s2)<α1>dΓ(s1, s2) where s<β> = sign(s).|s|β In case where X = (X 1, X 2) is complex i.e. X 1 = X 1
1 + ıX 1 2 and X 2 = X 2 1 + ıX 2 2 , then the covariation of
X 1 on X 2 is : [X 1, X 2]α = Z
S4
(s1
1 + ıs1 2).(s2 1 + ıs2 2)<α1>dΓX (s1 1, s1 2, s2 1, s2 2)
and the notation z<β> = |z|β1z. A useful result: For any SαS vector X on Rd with spectral measure ΓX then, " d X
i=1
aiXi,
d
X
i=1
biXi #
α
= Z
Sd
d X
i=1
aisi ! . d X
i=1
bisi !<α1> dΓX (s1, . . . , sd)
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
1
Linearity with respect to the first component i.e. for any SαS vector (X1, X2, Y ) we have, [X1 + X2, Y ]α = [X1, Y ]α + [X2, Y ]α.
2
if X and Y are independent jointly SαS variables, then [X, Y ]α = 0. the inverse is not true in general
3
The covariation is additive with respect to its second component, [X, Y1 + Y2]α = [X, Y1]α + [X, Y2]α if Y1 and Y2 are independents.
4
for any real or complexe a and b, [a.X, b.Y ]α = ab<α1>[X, Y ]α.
5
Let X = (Xt)t an SαS process and denote l(X) the space of finite linear combinations of X. The application, k.kα : l(X)
Y 7 ! kY kα , ([Y , Y ]α)
1
α
is a norm covariation norm. In this case (l(X), k.kα) is a Banach space
6
In (l(X), k.kα), the covariation is continuous moreover we have: |[Z1, Z2]α [Z1, Z3]α| 2kZ1kα.kZ2 Z3kα1
α
.
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
Let r(t) be the covariance function of a second order stationary process Xt,
r(t) is positive definite
n
X
i=1 n
X
j=1
cicjr(ti tj) 0
Bochner’s Th. r(t) = Z 1
1
eıtλF(d) F is a positive measure
Cramer-Kolmogorov Xt = Z 1
1
eıtλd⇠() ⇠ have orthogonal increments
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
now the covariance is bivariate r(s, t) and
the cov. is bilinear positive definite
F(A, B) = cov(⇠(A), ⇠(B)) F positive definite bi-measure
Cramer-Rao Xt = Z 1
1
f (t, )d⇠()
r(s, t) = Z Z f (s, λ)f (t, λ0)F(dλ, dλ0)
The bimeasure F is positive definite in the sense,
n
X
i=1 n
X
j=1
cicjF(Ai, Aj) 0,
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
⇠ have independent increments
i.e. if A \ B = ; then dξ(A) inndep. dξ(B)
µ(.) = k⇠(.)kα
α pos. measure
Cambanis-Miller Xt = Z 1
1
f (t, )d⇠() ⇠ is independent increments
In this case the covariation verify: [Xs, Xt]
α =
Z f (s, λ)(f (t, λ))
<α1>µ(dλ)
and for the harmonisable case: [Xs, Xt]
α =
Z eıλ(st)µ(dλ)
Azzaoui et al ... Special classes of stable Processes...
Motivation and application examples Overview on α-stable variables and processes
The idea: replace the Camabnis-Miller spectral representation by one similar to Cramer-Rao spectral representation = ) relax the independently scattered condition of the SαS measure. = ) find a weaker condition for the additivity of the covariation How to? Define a bimeasure from the covariation using the additivity = ) Show that this bimeasure characterises the studied process. = ) prove the Cramer-Rao type representation. Go further? Apply this to harmonisable processes
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
For the covariation to be additive with respect to its second variable i.e. 8i 2 {1, .., d}, 8θ1, ..., θd 2 C: [Xi, θ1X1 + ... + θdXd]α = [Xi, θ1X1]α + ... + [Xi, θdXd]α, it suffices that for all i, j and k 2 {1, ..., d} 8θ1, ..., θd 2 R, ∂3φ ∂θi∂θj∂θk (θ1, ...θd) = 0. (1) where φ is the Fourier transform of ΓX Examples: Independent variables verify this conditions φ(θ1, . . . , θd) = a1 cos(θ1) + · · · + ad cos θd) A more general example: φ(θ1, ..., θd) =
d
X
i6=j=1
ϕi,j(θi, θj) where ϕi,j are characteristic functions of finite measures on Sd.
