The structure of self-similar stable mixed moving averages

Transcription

1 To appear in the ANNALS OF POBABILITY, 2002 The structure of self-similar stable mixe moving averages Vlaas Pipiras an Mura S. Taqqu Boston University January 24, 2002 Abstract Let α (1, 2) an X α be a symmetric α-stable (SαS) process with stationary increments given by the mixe moving average X α (t) = (G(x, t + u) G(x, u))m α (x, u), t, X where (X, X,µ) is a stanar Lebesgue space, G : X is some measurable function an M α is a SαS ranom measure on X with the control measure m(x, u) =µ(x)u. Assume, in aition, that X α is self-similar with exponent H (0, 1). In this work, we obtain a unique in istribution ecomposition of a process X α into three inepenent processes X α = X (1) α + X (2) α + X (3) α. We characterize X α (1) an X α (2) an provie examples of X α (3). The first process X α (1) can be represente as e κs (F (y, e s (t + u)) F (y, e s u))m α (y, s, u), Y where κ = H 1/α, (Y,Y,ν) is a stanar Lebesgue space an M α is a SαS ranom measure on Y with the control measure m(y, s, u) =ν(y)su. Particular cases inclue the limit of renewal rewar processes, the so-calle ranom wavelet expansion an Takenaka process. The secon process X α (2) has the representation ( G1 (z)((t + u) κ + u κ +)+G 2 (z)((t + u) κ u κ ) ) M α (z, u), if κ 0, Z ( G1 (z) (ln t + u ln u )+G 2 (z)(1 (0, ) (t + u) 1 (0, ) (u)) ) M α (z, u), if κ =0, Z This research was partially supporte by the NSF Grant ANI at Boston University. AMS Subject classification. Primary 60G18, 60G52; seconary 28D, 37A. Keywors an phrases: stable, self-similar processes with stationary increments, issipative an conservative flows, cocycles, semi-aitive functionals. Short title: Self-similar mixe moving averages

2 where (Z, Z,υ) is a stanar Lebesgue space an M α is a SαS ranom measure on Z with the control measure m(z, u) = υ(z)u. Particular cases inclue linear fractional stable motions, log-fractional stable motion an SαS Lévy motion. An example of a process X α (3) is 1 ( (t + u) κ + 1 [0,1/2) ({x +ln t + u ) u κ +1 [0,1/2) ({x +ln u ) ) M α (x, u), 0 where M α is a SαS ranom measure on [0, 1) with the control measure m(x, u) =xu an { is the fractional part function. 1 Introuction Our goal is to unerstan the structure of ranom processes {X α (t) t with the following three characteristics: X α is symmetric α-stable with 1 <α<2, it is self-similar with inex H (0, 1) an it is a (stationary increments) mixe moving average. We shall first efine these terms. A ranom variable ξ is symmetric α-stable (SαS, in short) with α (0, 2] if its characteristic function satisfies E exp(iθξ) = exp( σ α θ α ) for some scale factor σ>0 an all θ. A real-value stochastic process {X α (t) t is SαS with α (0, 2] if all its linear combinations are SαS ranom variables. The most common way to represent a SαS process X α is through the so-calle spectral representation { {X α (t) t = f t (s)m α (s), (1.1) S t where = stans for the equality in the sense of the finite-imensional istributions (see Samoronitsky an Taqqu (1994)). Here, M α is a SαS ranom measure with the control measure m, (S, m) isa measure space an {f t t is a collection of eterministic functions. The ranom integrator M α (s), s S, can be viewe heuristically as a sequence of inepenent stable ranom variables with scale factor m(s). The representation (1.1) means that the characteristic function of X α is given by { E exp i n k=1 { θ k X α (t k ) = exp S n θ k f tk (s) α m(s), (1.2) where θ 1,t 1,...,θ n,t n. One must therefore require that {f t t L α (S, m). The kernel f t (s) in (1.1) characterizes the process X α an our goal is to unerstan the possible structures of such kernels when X α is, in aition, self-similar an a (stationary increments) mixe moving average. The process X α is sai to be self-similar with inex H>0, if, for any c>0, k=1 {X α (ct) t = {c H X α (t) t. (1.3) The conition 1 <α<2 implies 0 <H<1 (Corollary in Samoronitsky an Taqqu (1994)). A SαS process {X α (t) t is calle a (stationary increments) mixe moving average, if it can be represente as { {X α (t) t = (G(x, t + u) G(x, u))m α (x, u), (1.4) X t where (X, X,µ) is some measure space an SαS ranom measure M α has the control measure µ(x)u on X. We call processes of the form (1.4) (stationary increments) mixe moving averages because, 2

3 heuristically, if one took the erivative of X α, one woul get a stationary process Ẋα with the representation X Ġ(x, t+u)m α(x, u) which is of the so-calle mixe moving average type. The mixe part comes from the presence of the aitional variable x in the representation of Ẋ α ; without x, Ẋ α woul be the usual moving average. The class of (stationary) mixe moving averages was introuce by Surgailis, osiński, Manrekar an Cambanis (1993) as an important extension of the usual moving averages. It is close uner linear combinations an the elements of the class are ergoic. This class also appears in the ecomposition of stationary symmetric stable processes establishe by osiński (1995). One example of processes with the above characteristics is linear fractional stable motion (see Samoronitsky an Taqqu (1994)) given by { ( a (t + u) H 1 α + u H 1 ) ( α + + b (t + u) H 1 α u H 1 ) α M α (u), (1.5) where a, b, α (0, 2), H (0, 1), u + = max{u, 0, u = max{ u, 0 an SαS ranom measure M α has the Lebesgue control measure on. To obtain (1.5) from (1.4), take Y = {1, µ(x) =δ {1 (x) an G(1,u)=au H 1/α + + bu H 1/α. Another example is the limit of renewal rewar processes (see Pipiras an Taqqu (2000)) represente as {(t z v) + (0 z v) + (z v) H 2 α 1 + M α (z, v), (1.6) where M α is a SαS ranom measure on 2 with the Lebesgue control measure m(z, v) =zv. To obtain (1.6), take X =, µ(x) =x an G(x, u) =(u 0+x) + x H 2 α 1 + in (1.4), an then make a change of variables u = z,x = z v. We showe in Pipiras an Taqqu (2000) that the process (1.6) is not a linear fractional stable motion in (1.5). ecall that, in contrast to the Gaussian case (α =2) where fractional Brownian motion is the only H self-similar process with stationary increments, there are many ifferent H self-similar processes with stationary increments in the stable case α (0, 2). (See Samoronitsky an Taqqu (1994) for aitional examples.) We are intereste in the following problem: Is there a way to somehow classify an characterize all stable self-similar processes with stationary increments which have the representation (1.4)? We assume that the mixing space X in (1.4) is the so-calle stanar Lebesgue space with a σ-algebra X an a σ-finite measure µ, an that SαS measure M α has the control measure m(x, u) =µ(x)u on X. ecall that a stanar Lebesgue space is a stanar Borel space with a σ-finite measure an that a stanar Borel space is a measurable space measurably isomorphic (i.e. there is a oneto-one, onto an bimeasurable map) to a Borel subset of a complete separable metric space. For example, n with a measure consisting of Lebesgue measure an iscrete point masses is a stanar Lebesgue space. We also implicitly assume that G : X is a measurable function an that {G t t L α (X,µ(x)u), where G t (x, u) =G(x, t + u) G(x, u), t, u, x X, so that the process X α in (1.4) is well-efine. Observe also that stationarity of the increments is alreay embee in the representation (1.4) whereas self-similarity requires aitional assumptions on G. To answer the above question, we will use results obtaine in Pipiras an Taqqu (2001a). In that paper, we showe that, if α (1, 2), every self-similar stationary increments mixe moving average 3

