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Bispectrum estimation for a continuous-time stationary process from a random sampling

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Bispectrum estimation for a continuous-time stationary process from a random sampling
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  Bispectrum estimation for a continuous-timestationary process from a random sampling Karim Benhenni and Mustapha Rachdi Universit´e de GrenobleLaboratoire Jean KuntzmannUFR SHS, BP. 47, 38040 Grenoble Cedex 09, France(e-mail:  Mustapha.Rachdi@upmf-grenoble.fr andKarim.Benhenni@upmf-grenoble.fr ) Abstract.  We propose an asymptotically unbiased and consistent estimate of thebispectrum of a stationary continuous-time process  X   =  { X  ( t ) } t ∈ R . The estimateis constructed from observations obtained by a random sampling of the time by { X  ( τ  k ) } k ∈ Z , where  { τ  k } k ∈ Z  is a sequence of real random variables, generated froma Poisson counting process. Moreover, we establish the asymptotic normality of the constructed estimate. Keywords:  Periodogram, Cumulants, Quadratic-mean consistency, Bispectral den-sity, Point process. 1 Introduction The idea of constructing the Fourier transforms of high order cumulants wassuggested by Kolmogorov, and polyspectra were introduced in [Shiryaev, 1962].In [Brillinger, 1965] and [Brillinger and Rosenblatt, 1967] authors gave a com-prehensive treatment of the theoretical properties of polyspectra, and havediscussed also the estimation of polyspectra from sample records (these es-timation procedures are based on a generalization of the  window   techniqueapplied to products of the finite Fourier transform of the data).Bispectra was discussed in [Tukey, 1959] and [Akaike, 1966], and an appli-cation of the bispectral analysis to the study of ocean waves is given in[Hasselmann  et al. , 1963], to tides in [Cartwright, 1968], and to turbulencein [Lii  et al. , 1976] and in [Helland  et al. , 1979].These Fourier transforms -or rather, the Fourier transforms of the correspond-ing high order cumulants- are called  polyspectra  , and are defined formally asfollows.Let  { X  ( t ) } t ∈ R  be a weakly stationary process up to order  k , and let the realnumber  C  ( k ) X  ( s 1 ,s 2 ,...,s k − 1 ) denotes the joint cumulant of order  k  of the setof random variables { X  ( t ) ,X  ( t + s 1 ) ,...,X  ( t + s k − 1 ) } , i.e.  C  ( k ) X  ( s 1 ,s 2 ,...,s k − 1 )is the coefficient of ( z 1 ,...,z k ) in the expansion of the cumulant generatingfunction κ ( z  1 ,...,z  k ) = ln( IE  { exp( z  1 X  ( t ) + z  2 X  ( t + s 1 ) + ··· + z  k X  ( t + s k − 1 )) } ) (note that, by the stationarity condition,  C  ( k ) X  ( s 1 ,...,s k − 1 ) does not dependon  t ).  