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
For the covariation to be additive with respect to its second variable that is, for all i0 2 {1, . . . , d}: 8θ1, . . . , θd 2 C, [Xi0, θ1X1 + · · · + θdXd]α = [Xi0, θ1X1]α + · · · + [Xi0, θdXd]α. (2) It is sufficient that, for all i, j 2 {1, . . . , d} not all equal, the Fourier transform φ fulfill the condition that for all θ1, . . . , θd 2 C one has: θi ∂2φ ∂θ2
i
(θi, θj) + θj ∂2φ ∂θi∂θj (θi, θj) = θi ∂2φ ∂θ2
i
(θi, 0) + θj ∂2φ ∂θi∂θj (0, θj) θi ∂2φ ∂θi∂θj (θi, θj) + θj ∂2φ ∂θ2
j
(θi, θj) = θi ∂2φ ∂θi∂θj (θi, 0) + θj ∂2φ ∂θ2
j
(0, θj) (3)
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
Let si = cos(η) and sj = sin(η) and γ(η) the density of η satisfying the 3rd order conditions i.e. for all θi, θj 2 C, the condition that Z 2π cos2(η) sin(η) sin (θi sin(η) + θj cos(η)) γ(η)dη =
1
X
k=0
J2k+1(t) ⇢Z 2π h cos2(η) sin(η) sin ((2k + 1)(η + atan2(θi, θj))) i γ(η) dη
(4) where t = q θ2
i + θ2 j and atan2(θi, θj) = 2 arctan
q
θ2
i +θ2 j θi
θj
! . One way to obtain this result is to consider densities γ(η), which satisfy for all k 2 N the conditions: Z 2π cos ((2k + 1)η) [sin(η) + sin(3η)] γ(η) dη = 0, and Z 2π sin ((2k + 1)η) [sin(η) + sin(3η)] γ(η) dη = 0, (5)
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
In the case that γ(η) is symmetric on support [0, 2π] this requires the following recurrence relationships: 2π
2k
X
j2;j even
2jϕ(j) + 1 = 0 22k+1πϕ(2k + 1) + 4ϕ(2) + 1 = 0. (6) Where ϕ is Fourier transform of γ(.). In the case that γ(η) is not symmetric we consider γ(η) = γ(η)Iη2[0,π] + γ(η)Iη2[π,2π] with Fourier transform for interval [0, π] denoted by ϕ1(θ) and for interval [π, 2π] given by ϕ2(η) with ϕ(θ) = ϕ1(θ) + ϕ2(θ) which yields for the i-th component with i 2 {1, 2}: 2π
2k
X
j2;j even
2jϕi(j) + 1 = 0 22k+1πϕi(2k + 1) + 4ϕi(2) + 1 = 0. (7)
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
Let us come back to the random measure dξ
Condition (O): we will say that it verifies the additivity condition if for all n 2 and all disjoints Borelian sets {A1, ..., An} the SαS vector (dξ(A1), ..., dξ(An)) verify the condition (1). Let us now consider the set function F defined on B(R) ⇥ B(R) by : F : B(R) ⇥ B(R)
C (A, B) 7 ! [dξ(A), dξ(B)]α (8) F is additive with respect to its two variables:it is a bimeasure. For the second variable, if B1 and B2 two distinct Borel Sets, F(A, B1 [ B2) = [dξ(A), dξ(B1 [ B2)]α = [dξ(A), dξ(B1) + dξ(B2)]α (9) Since dξ satisfy the condition (O) then, [dξ(A), dξ(B1) + dξ(B2)]α = [dξ(A), dξ(B1)]α + [dξ(A), dξ(B2)]α = F(A, B1) + F(A, B2).