4 X α is generate by a nonsingular measurable multiplicative flow an by a relate cocycle an a semiaitive functional (see Section 2 below). Then, by using the so-calle Hopf s ecomposition of a flow into its issipative an conservative parts, we establishe a unique ecomposition in istribution of X α into two inepenent processes X α = X D α + X C α, (1.7) where the process Xα D is generate by a nonsingular issipative flow an Xα C is generate by a nonsingular conservative flow. In this paper, we stuy in greater etail the two components of the ecomposition (1.7). This will allow us to obtain a more refine ecomposition than (1.7). The paper is organize as follows. In Section 2, we recall some basic efinitions from Pipiras an Taqqu (2001a) which will be use here. Section 3, where we escribe cocycles an semi-aitive functionals for various types of flows, is of a technical nature. In Section 4, we show that processes X α, generate by issipative flows, can be completely characterize. Any such process can be represente in istribution as e κs (F (y, e s (t + u)) F (y, e s u))m α (y, s, u), (1.8) Y where (Y,Y,ν) is some stanar Lebesgue space, F : Y is a measurable function, M α is a SαS ranom measure on Y with the control measure ν(y)su an κ = H 1/α (see Theorem 4.1). We also mention some examples which, together with a further stuy of processes (1.8), can be foun in Pipiras an Taqqu (2001b). In Sections 5 8, we consier processes generate by conservative flows. Accoring to Theorem 5.1 in Section 5, any process X α, generate by the simplest conservative flow, namely an ientity flow, can be represente in istribution as (G 1 (z)((t + u) κ + u κ +)+G 2 (z)((t + u) κ u κ ))M α (z, u), if κ 0, (1.9) Z Z (G 1 (z)(ln t + u ln u )+G 2 (z)(1 (0, ) (t + u) 1 (0, ) (u)))m α (z, u), if κ =0, (1.10) where (Z, Z,υ) is a stanar Lebesgue space, G 1,G 2 : Z are some functions an a SαS ranom measure M α has the control measure υ(z)u. We call processes (1.9) mixe linear fractional stable motions (mixe LFSM, in short) because the usual linear fractional stable motions (that is, when Z = {1 an υ = δ {1 ) are special cases of processes (1.9). Since processes (1.10) can be viewe as limiting cases of processes (1.9) when κ 0 (for etails, see Definition 5.1 below), we shall call them mixe LFSM as well. Note that the usual log-fractional stable motion (when Z = {1, υ = δ {1 an G 2 0) an the stable Lévy motion (when Z = {1, υ = δ {1 an G 1 0) are special cases of (1.10). In Sections 6 an 7, we ecompose a process generate by a conservative flow into a mixe LFSM an, in analogy to osiński (1995), a process of the thir kin. An example of a process of the thir kin is given in Section 8. In Section 9, we obtain a ecomposition of self-similar processes with stationary increments, having the representation (1.4). This is in the spirit of the ecomposition obtaine by osiński in the context of stationary processes. By using (1.7) an results from Sections 4 8, we show that, for any α (1, 2), H>0, the H self-similar process X α with stationary increments, having the representation (1.4), can be uniquely ecompose in istribution into three inepenent processes X α = X (1) α + X (2) α + X (3) α. (1.11) 4

5 The process X α (1) in (1.11) is the first process Xα D in the ecomposition (1.7). It is generate by a issipative flow an, by results of Section 4, has the representation (1.8). The sum of the two processes X α (2) an X α (3) in (1.11) is the secon process Xα C in the ecomposition (1.7). The process X α (2) has the representation (1.9) or (1.10), epening on whether κ 0orκ = 0, that is, it is a mixe LFSM. The process X α (3) is H self-similar process with stationary increments, generate by a conservative flow, which oes not have a mixe LFSM component. An example is given in Section 8. The ecomposition (1.11), which is not the same as that of osiński, allows one to make a finer istinction between processes. For example, both linear fractional stable motion (1.5) an the limit (1.6) of the renewal rewar processes, were they to be ifferentiable, woul have a erivative which is issipative accoring to osiński s ecomposition. Accoring to our ecomposition however, while linear fractional stable motion is conservative, the limit of the renewal rewar processes is issipative. This allows us to istinguish these two processes an conclue, as note in Pipiras an Taqqu (2001a), that they have ifferent finite-imensional istributions. Finally, in Section 10, we raw connections between processes X (i) α, i =1, 2, 3, an the unerlying flow in the ecomposition (1.11) when the representation (1.4) of the process X α is minimal. emark. When we investigate the processes Xα D an Xα C in the ecomposition (1.7), we assume α (0, 2). The restriction α (1, 2) appears only in Sections 7 an 9, where we refine the ecomposition (1.7) to get (1.11). As argue in the remark following Theorem 4.2 of Pipiras an Taqqu (2001a), the ecomposition (1.7) is still vali when α (0, 1], provie that the integral t e (t s) X α (s)s is well-efine. Were the ecomposition (1.7) vali for α (0, 1] in general, the results of the present paper woul exten to α (0, 2) as well. 2 Multiplicative flows We first recall some basic efinitions from Pipiras an Taqqu (2001a). Let (X, X,µ) be a stanar Lebesgue space, as efine in Section 1, α (0, 2) an H>0. Set also If the mixe moving average (1.4) is self-similar, then { c H X α (ct) t = { X { = X { = X κ = H 1 α. (2.1) c κ (G(x, c(t + u)) G(x, cu))c 1 α Mα (x, cu) c κ (G(x, c(t + u)) G(x, cu))m α (x, u) (G(x, t + u) G(x, u))m α (x, u) t t t = {X α (t) t. (2.2) But while mixe moving averages, as efine by (1.4), are always SαS an have stationary increments, they are not necessarily self-similar. We nee to choose G such that X α efine by (1.4) is self-similar an we also want to associate a flow to the process X α. The following efinition achieves these two goals. 5