2 Benhenni and Rachdi The outstanding property of polyspectra is that all polyspectra of orderhigher than two vanish when  { X  ( t ) } t ∈ R  is a Gaussian process. This fol-lows immediately from the well known property that all joint cumulants of order higher than two vanish for multivariate gaussian distributions. Hence,the bispectrum, trispectrum, and all higher order polyspectra are identicallyzero if   { X  ( t ) } t ∈ R  is Gaussian, and these higher order spectra may thus beregarded as measures of the departure of the process from Gaussianity. Themain aim of this paper is then to construct an estimate of the bispectrum of continuous-time stochastic process from a random sampling and to study itsasymptotic properties, namely its mean-square convergence and its asymp-totic normality.In Section 2, we give some preliminaries on the time-sampling techniqueadopted here. Section 3 is concerned with the construction of the bispectrumestimate and the main results of its asymptotic properties. The last sectionis devoted to the proofs. 2 Preliminaries Let  X   =  { X  ( t ) } t ∈ R  be a 4th order stationary process, with mean zero andcontinuous integrable covariance function. From [Karr, 1991], there existsa counting process  N  , independent of   X  , which is associated to a sequence { τ  k } k ∈ Z  of random variables taking their values in  R . The process  N   isdefined by:  N  ( A,ω ) =   k ∈ Z  11 A  for ( A,ω )  ∈ B  R  × Ω  , where  B  R  is the realborelean  σ -algebra and  N  ( A,ω ) is the number of   τ  k ’s belonging to  A .We assume that, for every set  A  in  B  R , the random variable  N  ( A ) has aPoisson distribution  P  ( Λ ( A )), where  Λ ( A ) =  βµ ( A ) and  µ  is the Lebesguemeasure on  R  and  β   denotes the mean intensity which is assumed to beknown.We consider the sample process  Z   = { Z  ( t ) } t ∈ R  constructed from the sequence { X  ( τ  n ) } n ∈ N  and the counting process  N  ( t ) as follows. Definition 1  The sample process   Z   is defined by: Z  ( A ) =   A X  ( t )  N  ( dt ) =  k ∈ Z X  ( τ  k )11 A ( τ  k ) =  τ  k ∈ A X  ( τ  k ) ,  ∀ A  ∈ B R . The process  Z   is also called  increment-process   and can be written as: Z  ( t ) =   t 0  X  ( s ) d  N  ( s ) or in the differential representation:  dZ  ( t ) =  X  ( t ) d  N  ( t ) , which proves that Z   is a stationary process and that its covariance function  R Z  is such that: R Z ( du ) =  R X ( u )  βδ  ( u ) + β  2   du .Denote, respectively, by  φ (2) X  and  φ (2) Z  the spectral densities of the process  X  of the process  X   and the increment process  Z  .If   R X  and its Fourier transform  F  R X  are absolutely integrable then  φ (2) Z exists, is bounded, uniformly continuous and is given by φ (2) Z  ( λ ) =  β  2 φ (2) X  ( λ ) +  β  2 π R X (0) ,  ∀ λ  ∈ R  Bispectrum estimation 3 Thus, the estimate of   φ (2) X  can be deduced from the estimate of   φ (2) Z  and R X (0) (Cf [Lii and Helland, 1982], [Lii and Masry, 1994], [Monsan and Rachdi, 1999]and [Gallego and Ruiz, 2000]). 