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
the bimeasure F defined in (8) verifies a similar positive definitness property: for all complex z1, . . . , zn and for all distinct Borel sets A1, . . . , An , we have:
n
X
i=1 n
X
j=1
zi(zj)<α1>F(Ai, Aj) 0. (10) the proof of this property is easy. It suffices to use the condition (O). Indeed,
n
X
i=1 n
X
j=1
zi(zj)<α1>F(Ai, Aj) =
n
X
i=1 n
X
j=1
zi(zj)<α1>[dξ(Ai), dξ(Aj)]α = " n X
i=1
zidξ(Ai),
n
X
i=1
zidξ(Ai) #
α
=
X
i=1
zidξ(Ai)
α
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
Let’s consider ν : A 7 ! ν(A) = E(v(dξ, A)) where v(dξ, A) is the total variation of the random measure dξ. It is defined, for all borelian A, by : v(dξ, A) = sup
I finite
8 < : X
i2I
|dξ(Ai)|, (Ai)i2I partition of A 9 = ; (11) The total variation v(dξ, .) is a positive random measure. The application of expectation is linear and continuous, we deduce then that νis a positive measure. Since dξ is with bounded variation then ν a bounded measure. With respect to this measure ν, we consider the norm of L1(ν) of a complex function f defined by, N(f) = Z
R
|f|dν We denote by Λα(dξ) the completion, with respect to the norm N
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
For purpose of constructing the covariation spectral representation we need the next result that ensure the convergence theorems
Suppose that dξ satisfy the condition (O), then we have the next properties:
1
For all A 2 B(R) we have , kdξ(A)kα Ψα(1).ν(A) where Ψα(p) is equal to
1 Sα(p) when ξ is real and
is equal to
1 ˜ Sα(p) in the complex case. The quantities Sα(p) et ˜
Sα(p) depends only on α and p.
2
Let B 2 B(R) a fix Borel set. If A is set verifying ν(A) = 0 then the total variation variation of the complex measure FB in A is null , that is v(FB, A) = 0. This result is also true for the measure ˜ FA(B) and B fixed.
3
Let B 2 B(R) be a fix Borel set, then for all bounded function f 2 Λα(dξ) we have the inequality : | Z
R
fdFB| Ψα(1).kdξ(B)kα1
α
Z
R
|f|dν (12)
4
Let f 2 Λα(dξ) be a fixed bounded function and note G(B) = ˜ I(f, B). We have the next implication: (ν(B) = 0 implies v(G, B) = 0).
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Additivity of the covariation The spectral representation
Let f and g Two bonded function in Λα(dξ). The covariation of the stochastic integral Z
R
fdξ on Z
R
gdξ is given by: Z
R
fdξ, Z
R
gdξ
= Z
R
Z
R
f(λ)
<α1> F(dλ, dλ0). (13)
Suppose that ξ is a real or complex isotrope symmetric α-stable process. Then the bimeasure F defined in (8) is the unique bimeasure characterizing the process X and verifying the integral representation (13).
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Harmonisable processes Calibration problem?
We consider a harmonisable SαS processes X = (Xt, t 2 R), with 1 < α < 2, of the form: Xt = Z 1
1
eıtλdξ(λ) where ξ is a stable process verifying the additivity condition f(t, λ) = eıtλ Why harmonisable? Fourier analysis gives a natural description of many phenomena, in Physics, Communications, Astronomy, cosmology... Density of the Fourier base functions : any integrable function can be expressed in terms of Fourier series. The inverse Fourier transform : a function is completely determined by its Fourier Transform and vis-versa.
