Spectral Analysis of Multi-dimensional Self-similar Markov Processes
aa r X i v : . [ m a t h . P R ] M a r Spectral Analysis of Multi-dimensional Self-similar MarkovProcesses
N. Modarresi and S. Rezakhah ∗ Abstract
In this paper we consider a discrete scale invariant (DSI) process { X ( t ) , t ∈ R + } withscale l >
1. We consider to have some fix number of observations in every scale, say T ,and to get our samples at discrete points α k , k ∈ W where α is obtained by the equality l = α T and W = { , , . . . } . So we provide a discrete time scale invariant (DT-SI) pro-cess X ( · ) with parameter space { α k , k ∈ W } . We find the spectral representation of thecovariance function of such DT-SI process. By providing harmonic like representation ofmulti-dimensional self-similar processes, spectral density function of them are presented.We assume that the process { X ( t ) , t ∈ R + } is also Markov in the wide sense and pro-vide a discrete time scale invariant Markov (DT-SIM) process with the above schemeof sampling. We present an example of DT-SIM process, simple Brownian motion, bythe above sampling scheme and verify our results. Finally we find the spectral densitymatrix of such DT-SIM process and show that its associated T -dimensional self-similarMarkov process is fully specified by { R Hj (1) , R Hj (0) , j = 0 , , . . . , T − } where R Hj ( τ ) isthe covariance function of j th and ( j + τ )th observations of the process. AMS 2000 Subject Classification:
Keywords:
Discrete scale invariance; Wide sense Markov; Spectral representation;Multi-dimensional self-similar processes.
The concept of stationarity and self-similarity are used as a fundamental property to han-dle many natural phenomena. Lamperti transformation defines a one to one correspondencebetween stationary and self-similar processes. A function is scale invariant if it is identicalto any of its rescaled version, up to some suitable renormalization in amplitude. Discretescale invariance (DSI) process can be defined as the Lamperti transform of periodically corre-lated (PC) process [2]. Many critical systems, like statistical physics, textures in geophysics,network traffic and image processing can be interpreted by these processes [1]. Flandrin et. ∗ Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue,Tehran 15914, Iran. E-mail: [email protected] (N. Modarresi), [email protected] (S. Rezakhah). l >
1, and we get our samplesat points α k , where k ∈ W , l = α T , W = { , , . . . } and T is the number of samples in eachscale. By such sampling we provide a discrete time scale invariant process in the wide senseand find the spectral representation of the covariance function of such process.This paper is organized as follows. In section 2, we present stationary and self-similarprocesses by shift and renormalized dilation operators. Then we provide a suitable platformfor our study of discrete time self-similar (DT-SS) and discrete time scale invariant (DT-SI)processes by introducing quasi Lamperti transformation. Harmonizable representation ofthese processes are expressed in this section too. Also by using the spectral density matrix ofPC processes, the spectral representation of the covariance function of DT-SI processes aregiven. In section 3, a harmonic like representation of multi-dimensional self-similar processesand spectral density function of them are obtained. As an example we introduce a processcalled simple Brownian motion which is DSI and Markov too. Finally a discrete time scaleinvariant Markov (DT-SIM) process with the above scheme of sampling is considered insection 3 and the spectral density matrix of such process and its associated T -dimensionalself-similar Markov process are characterized. In this section, by using renormalized dilation operator, we define discrete time self-similarand discrete time scale invariant processes. The quasi Lamperti transformation and its prop-erties are introduced. We also present the harmonizable representation of stationary andharmonic like representation of self-similar processes. The spectral density of PC processesand the spectral representation of the covariance function of DT-SI processes are given.2 .1 Stationary and self-similar processes
Definition 2.1
Given τ ∈ R , the shift operator S τ operates on process { Y ( t ) , t ∈ R } ac-cording to ( S τ Y )( t ) := Y ( t + τ ) . (2.1) A process { Y ( t ) , t ∈ R } is said to be stationary, if for any t, τ ∈ R { ( S τ Y )( t ) } d = { Y ( t ) } (2.2) where d = is the equality of all finite-dimensional distributions.If (2 . holds for some τ ∈ R , the process is said to be periodically correlated. The smallestof such τ is called period of the process. Definition 2.2
Given some numbers
H > and λ > , the renormalized dilation operator D H,λ operates on process { X ( t ) , t ∈ R + } according to ( D H,λ X )( t ) := λ − H X ( λt ) . (2.3) A process { X ( t ) , t ∈ R + } is said to be self-similar of index H , if for any λ > { ( D H,λ X )( t ) } d = { X ( t ) } . (2.4) The process is said to be
DSI of index H and scaling factor λ > or (H, λ )-DSI , if (2 . holds for λ = λ . As an intuition, self-similarity refers to an invariance with respect to any dilation factor.However, this may be a too strong requirement for capturing in situations that scaling prop-erties are only observed for some preferred dilation factors.