3 Bispectrum estimation and its asymptotic properties Let us denote by  φ (3) X  and  φ (3) Z  the respective bispectrum of   X   and  Z  . Set C  (3) X  ( u 1 ,u 2 ) = cum { X  ( u 1  + t ) , X  ( u 2  + t ) , X  ( t ) }  =  IE  { X  ( u 1  + t ) X  ( u 2  + t ) X  ( t ) } , and  dC  (3) Z  ( u 1 ,u 2 ) = cum { dZ  ( u 1  + t ) , dZ  ( u 2  + t ) , dZ  ( t ) } then  φ (3) X  and  φ (3) Z  are defined by φ (3) X  ( λ 1 ,λ 2 ) = 1(2 π ) 2   R 2 C  (3) X  ( u 1 ,u 2 ) exp( − i ( λ 1 u 1  + λ 2 u 2 )) du 1 du 2 φ (3) Z  ( λ 1 ,λ 2 ) = 1(2 π ) 2   R 2 exp( − i ( λ 1 u 1  + λ 2 u 2 )) dC  (3) Z  ( u 1 ,u 2 ) . As in the previous section, using the independence between  X   and  N  , weestablish a relationship between  dC  (3) Z  and  C  (3) X  . First, we have dC  (3) Z  ( u 1 ,u 2 )=  C  (3) X  ( u 1 ,u 2 )  βδ  ( u 1 ) δ  ( u 2 ) + β  2 δ  ( u 1 ) + β  2 δ  ( u 2 ) + β  2 δ  ( u 1  − u 2 ) + β  3  du 1 du 2 , thus, if we define by: ψ ( λ ) =   + ∞−∞  exp( iu 1 λ ) C  (3) X  ( u 1 ,u 1 ) du 1  + C  (3) X  (0 , 0)  β  2 / (2 π ) 2 we can write φ (3) Z  ( λ 1 ,λ 2 ) =  β  3 φ (3) X  ( λ 1 ,λ 2 ) −  2 β  (2 π ) 2 C  (3) X  (0 , 0) + ψ ( λ 1 ) + ψ ( λ 2 ) + ψ ( − λ 1  − λ 2 ) and φ (3) X  ( λ 1 ,λ 2 ) = 1 β  3  φ (3) Z  ( λ 1 ,λ 2 ) + 2 β  (2 π ) 2 C  (3) X  (0 , 0) − ψ ( λ 1 ) − ψ ( λ 2 ) − ψ ( − λ 1  − λ 2 )  . (1) Equation (1) is the relationship which allows to estimate  φ (3) X  from discretedata as follows. Given the observations  X  ( τ  k )   N  ( T  ) k =1  ,  T >  0, where  N  ( T  ) isthe number of points  τ  k   falling in [0 ,T  ], we estimate the bispectrum  φ (3) X  byestimating the bispectrum of   Z  , the function  ψ  and the constant  C  (3) X  (0 , 0).For this, let us denote by   φ (3) X,T  ,   φ (3) Z,T   and   ψ T   the respective estimates of  φ (3) X  , φ (3) Z  and  ψ . Consider the three dimensional spectral window  W   andthe bandwidth  b T  , which verify the following assumptions. Assumptions 1  4 Benhenni and Rachdi (i)  W   ∈ L 1 ∩ L ∞ ( R 3 )  is a positive function such that:   R 2 | W  ( u 1 ,u 2 , − u 1  − u 2 ) | du 1 du 2  = 1 (ii)  | W  ( u 1 ,u 2 , − u 1 − u 2 ) |  and   ∂W  ( u 1 ,u 2 , − u 1 − u 2 ) ∂u j   ≤ C  (1 + || ( u 1 ,u 2 ) || 2 ) − 2 − ε ,  j  = 1 ,  2  where   C   and   ε  are two positive real numbers, and   || ( u 1 ,u 2 ) || 2  de-notes the euclidean norm of   ( u 1 ,u 2 ) .(iii)  ( b T  ) T  ∈ R  is a sequence of positive real numbers such that: b T   →  + ∞  and   b 2 T  T   →  0  as   T   →  + ∞ In order to construct   φ (3) Z,T  , we set:  W  T  ( u 1 ,u 2 ,u 3 ) =  b 2 T  W  ( b T   u 1 ,b T   u 2 ,b T   u 3 ) . The periodogram (empirical estimate of   φ (3) Z  ) is defined by   I  T  ( λ 1 ,λ 2 ) = 1(2 π ) 2 T    d Z,T  ( λ 1 )   d Z,T  ( λ 2 )   d Z,T  ( − λ 1  − λ 2 ) ,  (2) where   d Z,T  ( λ ) =    N  ( T  ) k =1  X  ( τ  k )exp( − iλτ  k ), is the finite Fourier transformof the observations  { X   ( τ  k ) }  N  ( T  ) k =1  .In order to obtain a consistent