6 Definition 2.1 A SαS, H self-similar process X α with stationary increments, having the representation (1.4), is sai to be generate by a nonsingular measurable multiplicative flow {ψ c c>0 on (X, X,µ) associate with G (or simply generate by a flow {ψ c c>0 ) if, for all c>0 an t, c κ (G(x, c(t + u)) G(x, cu)) { (µ ψc ) 1/α = b c (x) (x) (G(ψ c (x),t+ u + g c (x)) G(ψ c (x),u+ g c (x))) (2.3) a.e. µ(x)u, where {b c c>0 an {g c c>0 is a cocycle an a semi-aitive functional for the flow {ψ c c>0, respectively, an supp {G(x, t + u) G(x, u),t = X a.e. µ(x)u. (2.4) We first explain some of the terms use in this efinition an then provie some insight in subsequent remarks. A family {ψ c c>0 of measurable maps from X onto X is calle a multiplicative flow if ψ 1 (x) =x an ψ c1 (ψ c2 (x)) = ψ c1 c 2 (x), (2.5) for all c 1,c 2 > 0 an x X. Note that relations (2.5) an ψ 1 (x) =x imply that ψ c has an inverse ψ 1/c. (In the literature, one typically consiers aitive flows {φ t t which satisfy the property φ t1 (φ t2 (x)) = φ t1 +t 2 (x), for all t 1,t 2 an x X. Physically, think of a particle in position x at time t = 0. Then φ t (x) is the position of the particle at time t. To go from multiplicative to aitive flows, one sets φ t = ψ e t.) It is sai to be nonsingular if µ(ψc 1 (B)) = 0 if an only if µ(b) = 0, for every c>0 an B X, an measurable if the map ψ c (x) :(0, ) X X is measurable. (Nonsingularity of the map ψ c also means that this map is invertible an that its inverse is measurable. In the case of a flow, these two conitions follow from (2.5), since the inverse of ψ c is a measurable map ψ 1/c.) A measurable map b c (x) :(0, ) X { 1, 1 is sai to be a cocycle for a flow {ψ c c>0 if b c1 c 2 (x) =b c1 (x)b c2 (ψ c1 (x)), (2.6) for all c 1,c 2 > 0 an x X. A measurable map g c (x) :(0, ) X is sai to be a semi-aitive functional for a flow {ψ c c>0 if g c1 c 2 (x) = g c 1 (x) c 2 + g c2 (ψ c1 (x)), (2.7) for all c 1,c 2 > 0 an x X. The support supp {G(x, t + u) G(x, u),t in (2.4) is efine as the minimal (in the a.e. sense) set A X Bsuch that, for all t, G(x, t + u) G(x, u) =0for (x, u) X \ A a.e. µ(x)u. emarks 1. To get a feeling for conition (2.3), note that it implies the H-self-similarity of the process X α. Inee, for any θ 1,...,θ n, t 1,...,t n, n 1 an c>0, n α θ k c κ (G(x, c(t k + u)) G(x, cu)) µ(x)u X k=1 6

7 n α (µ ψc ) = θ X k (G(ψ c (x),t k + u + g c (x)) G(ψ c (x),u+ g c (x))) (x) µ(x)u k=1 n α = θ X k (G(ψ c (x),t k + u + g c (x)) G(ψ c (x),u+ g c (x))) (µ ψ c )(x)u k=1 n α = θ X k (G(x, t k + u) G(x, u)) µ(x)u, k=1 since ψ c (X) =X, which proves (2.2). 2. The conition (2.3) is equivalent to { (µ c κ ψc ) 1/α G(x, cu) =b c (x) (x) G(ψ c (x),u+ g c (x)) + J(x, c) a.e. µ(x)u, (2.8) where J : X (0, ) is some measurable function (Proposition 5.1 in Pipiras an Taqqu (2001a)). 3. To unerstan conition (2.4), suppose on the contrary that the support of the functions G t (x, u) =G(x, t+u) G(x, u), t, is a set A X with A X. Because of the mixe moving average structure of the kernel G t, there is a set X 0 X such that A = X 0 (see Lemma 4.2 in Pipiras an Taqqu (2001a)). Conition A X then translates into X 0 X. Since the functions G t, t, vanish on the complement (X \ X 0 ) of the support X 0, we may in principle efine the flow {ψ c c>0 on X \ X 0 arbitrarily because both sies of (2.3) are zero anyway (assuming that ψ c maps X \ X 0 into itself). But this part of the flow woul then have nothing to o with the process X α. Since we want our flow to capture the unerlying structure of the process X α, we assume conition (2.4). 4. To get a feeling for (2.5), (2.6) an (2.7), assume that J 0 in (2.8) an let c 1,c 2 > 0. On one han, by (2.8), { (µ (c 1 c 2 ) κ ψc1 c G(x, c 1 c 2 u)=b c1 c 2 (x) 2 ) 1/α (x) G(ψ c1 c (x),u+ g 2 c 1 c 2 (x)). (2.9) On the other han, by iterating (2.8) twice, one gets ( ) (c 1 c 2 ) κ G(x, c 1 c 2 u)=c κ 2 c κ 1 G(x, c 1(c 2 u)) { (µ = c κ 2 b ψc1 ) c 1 (x) { (µ ψc1 ) = b c1 (x)b c2 (ψ c1 (x)) 1/α (x) G(ψ (x),c c1 2(u + c 1 2 g c 1 (x))) (x) (µ ψ c 2 ) 1/α (ψ c1 (x)) G(ψ c2 (ψ c1 (x)),u+ c 1 2 g c 1 (x)+g c2 (ψ c1 (x))). (2.10) elations (2.5), (2.6) an (2.7) imply that the right-han sies of (2.9) an (2.10) are ientical. 7