Definition 2.3
A process { X ( k ) , k ∈ ˇT } is called discrete time self-similar (DT-SS) processwith parameter space ˇ T , where ˇ T is any subset of distinct points of positive real numbers, iffor any k , k ∈ ˇ T { X ( k ) } d = ( k k ) H { X ( k ) } . (2.5) The process X ( · ) is called discrete time scale invariance (DT-SI) with scale l > and param-eter space ˇ T , if for any k , k = lk ∈ ˇ T , (2 . holds. Remark 2.1
If the process { X ( t ) , t ∈ R + } is DSI with scale l = α T for fixed T ∈ N and α > , then by sampling of the process at points α k , k ∈ W where W = { , , . . . } , we have X ( · ) as a DT-SI process with parameter space ˇ T = { α k ; k ∈ W } and scale l = α T . If weconsider sampling of X ( · ) at points α nT + k , n ∈ W , for fixed k = 0 , , . . . , T − , then X ( · ) isa DT-SS process with parameter space ˇ T = { α nT + k ; n ∈ W } . L H to the classof wide-sense stationary processes. This class encompasses all strictly self-similar processeswith finite variance, including Gaussian processes such as fractional Brownian motion but noother alpha-stable processes. Definition 2.4
A random process { X ( t ) , t ∈ R + } is said to be wide sense self-similar withindex H, for some H > if the following properties are satisfied for each a > i ) E [ X ( t )] < ∞ , ( ii ) E [ X ( at )] = a H E [ X ( t )] , ( iii ) E [ X ( at ) X ( at )] = a H E [ X ( t ) X ( t )] .This process is called wide sense DSI of index H and scaling factor a > , if the aboveconditions hold for some a = a . Definition 2.5
A random process { X ( k ) , k ∈ ˇ T } is called DT-SS in the wide sense withindex
H > and with parameter space ˇ T , where ˇ T is any subset of distinct points of positivereal numbers, if for all k, k ∈ ˇ T and all a > , where ak, ak ∈ ˇ T :( i ) E [ X ( k )] < ∞ , ( ii ) E [ X ( ak )] = a H E [ X ( k )] , ( iii ) E [ X ( ak ) X ( ak )] = a H E [ X ( k ) X ( k )] .If the above conditions hold for some fixed a = a , then the process is called DT-SI in thewide sense with scale a . Remark 2.2
Let { X ( t ) , t ∈ R + } in Remark . be DSI in the wide sense. Then X ( · ) withparameter space ˇ T = { α k ; k ∈ W } for α > is DT-SI in the wide sense, where W = { , , . . . } and X ( · ) with parameter space ˇ T = { α nT + k ; n ∈ W } for fixed T ∈ N and α > is DT-SS inthe wide sense.
Through this paper we are dealt with wide sense self-similar and wide sense scale invariantprocess, and for simplicity we omit the term ”in the wide sense” hereafter.
We introduce the quasi Lamperti transformation and its properties by followings.
Definition 2.6
The quasi Lamperti transform with positive index H and α > , denoted by L H,α operates on a random process { Y ( t ) , t ∈ R } as L H,α Y ( t ) = t H Y (log α t ) (2.6)4 nd the corresponding inverse quasi Lamperti transform L − H,α on process { X ( t ) , t ∈ R + } actsas L − H,α X ( t ) = α − tH X ( α t ) . (2.7)One can easily verify that L H,α L − H,α X ( t ) = X ( t ) and L − H,α L H,α Y ( t ) = Y ( t ) . Note that in theabove definition, if α = e , we have the usual Lamperti transformation L H . Theorem 2.1
The quasi Lamperti transform guarantees an equivalence between the shiftoperator S log α k and the renormalized dilation operator D H,k in the sense that, for any k > L − H,α D H,k L H,α = S log α k . (2.8) Proof: L − H,α D H,k L H,α Y ( t ) = L − H,α D H,k ( t H Y (log α t )) = L − H,α ( k − H ( kt ) H Y (log α kt ))= L − H,α ( t H Y (log α kt )) = α − tH ( α t ) H Y (log α kα t ) = Y (log α k + t ) = S log α k Y ( t ) . (cid:3) Corollary 2.1 If { Y ( t ) , t ∈ R } is stationary process, its quasi Lamperti transform {L H,α Y ( t ) ,t ∈ R + } is self-similar. Conversely if { X ( t ) , t ∈ R + } is self-similar process, its inverse quasiLamperti transform {L − H,α X ( t ) , t ∈ R } is stationary. Corollary 2.2 If { X ( t ) , t ∈ R + } is ( H, α T )-DSI then L − H,α X ( t ) = Y ( t ) is PC with period T > . Conversely if { Y ( t ) , t ∈ R } is PC with period T then L H,α Y ( t ) = X ( t ) is ( H, α T )-DSI . Remark 2.3 If X ( · ) is a DT-SS process with parameter space ˇ T = { l n , n ∈ W } , then itsstationary counterpart Y ( · ) has parameter space ˇ T = { nT, n ∈ N } : X ( l n ) = L H,α Y ( l n ) = l nH Y (log α α nT ) = α nT H Y ( nT ) . Also it is clear by the following relation that if X ( · ) is a DT-SI process with scale l = α T , T ∈ N and parameter space ˇ T = { α k , k ∈ W } , then Y ( · ) is a discrete time periodicallycorrelated (DT-PC) process with period T and parameter space ˇ T = { n, n ∈ N } : Y ( n ) = L − H,α X ( n ) = α − nH X ( α n ) . .3 Harmonizable representation A stationary process Y ( t ), EY ( t ) = 0, can be represented as Y ( t ) = Z ∞−∞ e iωt dϕ ( ω ) (2.9)which is called harmonizable representation of the process, and ϕ ( ω ) is a right continuousorthogonal increment process, see [9]. Also the covariance function can be represented as R Y ( t, s ) = Z ∞−∞ Z ∞−∞ e itω − isω ′ d Φ( ω, ω ′ ) (2.10)where the spectral measure satisfies d Φ( ω, ω ′ ) = E [ dϕ ( ω ) dϕ ( ω ′ )] = (cid:26) ω = ω ′ d Ψ( ω ) ω = ω ′ (2.11)and d Ψ( ω ) = E [ | dϕ ( ω ) | ]. All the spectral mass is located on the diagonal line ω = ω ′ . WhenΦ( ω, ω ′ ) is absolutely continuous, we have spectral density φ ( ω, ω ′ ) such that d Φ( ω, ω ′ ) = φ ( ω, ω ′ ) dωdω ′ . A necessary and sufficient condition for this equality to hold, as Loeve’s condi-tion for harmonizability, is that Φ( ω, ω ′ ) must satisfy R R | d Φ( ω, ω ′ ) | < ∞ . The correspondingnotion for processes after a Lamperti transformation introduces a new representation for aclass of processes deviating from self-similarity, which is called multiplicative harmonizability.A self-similar process X ( t ) has harmonic like representation as an inverse Mellin transform,namely an integral of uncorrelated spectral increments dϕ ( ω ) on the Mellin basis [1]. X ( t ) = Z t H + iω dϕ ( ω ) (2.12)and the process has this property if it verifies as R X ( t, s ) = Z Z t H + iω s H − iω ′ d Φ( ω, ω ′ ) . (2.13)The inverse Mellin transformation gives the expression of the spectral function if the corre-lation is known as [2] φ ( ω, ω ′ ) = Z Z t − H − iω s − H + iω ′ R X ( t, s ) dtt dss . (2.14) The spectral density of a PC process is introduced by Gladyshev in [8]. If Y ( n ) is a DT-PC process, the spectral density matrix is Hermitian nonnegative definite T × T matrix offunctions f ( ω ) = [ f jk ( ω )] j,k =0 , ,...,T − , and the covariance function has the representation R n ( τ ) := Cov (cid:0) Y ( n ) , Y ( n + τ ) (cid:1) = T − X k =0 B k ( τ ) e kπin/T (2.15)6here B k ( τ ) = Z π e iτω f k ( ω ) dω. Also f k ( ω ) and f jk ( ω ) , j, k = 0 , , . . . , T − f jk ( ω ) = 1 T f k − j (cid:0) ( ω − πj ) /T (cid:1) , ω < π. For k < ω < ω > π , the functions f k ( ω ) are defined by the equality f k + T ( ω ) = f k ( ω )and f k ( ω ) = f k ( ω + 2 π ).Let { X ( t ) , t ∈ R + } be a zero mean DSI process with scale l . If l <
1, we reduce the timescale, so that l in the new time scale is greater than 1. Our sampling scheme is to get samplesat points α k , k ∈ W , where by choosing the number of samples in each scale, say T ∈ N , wefind α by l = α T . Therefore the process under study { X ( α n ) , n ∈ W } is DT-SI with scale l = α T . Proposition 2.2 If X ( α n ) is DT-SI with scale l = α T , T ∈ N , then we have the spectralrepresentation of the covariance function of the process as R Hn ( τ ) := Cov (cid:0) X ( α n ) , X ( α n + τ ) (cid:1) = α (2 n + τ ) H T − X k =0 B k ( τ ) e kπin/T (2.16) where B k ( τ ) = Z π e iτω f k ( ω ) dω (2.17) and f jk ( ω ) = 1 T f k − j (cid:0) ( ω − πj ) /T (cid:1) (2.18) for j, k = 0 , , . . . , T − and ω < π . Proof:
According to (2.6) and Corollary 2.1, for any n, τ ∈ W R Hn ( τ ) = E [ X ( α n ) X ( α n + τ )] = E [ L H,α Y ( α n ) L H,α Y ( α n + τ )]= α (2 n + τ ) H E [ Y ( n ) Y ( n + τ )]where Y ( n ) is DT-PC process with period T = log α l . Thus by (3.1) R Hn ( τ ) = α (2 n + τ ) H R n ( τ ) = α (2 n + τ ) H T − X k =0 B k ( τ ) e kπin/T . (cid:3) Characterization of the spectrum
In this section we provide spectral density matrix of multi-dimensional self-similar process W ( n ). By using harmonic like representation of a self-similar process, we characterize thespectral density matrix of DT-SI process in subsection 3.1. A discrete time scale invariantMarkov (DT-SIM) process with a new scheme of sampling is considered and the propertiesof an introduced example is verfied too. The spectral density matrix of such process and itsassociated T -dimensional self-similar Markov process are characterized in subsection 3.2. By Rozanov [12], if ξ ( t ) = { ξ k ( t ) } k =1 ,...,n be an n -dimensional stationary process, then ξ ( t ) = Z e iλt φ ( dλ ) (3.1)is its spectral representation, where φ = { ϕ k } k =1 ,...,n and ϕ k is the random spectral measureassociated with the k th component ξ k of the n -dimensional process ξ . Let B kr ( τ ) = E [ ξ k ( τ + t ) ξ r ( t )] , k, r = 1 , . . . , n and B ( τ ) = [ B kr ( τ )] k,r =1 ,...,n be the correlation matrix of ξ . The components of the correla-tion matrix of the process ξ can be represented as B kr ( τ ) = Z e iλτ F kr ( dλ ) , k, r = 1 , . . . , n (3.2)where for any Borel set ∆, F kr (∆) = E [ ϕ k (∆) ϕ r (∆)] are the complex valued set functionswhich are σ -additive and have bounded variation. For any k, r = 1 , . . . , n , if the sets ∆ and∆ ′ do not intersect, E [ ϕ k (∆) ϕ r (∆ ′ )] = 0. For any interval ∆ = ( λ , λ ) when F kr ( { λ } ) = F kr ( { λ } ) = 0 the following relation holds F kr (∆) = 12 π Z ∆ ∞ X τ = −∞ B kr ( τ ) e − iλτ dλ (3.3)= 12 π B kr (0)[ λ − λ ] + lim T →∞ π X < | τ | T B kr ( τ ) e − iλ τ − e − iλ τ − iτ in the discrete parameter case, and F kr (∆) = lim a →∞ π Z a − a e − iλ τ − e − iλ τ − iτ B kr ( τ ) dτ in the continuous parameter case. 8sing the above results of Rozanov for multi-dimensional stationary processes and usingthe Lamperti transformation, we present the definition of multi-dimensional self-similar pro-cess and obtain the properties of the corresponding multi-dimensional self-similar process bythe following theorem. Definition 3.1
The process U ( t ) = (cid:0) U ( t ) , U ( t ) , . . . , U q − ( t ) (cid:1) is a q-dimensional discretetime self similar process in the wide sense with parameter space ˇ T , which consists of finite orcountable many points of R + , if the followings are satisfied ( a ) { U j ( · ) } for every j = 0 , , · · · , q − is DT-SS process with parameter space ˇ T . ( b ) U i ( · ) and U j ( · ) for i, j = 0 , , · · · , q − have self-similar correlation, that is Cov (cid:0) U i ( ts ) , U j ( tr ) (cid:1) = t H Cov (cid:0) U i ( s ) , U j ( r ) (cid:1) . where s, r, ts, tr are in ˇ T . Theorem 3.1
Let W ( α k ) = (cid:0) W ( α k ) , W ( α k ) , . . . , W q − ( α k ) (cid:1) , k ∈ Z , α > be a discretetime q -dimensional self-similar process. Then (i) The harmonic like representation of W j ( α k ) is W j ( α k ) = α kH Z π e iωk dϕ j ω ) . (3.4) where ϕ j ( ω ) is the corresponding spectral measure, that E [ dϕ j ( ω ) dϕ r ( ω ′ )] = dD Hjr ( ω ) , j, r =0 , , . . . , q − when ω = ω ′ and is when ω = ω ′ . We call D Hjr ( ω ) the spectral distributionfunction of the process. (ii) The corresponding spectral density matrix of { W ( α k ) , k ∈ Z } is d H ( ω ) = [ d Hjr ( ω )] j,r =0 ,...,q − ,where d Hjr ( ω ) = 12 π ∞ X n = −∞ α − nH e − iωn Q Hjr ( α n ) (3.5) α > and Q Hjr ( α n ) is the covariance function of W j ( α n ) and W r (1) . Before proceeding to the proof of the theorem we remind that based on our samplingscheme at points α k , k ∈ Z , of continuous DSI process with scale l = α T , α ∈ R , T ∈ N . Sowe consider W ( · ) at points l n = α nT as the corresponding T -dimensional DT-SS process andapply this theorem in Lemma 3.4. 