estimate of   φ (3) Z  , we smooth   I  T   by  W  T  , whichyields to:   φ (3) Z,T  ( λ 1 ,λ 2 ) =   12 πT   2 + ∞  i,j = −∞ W  T   ( λ 1  − ω i ,λ 2  − ω j , − λ 1  − λ 2  − ω i + j )   I  T   ( ω i ,ω j ) where  ω j  = 2 πj/T   for  j  ∈ Z  denotes the Fourier frequencies. Assumptions 2 R X  and   C  (3) X  are absolutely integrable and for all   k  = 1 ,..., 5   R k || u || 1 | φ ( k +1) X  ( u ) | du <  + ∞ , where   u  = ( u 1 ,...,u k )  and   || u || 1  =   kj =1 | u j | , where   φ ( k ) X  ( u )  denotes the  k -th derivative of   φ X ( u ) . We have that under Assumptions 2, the measure  dC  (3) Z  is integrable and   R k || u || 1 | dC  ( k +1) X  ( u ) |  <  + ∞ ,  for  k  = 1 ,..., 5 from which we infer the following properties that are useful to establish theasymptotic behavior of    φ (3) Z,T  ( λ 1 ,λ 2 ) (Cf Theorem 1). For this aim, the follow-ing proposition gives the asymptotic behavior of the bias and the covarianceof the periodogram   I  T  .  Bispectrum estimation 5 Proposition 1  Let   λ 1 , λ 2 , µ 1  and   µ 2  be any real numbers, then  IE  {   I  T  ( λ 1 ,λ 2 ) }  =  φ (3) Z  ( λ 1 ,λ 2 ) + O (1 /T  ) where   O (1 /T  )  is uniform in   λ 1  and   λ 2 . Moreover, the covariance is  1 T   cov    I  T  ( λ 1 ,λ 2 ) ,   I  T  ( µ 1 ,µ 2 )   = 12 π  p ∈S 3 3  i =1   ∆ T  ( λ i  + µ p ( i ) ) φ (2) Z  ( λ i ) + O (1 /T  ) where   λ 3  =  − ( λ 1  +  λ 2 ) ,  µ 3  =  − ( µ 1  +  µ 2 ) ,  S  3  is the set of all permutations of   { 1 , 2 , 3 }  and    ∆ T  ( λ ) = sin( Tλ/ 2) / ( Tλ/ 2) . Theorem 1  If both assumptions 1 and 2 are satisfied, then the bias and the covariance of    φ (3) Z,T   are given by  IE  {   φ (3) Z,T  ( λ 1 ,λ 2 ) }  =  φ (3) Z  ( λ 1 ,λ 2 ) + O ( b T  T   ) and   lim T  → + ∞ T b 2 T  cov    φ (3) Z,T  ( λ 1 ,λ 2 ) ,   φ (3) Z,T  ( µ 1 ,µ 2 )  = 2 π 3  i =1 φ (2) Z  ( λ i )  p ∈S 3 3  i =1 δ  ( λ i  − µ p ( i ) )    + ∞−∞    + ∞−∞ W  2 ( u 1 ,u 2 , − u 1  − u 2 ) du 1 du 2 . In order to estimate the function  ψ , we propose the following estimate:   ψ T  ( λ ) = 1(2 π ) 2 T     + ∞−∞ W  T  ( λ − u )   D X,T  ( u )   d X,T  ( − u ) du, where   D X,T  ( λ ) =    N  ( T  ) j =1  exp( − iλτ  j ) X  2 ( τ  j ). Then, the asymptotic behav-ior of this estimate is studied in the following proposition. Proposition 2  Under Assumptions 1 and 2, we have  IE  {   ψ T  ( λ ) }  =  ψ ( λ ) + O  1 T   ,  (3) and cov  {   ψ T  ( λ ) ,   ψ T  ( µ ) }  =  C  5 δ  ( λ ) δ  ( µ ) D 21 (0)   R 2 D 1 ( x ) D 1 ( y ) D 1 ( − x − y ) dxdy + β  2 2 π  ( δ  ( λ ) + δ  ( µ )) W  (0) + O   1 b 2 T   For the term  C  (3) X  (0 , 0) β/ (2 π ) 2 , we propose the following estimate:   C  T   = 1(2 π ) 2 T   N  ( T  )  j =1 X  3 ( τ  j ) = 1(2 π ) 2 T     T  0 X  3 ( t ) d  N  ( t ) for which the asymptotic properties are given in the following proposition.

lição 4 jovens

Jan 8, 2019

BenhenniRachdi

Jan 8, 2019
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