8 Any measurable nonsingular multiplicative flow {ψ c c>0, as in (2.3), can be ecompose into its issipative an conservative parts. This ecomposition is efine as follows (see, for example, Krengel (1985), page 17, an osiński (1995), page 1171). Consier a single nonsingular map V : X X. Then, there exists a unique (in the a.e. sense) ecomposition of X into two isjoint sets C an D such that (i) C an D are V -invariant, that is, VC = C an VD= D, (ii) the restriction of V to C is conservative, that is, there is no set A C such that µ(a) > 0 an V k (A), k 1, are isjoint, an (iii) D = k= V k B for some set B X, where V k (B), k 1, are isjoint. This is calle the Hopf s ecomposition of the map V. The sets D an C are calle the issipative part an the conservative part, respectively. Since each map ψ c is nonsingular, it has the Hopf ecomposition D c an C c. One can show (see Krengel (1969), Lemma 2.7) that this ecomposition is the same (in the a.e. sense) for all maps ψ c, c 1, an that there is a set D, invariant uner the flow, such that D = D c, c 1. Then the ecomposition of X into D an C := X \ D is calle the Hopf s ecomposition of the flow {ψ c c>0. A flow is calle issipative if X = D a.e. an conservative if X = C a.e. In Pipiras an Taqqu (2001a), we use the Hopf s ecomposition to establish the representation (1.7). 3 Cocycles an semi-aitive functionals associate with various flows In this section we escribe cocycles an semi-aitive functionals for various types of multiplicative flows. (Unless state otherwise, all the flows are multiplicative.) We first consier a general issipative flow an then eal with the simplest conservative flow, namely an ientity flow. The results we obtain will be use in the following sections. To eal with issipative flows, we will often use the following result ue to Krengel (1969), page 19 (see also osiński (1995), page 1176). Theorem 3.1 (Krengel) Every measurable nonsingular issipative flow {ψ c c>0 on a stanar Lebesgue space is null-isomorphic (mo 0) to a flow { ψ c c>0 on some stanar Lebesgue space (Y, Y B,ν(y)u) efine by ψ c (y, u) =(y, u +lnc) for all (y, u) Y an c>0. A null-isomorphism is a measurable, nonsingular, one-to-one an onto map with a measurable inverse. While it may not be measure preserving, by being nonsingular, it preserves sets of measure zero ( null refers to nonsingular). Null-isomorphism (mo 0) in Theorem 3.1 means that there exist two null sets N X an N Y, an a null-isomorphism Φ : Y \ N X \ N such that ψ c (Φ(y, u)) = Φ( ψ c (y, u)), for all c>0 an (y, u) Y \ N. The sets X \ N an Y \ N above are invariant uner the flows {ψ c c>0 an { ψ c c>0, respectively, i.e. ψ c (X \ N) =X \ N an similarly for ψ c. We will now characterize cocycles an semi-aitive functionals relate to issipative flows. Lemma 3.1 Let {ψ c c>0 be a issipative flow, {g c c>0 an {b c c>0 be a semi-aitive functional an a cocycle for {ψ c c>0, respectively. Suppose that Φ:Y \ N X \ N is a null-isomorphism of Theorem 3.1 between the flows {ψ c c>0 an { ψ c c>0 escribe above. Let g c (y, u) =g c (Φ(y, u)), if (y, u) Y \ N, an g c (y, u) =0,if(y, u) N, an b c (y, u) =b c (Φ(y, u)), if(y, u) Y \ N, 8

9 an b c (y, u) =1,if(y, u) N. Then, { g c c>0 is a semi-aitive functional, { b c c>0 is a cocycle for { ψ c c>0, an g(y, u) g c (y, u) = g(y, u +lnc), c b u +lnc) c (y, u) = b(y, b(y,, u) for all c>0 an (y, u) Y, where g : Y an b : Y { 1, 1 are some measurable functions. Proof: In view of the invariance of X \ N an Y \ N uner the flows {ψ c c>0 an { ψ c c>0, respectively, we may, without loss of generality, suppose that N an N are empty sets. We will prove the lemma for semi-aitive functionals first. By efinition of a semi-aitive functional, for all c 1,c 2 > 0 an x X, g c1 c 2 (x) =c 1 2 g c 1 (x)+g c2 (ψ c1 (x)). Then g c1 c 2 (Φ(y, u)) = c 1 2 g c 1 (Φ(y, u)) + g c2 (ψ c1 (Φ(y, u))) an, since ψ c Φ=Φ ψ c, we obtain that g c1 c 2 (y, u) = c 1 2 g c 1 (y, u)+ g c2 ( ψ c1 (y, u)) (3.1) = c 1 2 g c 1 (y, u)+ g c2 (y, u +lnc 1 ). (3.2) The relation (3.1) shows that g c (y, u) =g c (Φ(y, u)) is a semi-aitive functional for { ψ c c>0. Moreover, by taking u = 0 in (3.2), we get that g c1 c 2 (y, 0) = c 1 2 g c 1 (y, 0) + g c2 (y, ln c 1 ) or, by letting ln c 1 = v an c 2 = c, that g c (y, v) = g e v+ln c(y, 0) c 1 g e v(y, 0) = g(y, v +lnc) c 1 g(y, v), where g(y, s) := g e s(y, 0). Although g c (y, s) is measurable in c, y an s, we nee to show that it is measurable in c an y when s = 0. To o so, set u = ln c 1 in (3.2). Then g c2 (y, 0) = g c1 c 2 (y, ln c 1 ) c 1 2 g c 1 (y, ln c 1 ) =: g(c 1,c 2,y) an the function g(c 1,c 2,y) is measurable in its arguments. Hence, by fixing c 1 for which it is measurable in (c 2,y), we obtain the measurability of g c2 (y, 0). Let us now turn to cocycles. Since b c1 c 2 (y, u) =b c1 c 2 (Φ(y, u)) = b c1 (Φ(y, u))b c2 (ψ c1 (Φ(y, u))) = b c1 (Φ(y, u))b c2 (Φ( ψ c1 (y, u)) = b c1 (Φ(y, u))b c2 (Φ(y, u +lnc 1 )) = b c1 (y, u) b c2 (y, u +lnc 1 ), { b c c>0 is a cocycle for { ψ c c>0, an, by letting u =0,c 2 = c an ln c 1 = v, weget b c (y, v) = b e v+ln c(y, 0) b e v(y, 0) = b(y, v +lnc) b(y,, v) where b(y, u) = b e u(y, 0). (One may argue that the function b(y, u) is measurable as in the case above of the semi-aitive functional.) We now turn to ientity flows which are the simplest of conservative flows. A flow {ψ c c>0 on a space (X, X,µ) is calle an ientity flow if ψ c (x) =x, for all c>0 an x X. In the following lemma we characterize semi-aitive functionals an cocycles for an ientity flow. 9