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Harmonisable processes Calibration problem?
The covariation function of a harmonisable stable process: [Xs, Xt]α = Z
R
Z
R
eı(sλtλ0)F(dλ, dλ0) F is characterizing spectral bimeasure on R2 kVFk(R ⇥ R) , sup 8 < :
n
X
i=1 n
X
j=1
|F(Ai, Bj)|, such that, (Ai ⇥ Bj)i,j=1..n are disjoints 9 = ; < 1.
1
We suppose that F has a density h, [Xs, Xt]α = Z
R
Z
R
eı(sλtλ0)h(λ, λ0)dλdλ0 = ) The calibration and identification of the process X is reduced to the estimation of the spectral density h from the process observations.
1see [Rao(1982)] Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion Harmonisable processes Calibration problem?
Suppose that F has a density h, C(s, t) , [Xs, Xt]α = Z 1
1
Z 1
1
eı(sλtλ0)h(λ, λ0)dλdλ0. if the covariation function C is integrable then : h(λ, λ0) = 1 4π2 Z 1
1
Z 1
1
eı(sλtλ0).C(s, t)dsdt The estimation of the spectral density h can be obtained through an efficient estimation of C. = ) Let us first see what has been done in the second order case Remark: We will not focus in the stationary2 case in this talk.
2see [Cambanis(1983)], [Masry and Cambanis(1984)] Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion second order case difficulties of the stable case
Let X = (Xt, t 2 R) a centered second order harmonisable process, its covariation function C(s, t) = E(XsXt) = Z Z eı(sλtλ0)h(λ, λ0)dλdλ0 = ) Fourier and Fubini theorem implies that: h(λ, λ0) = 1 4π2 E ✓Z eısλXsds Z eıtλ0Xtdt ◆ , E(I(λ) I(λ0)) This leads to the periodogram (used in the 18th century) which is a Fourier transform of the process I(λ) = Z eısλXsds For example in the stationary case h is reduced to a single variable function, h(λ) = 1 4π2 E|I(λ)|2 which is the well known power spectral density
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion second order case difficulties of the stable case
The periodogram was enhanced by kernel estimation 3, which consist in smoothing the periodogram using a convolution kernel. (KbA ⇤ I)(λ, λ0) where bA is a smoothing bandwidth and KbA (x) =
1 bA K( x bA ) where K is a kernel 4.
Many other enhancements has been introduced in literature Looking for the optimal choice of the kernel via cross validation, MISE, ... finding the best window bandwidth for optimal smoothing 5 Spectral analysis and estimation of some special classes, periodically and almost periodically correlated processes, has been investigated [Lii and Rosenblatt(2006)]
3[Rosenblatt(1956), Parzen(1961)] 4K is positive continuous even function defined on R2 such that
R R K(s, t)dsdt = 1
5see [Priestley(1981), Priestley(1988)] Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion second order case difficulties of the stable case
Moments of order greater or equal to α are infinite = ) infinite variance and covariance E|X|p = Sα(p)kXkp
α
(14) where
Sα(p) = 8 > > > > < > > > > : 2p Γ( 1+p
2 ).Γ(1 p α )
Γ(1 p
2 )Γ( 1 2 )
, %p.Γ(1 p ↵ ) if X is real Γ( 2+p
2 ).Γ(1 p α )
Γ(1 p
2 )
, ⇢p.Γ(1 p ↵ ) if X is isotropic complex
A " bad " link to fractional moments through: [X, Y ]α kY kα
α
= E(X.Y <p1>) E|Y |p For non stationary processes the covariation and moments are time dependents = ) So what to do?
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion A theoretical toy example Estimation from continuous observations of the process Discrete Estimation...