9 roof of (i): W j ( α k ) for j = 0 , , . . . , q − ξ j ( k )has spectral representation ξ j ( k ) = R π e iωk dϕ j ( ω ). Thus by (2.6) W j ( α k ) = L H,α ξ j ( α k ) = α kH ξ j ( k ) = α kH Z π e iωk dϕ j ( ω ) . Proof of (ii):
The covariance matrix is denoted by Q H ( n, τ ) = [ Q Hjr ( α n , α τ )] j,r =0 ,...,q − where its elements have the spectral representation Q Hjr ( α m , α τ ) = E [ W j ( α m α τ ) W r ( α m )] = α mH E [ W j ( α τ ) W r (1)] = α mH Q Hjr ( α τ ) . (3.6)Also by (3.4) Q Hjr ( α τ ) = α τH E [ Z π e iωτ dϕ j ( ω ) Z π dϕ r ( ω ′ )] = α τH Z π e iωτ dD Hjr ( ω ) (3.7)where E [ dϕ j ( ω ) dϕ r ( ω ′ )] = dD Hjr ( ω ) when ω = ω ′ and is 0 when ω = ω ′ .The spectral distribution function of the correlation matrix Q H ( α k ) = [ Q Hjr ( α k )] j,r =0 ,...,q − is D H ( dω ) = [ D Hjr ( dω )] j,r =0 ,...,q − . (3.2)-(3.3), (3.7) and appropriate transformation we have D Hjr ( A ) = 12 π Z A ∞ X n = −∞ α − nH e − iλn Q Hjr ( α n ) dλ. (3.8)Let A = ( ω, ω + dω ], then we have the spectral density matrix as d H ( ω ) = [ d Hjr ( ω )] j,r =0 ,...,q − where d Hjr ( ω ) := D Hjr ( dω ) dω = 12 π ∞ X n = −∞ (cid:0) dω Z ω + dωω e − iλn dλ (cid:1) α − Hn Q Hjr ( α n )= 12 π ∞ X n = −∞ (cid:0) − in lim dω → e − i ( ω + dω ) n − e − iωn dω (cid:1) α − Hn Q Hjr ( α n )= 12 π ∞ X n = −∞ (cid:0) ( − in )( − in ) e − iωn (cid:1) α − Hn Q Hjr ( α n ) = 12 π ∞ X n = −∞ α − nH e − iωn Q Hjr ( α n ) . Existence of d Hjr ( ω ) follows from part (i) of the theorem as W k ( α n ) is the Lamperti counter-part of the stationary process ξ k ( n ), k = 0 , . . . , q − (cid:3) .2 Spectral density of DT-SIM process Let { X ( t ) , t ∈ R } be a DSI process with scale l and Markov in the wide sense. Using oursampling scheme described in this section, we assume l and α to be greater than one. Thus { X ( α n ) , n ∈ W } is a discrete time scale invariant Markov (DT-SIM) process with scale l = α T .Let R ( t , t ) be some function defined on T ×T and suppose that R ( t , t ) = 0 everywhereon T ×T , where T is an interval. Borisov [3] showed that the necessary and sufficient conditionfor R ( t , t ) to be the covariance function of a Gaussian Markov process with time space T is that R ( t , t ) = G (cid:0) min( t , t ) (cid:1) K (cid:0) max( t , t ) (cid:1) (3.9)where G and K are defined uniquely up to a constant multiple and the ratio G/K is a positivenondecreasing function on T .It should be noted that the Borisov result on Gaussian Markov processes can be easilyderived in the discrete case for second order Markov processes in the wide sense, by usingTheorem 8.1 of Doob [5].Here we present a closed formula for the covariance function of the DT-SIM processand characterized the covariance matrix of corresponding T -dimensional self-similar Markovprocess by theorems 3.2 and 3.3 [10]. Theorem 3.2
Let { X ( α n ) , n ∈ Z } be a DT-SIM process with scale l = α T , α > , T ∈ N ,then the covariance function R Hn ( τ ) = E [ X ( α n + τ ) X ( α n )] (3.10) where τ ∈ Z , n = 0 , , . . . , T − , R Hn + T ( τ ) = α T H R Hn ( τ ) and R Hn ( τ ) = 0 is of the form R Hn ( kT + v ) = [˜ h ( α T − )] k ˜ h ( α v + n − )[˜ h ( α n − )] − R Hn (0) (3.11) R Hn ( − kT + v ) = α − kT H R Hn + v (( k − T + T − v ) where k ∈ Z , v = 0 , , . . . , T − , ˜ h ( α r ) = r Y j =0 h ( α j ) = r Y j =0 R Hj (1) /R Hj (0) , r ∈ Z (3.12) and ˜ h ( α − ) = 1 . Proof:
Here we present the sketch of the proof. From the Markov property (3.9), for α > R Hn ( τ ) = G ( α n ) K ( α n + τ ) τ ∈ W . (3.13)11nd R Hn (0) = G ( α n ) K ( α n ) . Thus K ( α n + τ ) = R Hn ( τ ) R Hn (0) K ( α n ) . (3.14)For τ = 1, by a recursive substitution in (3.14) one can easily verify that K ( α n ) = K (1) n − Y j =0 h ( α j ) (3.15)where h ( α j ) = R Hj (1) /R Hj (0) . Hence for n = 0 , , . . . , T − , k ∈ W K ( α kT + n ) = K (1) kT + n − Y j =0 h ( α j ) . As X ( · ) is DT-SI with scale α T by (3.10) h ( α T + i ) = R HT + i (1) R HT + i (0) = R Hi (1) R Hi (0) = h ( α i ) , i ∈ W . Therefore using (3.12) we have kT + n − Y j =0 h ( α j ) = [˜ h ( α T − )] k ˜ h ( α n − ) . (3.16)Consequently for n = 0 , , . . . , T − K ( α kT + n ) = K (1) (cid:20) ˜ h ( α T − ) (cid:21) k ˜ h ( α n − ) . (3.17)Let τ = kT + v , then it follows from (3.14) and (3.17) that R Hn ( kT + v ) = K ( α n + kT + v ) K ( α n ) R Hn (0) = K (1)[˜ h ( α T − )] k ˜ h ( α v + n − ) K (1)˜ h ( α n − ) R Hn (0)= [˜ h ( α T − )] k ˜ h ( α v + n − )[˜ h ( α n − )] − R Hn (0)for k = 0 , , . . . , α > n, v = 0 , , . . . , T −
1. Similar to the above proof for τ = − kT + v we have R Hn ( − kT + v ) = [˜ h ( α T − )] − k ˜ h ( α v + n − )[˜ h ( α n − )] − R Hn (0)and also note that R Hn ( − kT + v ) = E [ X ( α − kT + n + v ) X ( α n )] = α − kT H E [ X ( α n + v ) X ( α kT + n )]= α − kT H R Hn + v ( kT + v ) = α − kT H R Hn + v (( k − T + T − v ) . (cid:3) xample 3.1 We consider moving of a particle in different environment A , A , . . . basedon Brownian motion with different rates. Specially we consider this movement by X ( t ) withindex H > and scale λ > as X ( t ) = ∞ X n =1 λ n ( H − ) I [ λ n − ,λ n ) ( t ) B ( t ) where B ( · ) , I ( · ) are Brownian motion and indicator function respectively and we call thisprocess, simple Brownian motion. Let A = [1 , λ ), A = [ λ, λ ) and A n = [ λ n − , λ n ) as disjoint sets. The process X ( t ) isDSI and Markov too. For checking these properties, first we find the covariance function ofit. The covariance function of the process for t ∈ A n , s ∈ A m and s t isCov (cid:0) X ( t ) , X ( s ) (cid:1) = λ ( n + m )( H − ) Cov (cid:0) B ( t ) , B ( s ) (cid:1) = λ ( n + m )( H − ) s (3.18)since as we know Cov (cid:0) B ( t ) , B ( s ) (cid:1) = min { t, s } . Therefore by the condition (3 . t is in ( λ n − , λ n ] then λt is in ( λ n , λ n +1 ]. Thus for t ∈ A n +1 and s ∈ A m +1 we haveCov (cid:0) X ( λt ) , X ( λs ) (cid:1) = λ ( n + m +2)( H − ) Cov (cid:0) B ( λt ) , B ( λs ) (cid:1) = λ ( n + m +2)( H − ) λs = λ H λ ( n + m )( H − ) s = λ H Cov (cid:0) X ( t ) , X ( s ) (cid:1) . Then X ( t ) is (H, λ )-DSI.By sampling of the process X ( · ) at points α n , n ∈ W , where λ = α T , T ∈ N and λ > .
2. For j = kT + i where i = 0 , , . . . , T − k = 0 , , . . . by (3 .
18) we have that h ( α j ) = R Hj (1) R Hj (0) = Cov (cid:0) X ( α j +1 ) , X ( α j ) (cid:1) Cov (cid:0) X ( α j ) , X ( α j ) (cid:1) = α k +1) T H ′ + j α k +1) T H ′ + j = 1as α j , α j +1 ∈ A k +1 and H ′ = H − . Also for j = kT + T − h ( α j ) = R Hj (1) R Hj (0) = Cov (cid:0) X ( α j +1 ) , X ( α j ) (cid:1) Cov (cid:0) X ( α j ) , X ( α j ) (cid:1) = α (2 k +3) T H ′ + j α (2 k +2) T H ′ + j = α T H ′ as α j ∈ A k +1 and α j +1 ∈ A k +2 . Thus for j = kT + i , i = 0 , , . . . , T − k = 0 , , . . . ˜ h ( α kT + i ) = kT + i Y r =0 h ( α r ) = k Y r =0 α T H ′ = α kT H ′ j = kT + T − h ( α kT + T − ) = kT + T − Y r =0 h ( α r ) = k +1 Y r =0 α T H ′ = α ( k +1) T H ′ . Finally as ˜ h ( α T − ) = α T H ′ ,˜ h ( α v + n − ) = (cid:26) v + n − T − α T H ′ v + n − > T − h ( α n − ) = 1, R Hn (0) = E [ X ( α n ) X ( α n )] = α T H ′ + n , n = 0 , , . . . , T −