10 Lemma 3.2 Let {ψ c c>0 be an ientity flow on a space X. If {b c c>0 is a cocycle an {g c c>0 is a semi-aitive functional for the flow {ψ c c>0, then b c (x) =1, g c (x) =(c 1 1)g(x), for all c>0 an x X, where g : X is a measurable map. Proof: The proof is straightforwar. Since {b c c>0 is a cocycle for an ientity flow, by (2.6), b c1 c 2 (x) =b c1 (x)b c2 (x) for all c>0 an x X. In particular, b c (x) =b c(x)b c(x) =(b c(x)) 2 an, since b c(x) { 1, 1, b c (x) = 1. For the semi-aitive functional {g c c>0, we have g c1 c 2 (x) =c 1 2 g c 1 (x)+g c2 (x) =c 1 1 g c 2 (x)+g c1 (x), for all c 1,c 2 > 0 an x X. This implies or, by letting c 1 = c an c 2 = e, where g(x) =(e 1 1) 1 g e (x). (c 1 2 1)g c1 (x) =(c 1 1 1)g c2 (x), g c (x) =(c 1 1)g(x), 4 Processes generate by issipative flows In the next theorem, we show that SαS processes generate by issipative multiplicative flows have a canonical representation. Theorem 4.1 Let α (0, 2), H>0an κ = H 1/α. Let also X α be a SαS processes generate by a issipative multiplicative flow as in Definition 2.1. Then there is a stanar Lebesgue space (Y,Y,ν) an a measurable function F : Y such that { {X α (t) t = e κs (F (y, e s (t + u)) F (y, e s u))m α (y, s, u), (4.1) Y where M α is a SαS ranom measure on Y with the control measure m(y, s, u) =ν(y)su. Conversely, if the process X α has the representation (4.1), then it is generate by a issipative flow. Proof: Accoring to Theorem 3.1, there is a stanar Lebesgue space (Y, Y, ν(y)s) an a null-isomorphism Φ : Y X such that ψ c (x) =ψ c (Φ(y, s)) = Φ(y, s +lnc), (4.2) for all c>0 an (y, s) Y. In other wors, the flow {ψ c c>0 on (X, X,µ) is null-isomorphic to aflow{ ψ c c>0 on (Y,ν(y)s) efine by ψ c (y, s) =(y, s +lnc). (We may suppose that the null sets in Theorem 3.1 are empty because, otherwise, we can replace X by X \ N in the efinition (1.4) of X α without changing its istribution.) By replacing x by Φ(y, s) in (2.8) an using (4.2), we get for all c>0 { (µ c κ ψc ) 1/α G(Φ(y, s),cu)=b c (Φ(y, s)) (Φ(y, s)) G(Φ(y, s+lnc),u+ g c (Φ(y, s))) + J(Φ(y, s),c) (4.3) 10 t

11 a.e. ν(y)su. By Lemma 3.1, we also have that a cocycle { b c c>0 = {b c Φ c>0 an a semi-aitive functional { g c c>0 = {g c Φ c>0 for the multiplicative flow { ψ c c>0 can be expresse as b c (Φ(y, s)) = b(y, s +lnc)( b(y, s)) 1, g c (Φ(y, s)) = g(y, s +lnc) g(y, s) c for some measurable functions b, taking values in { 1, 1, an g. Moreover, we have that (L enotes the Lebesgue measure) (µ ψ c ) (Φ(y, s)) = { (µ Φ) (ν L) ( ψ c (y, s)) To show (4.4), observe first that, since ψ c Φ=Φ ψ c,wehave 1 (ν L) Φ 1 (Φ(y, s)). (4.4) (µ ψ c Φ) (ν L) = (µ Φ ψ c ). (4.5) (ν L) elation (4.4) is merely a ifferent expression for (4.5), since on one han, (µ ψ c Φ) (ν L) an on the other han, = (µ ψ c Φ) (µ Φ) (µ Φ) (ν L) = (µ ψ c) Φ Φ, (ν L) Φ 1 (µ Φ ψ c ) (ν L) = (ν L) ψ c (ν L) (µ Φ ψ c ) (ν L) ψ c = (µ Φ) (ν L) ψ c, where ((ν L) ψ c )/(ν L) = 1 because the first component in ψ c (y, s) =(y, s +lnc) remains the same an the secon is a translation. Now, by setting { 1/α G(y, s, u) = b(y, s) (ν L) Φ 1 (Φ(y, s)) G(Φ(y, s),u) (4.6) an { (µ Φ) 1/α F (y, s, u) = b(y, s) (ν L) (y, s) G (Φ(y, s),u) in (4.3), we obtain that, for all c>0, c κ G(y, s, cu) = F ( y, s +lnc, u + g(y, s +lnc) ) g(y, s) + c J(y, s, c) a.e. ν(y)su, where J is some measurable function. By making the change of variables cu = z, we get ( G(y, s, z) =c κ F y, s +lnc, z ) g(y, s) + g(y, s +lnc) + c κ c c J(y, s, c) a.e. ν(y)sz. By the Fubini s theorem, this relation hols a.e ν(y)szc as well. Then, by setting u = s +lnc, wegetc = e u s an G(y, s, z) =e κ(u s) F (y, u, e s u z + g(y, u) e s u g(y, s)) + e (s u)κ J(y, s, e s u ) 11