Let X and Y be real or isotropic complex SαS random variables. When 0 < p < α increases to α, then there exist a constant M such that:
αCαkXkα
α
(15) where ψ1(x) = xα(1 + | log(x)| max(x1α, 1)) and Cα =
1α Γ(2α) cos( πα
2 ) . In addition the covariation of X on Y
verifies:
⇣ (α p)XY <p1>⌘ αCα[X, Y ]α
(16) where Y <p1> = |Y |p2.Y and ψ2(x, y) = y(1 + | log(x)| max(x1α, 1)). = ) This suggest that we can construct unbiased estimator of the covariation. But We have no chance to prove the L1 convergence.
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion A theoretical toy example Estimation from continuous observations of the process Discrete Estimation...
To construct an L1 convergent sequence of kXkα and the covariation [X, Y ]α. The idea is to take p = α un where (un)n is decreasing to 0 and temper with a triangular array ((w (n)
k )kn)
and we define, Υn = 1 n
n
X
k=1
w (n)
k .uk.|X|αuk
(17) that converges in L1 to αCαkXkα
α.
For the proof of the L1 convergence to [X, Y ]α we will need the notation: Υ0
n = 1
n
n
X
k=1
w (n)
k .uk.(Y )
<αuk 1>,
(18) which is defined for all real or isotropic complex SαS random variables Y .
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion A theoretical toy example Estimation from continuous observations of the process Discrete Estimation...
Let (un)n decreasing to 0 and a triangular array ((w (n)
k )kn) verifying 1
n
n
X
k=1
w (n)
k
= 1 and: (A1) lim
k!1(w (n) k .uk) = 0, we denote ∆n , 1
n
n
X
k=1
w (n)
k .uk
(A2) Suppose that the sequence En , 1 n
n
X
k=1
(w (n)
k )
α αuk
n!1 0,
The sequence of r.v’s Υn converges in L1 to αCα.kXkα
α. Precisely :
α
(19) For the covariation of X on Y we have:
n
(20)
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Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion A theoretical toy example Estimation from continuous observations of the process Discrete Estimation...
Let X and Y be tow jointly SαS random variables real or isotropic complexes and (un), (w (n)
k ) defined
E |XΥn αCα.[X, Y ]α| MkXkαkY kα1
α
. (∆n + (∆n + En) log(kY kα)) (21) Remarks: This theorem suggest that we can find optimal set of sequences (uk) and (w n
k ) such that the rate of
convergence is optimal in the sense of L1. The tail index α weaken this set through the conditions (A1) and (A2) on ∆n and En. The bound rate will also depend on the scale parameters of X and Y .
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for all 1 < p < α, the covariation function C may be written as: C(s, t) = α p αCα E(XsX <p1>
t
) + Rp(s, t), (22) where |Rp(s, t)| M.(α p)|.C(s, t).(1 + 1 α log(C(t, t)))| and M is a positive constant. Replacing the covariation in the formula of h(λ, λ0), we obtain: h(λ, λ0) = α p 4π2αCα Z Z eı(sλtλ0)E(XsX <p1>
t
)dsdt + 1 4π2 Z Z eı(sλtλ0)Rp(s, t)dt , J1 + J2
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When the Fubini theorem is allowed for the integral J1, we can write, J1 = α p 4π2αCα E ✓Z Z 1
1
eı(sλtλ0)Xs.X <p1>
t
dsdt ◆ , = α p 4π2αCα E ✓Z 1
1
eısλXsds. Z 1
1
eıtλ0X <p1>
t
dt ◆ , = α p 4π2αCα E ⇣ I(λ)Ip(λ0) ⌘ , (23) with I(λ) = Z 1
1
eısλXsds and Ip(λ0) = Z 1
1
eıtλ0X <p1>
t
dt This implies that, h(λ, λ0) = α p 4π2αCα E ⇣ I(λ)Ip(λ0) ⌘ + 1 4π2 Z Z 1
1