1. Thus R Hn ( kT + v ) = (cid:26) α ( k +2) T H ′ + n v + n − T − α ( k +3) T H ′ + n v + n − > T − .
18) we have the same result. (cid:3)
Corresponding to the DT-SIM process, { X ( α k ) , k ∈ Z } with scale l = α T , α > T ∈ N there exists a T -dimensional discrete time self-similar Markov process W ( t ) = (cid:0) W ( t ) , W ( t ) , . . . , W T − ( t ) (cid:1) with parameter space ˇ T = { l n ; n ∈ W , l = α T } , where W k ( l n ) = W k ( α nT ) = X ( α nT + k ) , k = 0 , . . . , T − . (3.19)The elements of the covariance matrix which is defined by (3.6) at points l n and l τ by (3.10)and (3.11) can be written as Q Hjk ( l n , l τ ) = E [ W j ( l n + τ ) W k ( l n )] = α nHT E [ X ( α τT + j ) X ( α k )]= α nHT R Hk ( τ T + j − k ) = α nHT [˜ h ( α T − )] τ C Hjk R Hk (0) (3.20)in which C Hjk = ˜ h ( α j − )[˜ h ( α k − )] − and R Hk ( · ) is defined in (3.10). Theorem 3.3
Let { X ( α n ) , n ∈ W } be a DT-SIM process with the covariance function R Hn ( τ ) . Also let { W ( l n ) , n ∈ W } , defined in (3 . , be its associated T -dimensional dis-crete time self-similar Markov process with covariance function Q H ( l n , l τ ) . Then Q H ( l n , l τ ) = α nHT C H R H [˜ h ( α T − )] τ , τ ∈ W (3.21) where ˜ h ( · ) is defined by (3 . and the matrices C H and R H are given by C H = [ C Hjk ] j,k =0 , ,...,T − ,where C Hjk = ˜ h ( α j − )[˜ h ( α k − )] − , and R H = R H (0) 0 · · · R H (0) · · · ... ... ... ... · · · R HT − (0) . (cid:3) emark 3.1 It follows from Theorem . that for each k = 0 , , . . . , T − the process W k ( l n ) = X ( α nT + k ) is a self-similar Markov process for n ∈ W . The covariance function ofthe process is Γ Hk ( l n , l τ ) = E [ W k ( l n + τ ) W k ( l n )] = α nHT [˜ h ( α T − )] τ R Hk (0) , τ ∈ W where C Hkk = ˜ h ( α k − )[˜ h ( α k − )] − = 1 . The introduced T -dimensional self-similar Markov process W ( t ) with parameter spaceˇ T = { l n , n ∈ W } , l = α n , T ∈ N , is the counterpart of the T -dimensional stationaryMarkov process Y ( t ) = (cid:0) Y ( t ) , Y ( t ) , . . . , Y T − ( t ) (cid:1) . The spectral density matrix of such T -dimensional self-similar process is characterized by the following lemma. Lemma 3.4
The spectral density matrix d H ( ω ) = [ d Hjr ( ω )] j,r =0 ,...,T − of the T -dimensionalself-similar process { W ( l n ) , n ∈ W } , defined by (3 . , where l = α T has the Markov propertyand is specified by d Hjr ( ω ) = 12 π " ˜ h ( α j − ) R Hr (0)˜ h ( α r − )(1 − e − iωT α − HT ˜ h ( α T − )) − ˜ h ( α r − ) R Hj (0)˜ h ( α j − ) (cid:0) − e − iωT α HT ˜ h − ( α T − ) (cid:1) where R Hk (0) is the variance of the process X ( · ) at point α k and ˜ h ( α k ) is defined by (3 . . Proof:
As we mentioned prior to the proof of the Theorem 3.1, we consider Q Hjr ( · , · ) atdiscrete points l m and l s where m, s ∈ Z , then Q Hjr ( l m , l s ) = E [ W j ( l m + s ) W r ( l m )] = l mH E [ W j ( l s ) W r (1)] = l mH Q Hjr ( l s ) . If the T -dimensional discrete time self-similar Markov process W ( · ) is sampled at points l n = α nT , then in (3.7) we have τ T instead of τ , thus in (3.5) we have nT instead of n and thecorresponding spectral density matrix of the covariance matrix Q H ( l s ) = [ Q Hjr ( l s )] j,r =0 , ,...,T − is d H ( ω ) = [ d Hjr ( ω )] j,r =0 , ,...,T − where d Hjr ( ω ) = 12 π ∞ X s =0 l − Hs e − iωsT Q Hjr ( l s ) + 12 π − X s = −∞ l − Hs e − iωsT Q Hjr ( l s ) := d Hjr ( ω ) + d Hjr ( ω ) . Using (3.20), we evaluate d Hjr ( ω ) as d Hjr ( ω ) = 12 π ∞ X s =0 e − iωsT l − Hs R Hr ( sT + j − r ) = 12 π ∞ X s =0 e − iωsT α − HsT [˜ h ( α T − )] s C Hjr R Hr (0)= ˜ h ( α j − ) R Hr (0)2 π ˜ h ( α r − ) ∞ X s =0 (cid:0) e − iωT α − HT ˜ h ( α T − ) (cid:1) s . (3.22)15ow we verify the convergence of the above summation. By (3.12) we have | e − iωT ˜ h ( α T − ) | = | ˜ h ( α T − ) | = T − Y j =0 | h ( α j ) | = T − Y j =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Hj (1) R Hj (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = T − Y j =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E [ X ( α j +1 ) X ( α j )] p E [ X ( α j +1 )] E [ X ( α j )] × E [ X ( α T ) X ( α T − )] p E [ X (1)] E [ X ( α T − )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By scale invariance of X ( · ) we have that E [ X ( α T )] = α T H E [ X (1)]. Now for j = 0 , . . . ,T − X ( α j +1 ) X ( α j )] < h ( α T − ) < α T H , and | e − iωT α − HT ˜ h ( α T − ) | < . Therefore the summation on the right side of (3.22) is convergent. Thus the spectral densityis d Hjr ( ω ) = ˜ h ( α j − ) R Hr (0)2 π ˜ h ( α r − ) × − e − iωT α − HT ˜ h ( α T − ) . Now we are to evaluate d Hjr ( ω ) as d Hjr ( ω ) = 12 π ∞ X s =1 l Hs e iωsT Q Hjr ( l − s ) . As Q Hjr ( l − s ) = E [ W j ( l − s ) W r (1)] = l − sH E [ W j (1) W r ( l s )] = l − sH Q Hrj ( l s ), so by a similarmethod, one can easily verify that d Hjr ( ω ) = ˜ h ( α r − ) R Hj (0)2 π ˜ h ( α j − ) × e iωT α − HT ˜ h ( α T − )1 − e iωT α − HT ˜ h ( α T − ) . So we arrive at the assertion of the lemma. (cid:3)
Remark 3.2
Lemma . provides the spectral density of discrete time self-similar Markovprocess { W k ( l n ) , n ∈ Z } , defined by (3 . , for k = 0 , , . . . , T − as d Hkk ( ω ) = R Hk (0) (cid:0) − α − HT ˜ h ( α T − ) (cid:1) π (cid:0) − ωT ) α − HT ˜ h ( α T − ) + α − HT ˜ h ( α T − ) (cid:1) . Remark 3.3
Using Lemma . , relations (2 . , (2 . and (3 . , we see that the spectraldensity matrix f ( ω ) = [ f jr ( ω )] j,r =0 , ,...,T − of a DT-SIM process which is defined by (2 . isfully specified by { R Hj (1) , R Hj (0) , j = 0 , , . . . , T − } . xample 3.2 Here we present the T-dimensional discrete time self-similar Markov processcorresponding to the simple Brownian motion, described in Example . , as W ( l n ) = ( W ( l n ) ,W ( l n ) , . . . , W T − ( l n )) , where W k ( l n ) = X ( α nT + k ) . Now as we mentioned in Lemma . we obtain spectral density matrix of W ( l n ) . In Example . we find that ˜ h ( α j − ) = 1 and ˜ h ( α r − ) = 1 as j, r = 0 , , . . . , T − , ˜ h ( α T − ) = α T H ′ , H ′ = H − and R Hr (0) = α T H ′ + r , R Hj (0) = α T H ′ + j thus the spectral density matrix of W ( l n ) is d Hjr ( ω ) = α T H ′ π (cid:20) α r − e − iωT α − T/ − α j − e − iωT α T/ (cid:21) . Acknowledgements
The authors would like to express their thanks to both anonymous referees for valuablecomments and suggestions which improve the original manuscript.
References [1] P. Borgnat, P.O. Amblard, P. Flandrin, ”Scale invariances and Lamperti transforma-tions for stochastic processes”,
Journal of Physics A: Mathematical and General , Vol.38,pp.20812101, 2005.[2] P. Borgnat, P. Flandrin, P.O. Amblard, ”Stochastic discrete scale invariance”,
IEEESignal Processing Letters , Vol. 9, No. 6, pp. 181-184, 2002.[3] I.S. Borisov, ”On a criterion for Gaussian random processes to be Markovian”,
TheoryProbab. Appl. , No.27, pp.863-865, 1982.[4] M.E. Caballero, L. Chaumont, ”Weak convergence of positive self-similar Markov pro-cesses and overshoots of Levy processes”, The annals of probability, Vol.34, No.3,pp.1012-1034, 2006.[5] J.L. Doob, ”Stochastic Processes”, Wiley, New York 1953.[6] P. Flandrin, P. Borgnat, P.O. Amblard, ”From stationarity to selfsimilarity, and back: Variations on the Lamperti transformation”,
Lecture notes in Physics, Processes withLong-Range Correlations , Springer, 621, pp.88-117, 2003.[7] I.I. Gikhman, A.V. Skorkhod, ”The Theory of Stochastic Processes II”, Springer, 2004.[8] E.G. Gladyshev, ”Periodically correlated random sequences”,
Soviet Math. Dokl. , No.2,pp.385-388, 1961.[9] M. Loeve, ”Probability Theory”, New York: D. Van Nostrand, Princeton, 3rd ed. 1963.[10] N. Modarresi, S. Rezakhah, ”Discrete time scale invariant Markov processes”,arxiv.org/pdf/0905.3959v3, 2009.[11] A.R. Nematollahi, A.R. Soltani, ”Discrete time periodically correlated Markov pro-cesses”,
Probab. Math. Statist , Vol.20, Fasf.1, pp.127-140, 2000.[12] Y.A. Rozanov, ”Stationary Random Processes”, Holden-Day, San Francisco, 1967.[13] B. Yazici, R.L. Kashyap, ”A class of second-order stationary self-similar processes for1/f phenomena”,