12 a.e. ν(y)szu. Fix u = u 0 for which this relation hols a.e. ν(y)sz. Then G(y, s, z) =e κs F (y, e s (z g(y, s))) + J(y, s), (4.7) a.e. ν(y)sz for some measurable functions F an J. Now, by writing own the characteristic functions, it is easy to see that (4.6) implies { {X α (t) t = (G(x, t + u) G(x, u))m α (x, u) X { = ( G(y, s, t + u) G(y, s, u)) M α (y, s, u), Y t where M α is a SαS ranom measure on Y with the control measure m(y, s, u) =ν(y)su. Then, by using (4.7), we get that { {X α (t) t = e κs (F (y, e s (t + u)) F (y, e s u)) M α (y, s, u). Y t To prove the converse, suppose that the process X α has the representation (4.1). By using an argument similar to the one in the proof of Lemma 4.2 in Pipiras an Taqqu (2001a), one can conclue that there is a set Y 0 Y such that supp{e κs (F (y, e s (t + u)) F (y, e s u)), t = Y 0 a.e. ν(y)su. Hence, by replacing Y with Y 0 in (4.1), we may suppose that the conition (2.4) hols. We may o this without loss of generality because this replacement oes not change the istribution of X α. We shall now show that relation (2.8) is satisfie with G(y, s, u) =e κs F (y, e s u), where the x in this relation stans for (y, s). Observe that, since c κ G(y, s, cu) =G(y, s +lnc, u) for any c>0, the conition (2.8) is satisfie with ψ c (y, s) =(y, s +lnc), b c (y, s) =1,g c (y, s) = 0 an J(y, s, c) =0. One still nees to verify that {ψ c c>0 is a issipative multiplicative flow. It is obviously a (measurable, nonsingular) multiplicative flow. To see why it is issipative, recall the efinition given at the en of Section 2 an observe that, say for c>1, ψ c (y, s) =(y, s +lnc), the sets ψc k (Y [0, ln c)) = Y [k ln c, (k +1)lnc), k Z, are isjoint an Y = k Z ψc k (Y [k ln c, (k +1)lnc)). emark. Further stuy of mixe moving averages generate by issipative flows can be foun in Pipiras an Taqqu (2001b). In particular, we provie there many examples of such processes, for example, the limit process (1.6) of the renewal rewar problem iscusse above, the so-calle ranom wavelet expansion of Chi (2001), the Takenaka process of Takenaka (1991) an the new self-similar processes (3.1) of Samoronitsky an Taqqu (1990). 5 Processes generate by ientity flows We now turn to processes generate by conservative flows, that is, processes Xα C in the ecomposition (1.7). In contrast to Section 4, we will not provie a canonical representation of such processes because conservative flows cannot be characterize as simply as issipative flows are by the Krengel s Theorem 3.1. Instea, in this an the following three sections, we will pursue a ifferent iea. We begin by showing that one can characterize processes generate by the simplest conservative flows, namely, ientity flows. ecall that a multiplicative flow {ψ c c>0 is an ientity flow on (X, X,µ)ifψ c (x) =x for all x X an c>0. The processes that we get will enter in our finer ecomposition of the process Xα C consiere in Section 7 below. 12 t

13 Theorem 5.1 Let α (0, 2) an H>0. Suppose that the process X α is generate by an ientity flow in the sense of Definition 2.1. Then there exist measurable functions G 1,G 2 : X such that the process {X α (t) t can be represente as ( G1 (x)((t + u) κ + u κ +)+G 2 (x)((t + u) κ u κ ) ) M α (x, u), if κ 0, (5.1) X X ( ) G 1 (x)(ln t + u ln u )+G 2 (x)(1 (0, ) (t + u) 1 (0, ) (u)) M α (x, u), if κ =0, (5.2) where M α is a SαS ranom measure on X with the control measure m(x, u) =µ(x)u. Conversely, if the process X α has representation (5.1) or (5.2) with supp{g 1,G 2 = X a.e. µ(x), then it is generate by an ientity flow. Proof: If {ψ c c>0 is an ientity flow, then relation (2.8), which is equivalent to (2.3), becomes c κ G(x, cu) =b c (x)g(x, u + g c (x)) + J(x, c) (5.3) for all c>0 a.e. µ(x)u. Since {b c c>0 is a cocycle an {g c c>0 is a semi-aitive functional for an ientity flow, we obtain from Lemma 3.2 that b c (x) =1ang c (x) =(c 1 1)g(x) for some measurable function g. Then, relation (5.3) becomes c κ G(x, cu) =G(x, u +(c 1 1)g(x)) + J(x, c) (5.4) for all c>0 a.e. µ(x)u. As in Cambanis, Maejima an Samoronitsky (1992), page 104, it follows from (5.4) that, for all c 1,c 2 > 0, J(x, c 1 c 2 )=c κ 1 J(x, c 2)+J(x, c 1 )=c κ 2 J(x, c 1)+J(x, c 2 ) (5.5) a.e. µ(x). We shall now consier the cases κ 0 an κ = 0 separately. Case κ 0. It follows from the secon equality in (5.5) that, for all c 1,c 2 > 0 an c 1,c 2 1, J(x, c 1 ) 1 c κ 1 = J(x, c 2) 1 c κ 2 a.e. µ(x). Then, for all c>0, J(x, c) =J(x)(1 c κ ) a.e. µ(x) for some measurable function J. elation (5.4) now becomes c κ G(x, cu) =G(x, u +(c 1 1)g(x)) + J(x)(1 c κ ) (5.6) for all c>0 a.e. µ(x)u, which by the Fubini s theorem hols also a.e. µ(x)uc. By making a change of variables, we obtain that G(x, z) =c κ G(x, c 1 (z + g(x)) g(x)) + J(x)(c κ 1) an hence G(x, z)+j(x) = c κ (G(x, c 1 (z + g(x)) g(x)) + J(x)) = c κ F (x, c 1 (z + g(x))) a.e. µ(x)zc for some function F. By letting G(x, z) =G(x, z g(x)) + J(x), we get that G(x, z) = c κ F (x, c 1 z) a.e. µ(x)zc. In particular, G(x, z) =z κ (c 1 z) κ F (x, c 1 z) a.e. µ(x)zc when z>0 an hence G(x, z) =z κ v κ F (x, v) a.e. µ(x)vz when z>0. By fixing v = v 0, for which this equation 13