eı(sλtλ0)Rp(s, t)dsdt (24)
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What about the term J2 ? Suppose that C and the map (s, t) 7 ! C(s, t) log(C(t, t)) are integrable on R2. With the fact that, |Rp(s, t)| M.(α p).|C(s, t).(1 + 1 α log(C(t, t)))| We deduce that the term Z Z 1
1
eı(sλtλ0)Rα(s, t)dsdt converge, uniformly with respect to (λ, λ0), toward 0 when p increases to α. = ) Then uniformly to (λ, λ0), we have : h(λ, λ0) = lim
p%α
α p 4π2αCα E ⇣ I(λ)Ip(λ0) ⌘ (25) This suggest an asymptotically unbiased estimator h by taking p too close to α. ˆ h(λ, λ0) = α p 4π2αCα I(λ).Ip(λ0). (26)
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Suppose we observe the process on the time interval [A, A], a natural estimator is given by: ˜ h(λ, λ0) = UA 4π2αCα ˜ I(λ).˜ IαUA(λ0) (27) with ˜ I(λ) = Z A
A
eısλXsds and ˜ Ip(λ0) = Z A
A
eıtλ0X <p1>
t
dt
6
where UA decrease to 0 when A goes to infinity. This estimator is asymptotically unbiased and for all λ and λ0, we have,
⇣ ˜ h(λ, λ0) ⌘ h(λ, λ0)
(28) where the term GA depend only on the covariation; it tend to 0 when A goes to infinity.
6Existence of integrals ˜
I(λ) and ˜ Ip(λ0) comes from the integrability of the covariation function, see [Samorodnitsky and Taqqu(1994)].
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Obviously ˜ h does not converge in L1 to h since its constructions comme from the on convergent estimator the
ˆ h(λ, λ0) = 1 4π2αCα ˆ I(λ). @ 1 n(A)
n(A)
X
k=1
w (n)
k .uk.ˆ
Iαuk 1 A (29) The sequences (uk) and (w (n)
k ) are those given previously and n(A)
! 0
A!1. We can derive "easily" that ˆ
h is consistent (in L1 norm).
suppose that the covariation function C and the map defined by (s, t) 7 ! C(s, t) log(C(t, t)) are integrable functions on R2. We also assume that (uk), (w (n)
k ) fulfill the previous L1 convergence theorem, then for all λ
and λ0, ˆ h(λ, λ0) is asymptotically unbiased and it converges to h(λ, λ0) in the L1 norm, E
h(λ, λ0) h(λ, λ0)
A!1 0.
(30)
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The smoothing idea consist in regularizing the estimator ˆ h by a convolution kernel KbA : ˆ ˆ h(λ, λ0) = (KbA ⇤ ˆ h)(λ, λ0) (31) where bA is a smoothing bandwidth verifying bA ! 0 and AbA ! 0. The window KbA is defined as KbA (x) =
1 bA K( x bA ) where K is a kernel 7.
To clarify this, let us denote ZA(s, t) = 1 αCαn(A)
n(A)
X
k=1
w (n)
k .uk.Xs.(Xt)
<αuk 1>
We remark that ˆ h = F1(ZA) where F1 is inverse Fourier transform operator. By using the Fourier transform to ˆ ˆ h we have: F(ˆ ˆ h)(s, t) = F ⇣ KbA ⇤ F1(ZA) ⌘ (s, t) = ZA(s, t).F(KbA )(s, t). (32)
7K is positive continuous even function defined on R2 such that
R R K(s, t)dsdt=1
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Let k(s, t) the Fourier transform of the kernel K then, F(KbA )(s, t) = k(bAs, bAt) .Thus applying again the inverse Fourier transform, ˆ ˆ h(λ, λ0) = F1(kbA .ZA)(λ, λ0) = 1 4π2 Z A
A
Z A
A
k(bAs, bAt).ZA(s, t)eı(sλitλ0)dsdt This formula is simple to deal with in practice because it use directly the observation of the process. We can easily show that ˆ ˆ h is asymptotically unbiased under some conditions on the (uk), (w n
k ) and the
bandwidth bA and the smoothness of the spectral density h.