14 hols a.e. µ(x)z, we get that G(x, z) =z κ G 1 (x) a.e. µ(x)z for some function G 1 when z>0. Similarly, G(x, z) =z κ G 2 (x) a.e. µ(x)z for some function G 2 when z<0. Hence G(x, z) =G 1 (x)(z + g(x)) κ + + G 2 (x)(z + g(x)) κ J(x) (5.7) a.e. µ(x)z. The result (5.1) of the theorem now follows. Case κ = 0. Since the secon equality in (5.5) is now trivial, we shall use the first equality instea, namely that, for all c 1,c 2 > 0, J(x, c 1 c 2 )=J(x, c 2 )+J(x, c 1 ) a.e. µ(x). By setting k(x, s) =J(x, e s ), s, we obtain that, for all s, t, k(x, s + t) =k(x, s)+k(x, t) (5.8) a.e. µ(x). It is easy to see that for a.e. x X, the function k(x, s), s, satisfies the conitions (i)-(iv) of Proposition A.1 in the appenix. To unerstan, for example, why the conition (ii) is satisfie, observe that, since Q is countable, one may first conclue that, for s, the relation (5.8) hols also for a.e. x X an for all t Q (the a.e. set here epens on s only). Then, by using the Fubini s theorem, the relation (5.8) hols also for a.e. (x, s) an for all t Q, which now implies that, for a.e. x X, the function k(x, s) satisfies the conition (ii). Proposition A.1 now implies that for a.e. x X, k(x, s) =k(x, 1)s a.e. s. We may suppose without loss of generality that the function k(x, 1) is measurable (otherwise, consier k 0 (x, s) =k(x, s 0 s) where s 0 is such that k(x, s 0 ) is measurable). Therefore, we euce that for some measurable function G 1 : X, J(x, c) =k(x, ln c) =G 1 (x)lnc a.e. µ(x)c. elation (5.4) can now be written as G(x, cu) =G(x, u +(c 1 1)g(x)) + G 1 (x)lnc a.e. µ(x)cu. By making the change of variables u + c 1 g(x) =v, wehave or, by setting G(x, z) =G(x, z g(x)), G(x, cv g(x)) = G(x, v g(x)) + G 1 (x)lnc G(x, cv) = G(x, v)+g 1 (x)lnc a.e. µ(x)cv. Consier now v>0 an write the above relation as G(x, cv) G 1 (x)lncv = G(x, v) G 1 (x)lnv a.e. µ(x)cv. By setting Ĝ(x, z) = G(x, z) G 1 (x)lnz for x X, z>0, we then have Ĝ(x, cv) = Ĝ(x, v) a.e. µ(x)cv. By making the change of variables c = z/v an then fixing v, we euce that Ĝ(x, z) =G 2,1 (x)1 (0, ) (z) a.e. µ(x)z for some G 2,1. Going backwars, we get for z + g(x) > 0, G(x, z) = G(x, z + g(x)) = G 2,1 (x)1 (0, ) (z + g(x)) + G 1 (x) ln(z + g(x)) a.e. µ(x)z. When z + g(x) < 0, one may euce similarly that, for some function G 2,2, G(x, z) = G 2,2 (x)1 (,0) (z + g(x)) + G 1 (x)ln z + g(x) = G 2,2 (x) G 2,2 (x)1 (0, ) (z + g(x)) + G 1 (x)ln z + g(x) 14

15 a.e. µ(x)z. By combining the previous two relations we get G(x, z) =(G 2,1 (x) G 2,2 (x))1 (0, ) (z + g(x)) + G 1 (x)ln z + g(x) + G 2,2 (x) (5.9) a.e. µ(x)z an hence representation (5.2). Suppose now that κ 0 an the process X α has representation (5.1) which is also representation (1.4) with the kernel function G(x, u) =G 1 (x)u κ + + G 2 (x)u κ, x X, u. Since c κ G(x, cu) = G(x, u), we see that conition (2.3) hols with the ientity flow ψ c (x) =x (an the corresponing cocycle b c (x) = 1 an the semi-aitive functional g c (x) = 0). To conclue that X α is generate by an ientity flow, one still nees to check conition (2.4). Assume that (2.4) oes not hol. Then, by Lemma 4.2 in Pipiras an Taqqu (2001a), there is a measurable set X 0 X with µ(x 0 ) > 0 such that, for all t, G(x, t + u) G(x, u) = 0 a.e. for x X 0 an u. Hence, by the Fubini s theorem an a change of variables, G(x, v) =G(x, u) a.e. for x X 0 an u, v. This conition implies that G 1 (x) = 0 an G 2 (x) = 0 a.e. for x X 0. Since µ(x 0 ) > 0, this contraicts supp{g 1,G 2 = X. The case κ = 0 may be prove in a similar way. emarks 1. The process (5.1) is well-efine for G 1,G 2 L α (X, X,µ), α (0, 2) an H (0, 1), since by using the inequality a + b α const( a α + b α ), G 1 (x) ( (t + u) κ + u κ +) + G2 (x) ( (t + u) κ u κ ) α µ(x)u <, (5.10) X for all t. It is not efine for H 1 since the integral in (5.10) equals +. This also means that the process X α cannot be generate by an ientity flow when H The process (5.1) is also a mixe fractional motion in the sense of Burnecki et al. (1998), since for t>0 it can be represente in istribution as ( ) u κ G 1 (x) (u 1 t +1) κ + 1 M α (x, u) X 0 0 ( ) + ( u) κ G 1 (x)(( u) 1 t 1) κ + + G 2 (x)((( u) 1 t 1) κ 1) M α (x, u) X ( = z κ G x, t ) M α ( x, z), X 0 z where X = X {1, 2, x =(x, j, z) an G( x, s) = G(x, j, s) =1 {j=1 G 1 (x)((s +1) κ + 1) + 1 {j=2 (G 1 (x)(s 1) κ + + G 2 (x)((s 1) κ 1)) for x X, s>0 an j {1, 2. Example 5.1 By setting X = {1 an µ(x) =δ {1 (x) in (5.1), one gets ( ( X α (t) = a (t + u) κ + u κ ) ( + + b (t + u) κ u κ )) Mα (u), t, (5.11) that is, the usual linear fractional stable motions. We refer the reaer to Chapter 7 in Samoronitsky an Taqqu (1994) for more information about these processes. Note also that, by setting X = {1 an µ(x) =δ {1 (x) in (5.2), one gets X α (t) = ( ) a(ln t + u ln u )+b(1 (0, ) (t + u) 1 (0, ) (u)) M α (u), t. (5.12) 15

16 When b = 0, the process (5.12) is the so-calle log-fractional stable motion introuce by Kasahara, Maejima an Vervaat (1988). If a = 0, then (5.12) becomes the usual SαS Lévy motion because X α (t) =b (1 (0, ) (t + u) 1 (0, ) (u))m α (u) = b 1 (0,t) (u)m α (u). Definition 5.1 We will call processes (5.1) an (5.2) mixe linear fractional stable motions (mixe LFSM, in short). We give that name to the processes (5.1) because they are extensions of the linear fractional stable motions (5.11). We also call the processes (5.2) mixe LFSM because they can be viewe as limiting cases of mixe LFSM (5.1) when κ 0. To see this, observe that X (G 1 (x) t + u κ u κ κ ) + G 2 (x)((t + u) κ + u κ +) M α (x, u) is a mixe LFSM, an that ( t + u κ u κ )/κ ln t + u ln u an (t + u) κ + u κ + 1 (0, ) (t + u) 1 (0, ) (u) asκ 0. We conclue this section by applying Theorem 5.1 to characterize linear fractional stable motions, log-fractional stable motion an Lévy motion. This characterization extens Theorem 3 in Cambanis et al. (1992) because it oes not assume the local L 1 integrability of the kernel function G an also inclues the case H =1/α. Corollary 5.1 Let α (0, 2) an H (0, 1). Suppose that a SαS nonegenerate process {X α (t) t with stationary increments, having the representation { {X α (t) t = (G(t + u) G(u))M α (u), t where M α has the Lebesgue control measure, is self-similar with exponent H. Then, if κ = H 1/α 0, X α is a linear fractional stable motion (5.11). Ifκ =0, then X α is the sum of the log-fractional stable motion an the Lévy stable motion in (5.12). Proof: Example 4.1 an Theorem 4.1 in Pipiras an Taqqu (2001a) show that the kernel function G satisfies conitions (2.3) an (2.4). Since X = {1, the flow can only be an ientity flow. The conclusion then follows from Theorem The mixe LFSM component set Accoring to Theorem 5.1, mixe LFSM s are mixe moving averages X α characterize by ientity flows. Another way to get a mixe LFSM is through the kernel G of the process X α irectly, without using flows. This iea is base on the mixe LFSM component set which we introuce an whose properties we explore in this section. The set will be use in the next section to ecompose the conservative component process Xα C. 16