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Let K be a Parzen-Rosenblatt kernel with integrable Fourier transform k. Suppose that the covariation C and the map s 7 ! kXskα is continuous and bounded in neighborhood of 0. Let (uk), (w (n)
k ) as defined previously
and that the bandwidth bA satisfy: lim ∆n(A) b2
A
= 0. Then, for all λ, λ0 of R, the estimator ˆ ˆ h converge in the L1 norm to the spectral density h.
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Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion A theoretical toy example Estimation from continuous observations of the process Discrete Estimation...
Suppose that we can observe the process at any time in [A, A]. The idea is to simulate (s1, t1), . . . , (sN, tN) i.i.d. random instants having a uniform distribution on the rectangle [A, A] ⇥ [A, A]. We also suppose that this instants couples are independent of the process X. The discrete estimation consist then in replacing the integrals R A
A ... by empirical means
˜ I(λ) = 2A N
N
X
k=1
eıskλXsk and ˜ Ip(λ0) = 2A N
N
X
k=1
eıtkλ(Xtk )<p1> = ) these are discrete fractional periodograms. Using the independence of the time sample from the process observations show easily that
1
The bias of the discrete estimator is the same as the continuous time estimator.
2
The convergence properties shown for the continuous time estimator remain true for the discrete estimator.
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Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion
Suppose that C is a continuous function. We will say that X is covariation stationary if C(s, t) depend only on s t. The process X is periodically covariated with period T > 0 if C(s + T, t + T) = C(s, t) for all s et t . In this case C(t, t + τ) ⇠
1
X
1
ak(τ)eı 2πk
T
t.
where ak(τ) = 1 T Z T C(t, t + τ).eı 2πk
T
tdt.
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Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion
1
The process X is covariation stationary if and only if the bimesure F is concentrated on the diagonal line.
2
It is periodically covariated with period T if and only if F is concentrated on parallel lines (Sk)k , Sk = {(λ, λ0) 2 R ⇥ R, /λ λ0 = 2πk
T }.
In this case, ak(τ) = Z
R
eıτλφk(λ)dλ.
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Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion
Like in the general case we propose: ˆ ak(τ) = 8 > > > > < > > > > : UA αACα Z Aτ Xt.(Xt+τ)
<α1UA>eı 2πk T
tdt
if 0 < τ < A UA αCαA Z A
τ
Xt(Xt+τ)
<α1UA>eı 2πk T
tdt
if A < τ < 0
where UA decrease to 0, when A goes to infinity. This estimator is asymptotically unbiased and we have: |E(ˆ ak(τ)) ak(τ)| M2UA + g(A, τ) A , where |g(A, τ)| M3 + |τak(τ)|.
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Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion
Suppose that X is strongly mixing with a mixing function χ verifying:
1
the function χ(t) is a decreasing integrable even function on R where χ(t) = O(tγ) and γ > 0
2
the sequence (UA)A have the form UA =
α log log(A) ,
then the estimator ˆ ak(τ) converge almost surely to ak(τ).
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion
Central limit theorems for the estimators ! important for the statistical inferences and tests study the invariance principe and functional limit theorems Find the optimal choice of the kernel smoother and its optimal corresponding bandwidth Find the optimal sampler for the discrete estimation. Apply these results to practical problems, communications, finances...
Azzaoui et al ... Special classes of stable Processes...
Spectral representation Harmonisable processes Overview and difficulties Our Estimation approach PC processes Conclusion
Bochner) 28 (1982) 295–351.
Contribution to statistics (Essays in honor of Norman L. Johnson) (1983) 63–79.
applications 18 (1) (1984) 1–31.
Statistics 27 (3) (1956) 832–837.
Chapman & Hall, 1994.
Azzaoui et al ... Special classes of stable Processes...