17 Definition 6.1 Let X α be a mixe moving average (1.4) with the kernel function G. The mixe LFSM component set of the process X α is the set { E := x X : G i = G i (x),i=1, 2, 3, g= g(x) such that G(x, u) =G 1 (x)ln u + g(x) + G 2 (x)1 (0, ) (u + g(x)) + G 3 (x) a.e. u (6.1) when κ = 0, an the set { E := x X : G i = G i (x),i=1, 2, 3, g= g(x) such that G(x, u) =G 1 (x)(u + g(x)) κ + + G 2 (x)(u + g(x)) κ + G 3 (x) a.e. u (6.2) when κ 0. Lemma 6.1 The mixe LFSM component set E is measurable. The functions G 1, G 2, G 3 an g, when restricte to E are measurable as well. Moreover, if the function G satisfies conition (2.4), then supp{g 1,G 2 = E a.e. Proof: We prove the lemma only in the case κ = 0. The case κ 0 may be prove in a similar way. The iea is to express this set an the functions G i, i =1, 2, 3, an g in terms of the given measurable function G. Define first the set X 1 = {x X : G(x, u)(ln u) 1 G 1 (x) a.e. as u +, where by h(u) h a.e. as u +, we mean that ɛ >0 u 0 = u 0 (ɛ) such that h(u) h <ɛ a.e. for u>u 0. We first show that the set X 1 is measurable. By the Cauchy s criterion, X 1 is equal to the set n 1 m 1 A n,m, where A n,m = {x : K(x, u, v) < 1/n a.e. uv for u, v > m an K(x, u, v) = G(x, u)(ln u) 1 G(x, v)(ln v) 1. Note that the set A n,m is measurable since it equals {x : K(x) =0, where the function K(x) = 1 {K(x,u,v) 1/n1 {u,v>m uv is measurable by the Fubini s theorem. Therefore, X 1 is measurable. Then, for x X 1, we have G(x, u + n)(ln(u + n)) 1 G 1 (x) for a.e. u>0asn, an, by the Fubini s theorem, this convergence hols a.e. for (x, u) X 1. Since G 1 is the limit of measurable functions, it has to be measurable as well. Moreover, we have E = X 1 E(X 1, G 1,G 2,G 3,g), where the set E(...) is efine as in (6.1) but with X an G 1 replace by X 1 an G 1, respectively, an requiring the existence of G 2, G 3 an g only. (We shall continue to use the E(...) type notation below.) Now, efine the set X 2 = {x X 1 : G(x, u) G 1 (x)ln u G 3 (x) a.e. as u. Arguing as above, we can euce that the set X 2 is measurable an that the function G 3, restricte to X 2, is measurable as well. Moreover, E(X 1, G 1,G 2,G 3,g) = X 2 E(X 2, G 1,G 2, G 3,g), since ln u + g(x) ln u 0 an 1 (0, ) (u + g(x)) 0asu. Next, efine the set X 3 = {x X 2 : G(x, u) G 1 (x)ln u G 3 (x) G 2 (x) a.e. as u +. 17

18 Again, the set X 3 is measurable, the function G 2, when restricte to X 2, is measurable as well, an E(X 2, G 1,G 2, G 3,g)=X 3 E(X 3, G 1, G 2, G 3,g). Finally, efining the set { { X 4 = x X 3 : exp (G(x, u) G 3 (x))( G 1 (x)) 1 1 { G1 + u g(x) a.e. as u, (x) 0 we have the measurability of g an X 4, an E(X 3, G 1, G 2, G 3,g)=X 4 E(X 4, G 1, G 2, G 3, g). Hence, since E = X 1 X 2 X 3 X 4 E(X 4, G 1, G 2, G 3, g), where each set in the intersection is measurable, we get that E is measurable as well. It is also clear that G i = G i E, i =1, 2, 3, an g = g E, which implies the measurability of the functions. If supp{g 1,G 2 E a.e., then there is a set X 0 with µ(x 0 ) > 0 such that G 1 (x) = 0 an G 2 (x) =0 for x X 0. Hence, for any t, G(x, t + u) G(x, u) = 0 for x X 0 an u. Since µ(x 0 ) > 0, this contraicts (2.4). The following lemma shows that the mixe LFSM component set E is a subset of the conservative part of the flow {ψ c c>0. Proposition 6.1 Suppose that the mixe moving average X α is generate by a flow in the sense of Definition 2.1. Then the mixe LFSM component set E is a subset (a.e.) of the conservative part of the flow {ψ c c>0. Moreover, the set E is invariant uner the flow a.e., that is, for any c>0, µ(e ψ 1 c (E)) = 0. Proof: We prove the proposition only for the case κ 0. The case κ = 0 may be prove in a similar way. By (2.3) an relation (5.11) in Pipiras an Taqqu (2001a), the conservative part of the flow {ψ c c>0 is C = {x X : I(x) = a.e. µ(x), where I(x) = G(ψ c (x), 1+u) G(ψ c (x),u) α u (µ ψ c) (x) c 1 c 0 = G(x, c(1 + u)) G(x, cu) α u c Hα c. 0 The inclusion E C a.e. then follows from (6.2), since, for a.e. x E, I(x) = 0 c 1 c G 1 (x) ( (1 + u) κ + u κ +) + G2 (x) ( (1 + u) κ u κ ) α u =. Let us now show that µ(e ψc 1 (E)) = 0 for all c>0. Fix c>0 an consier x E. By using (2.8) an (6.2), we get for a.e. µ(x), { (µ G(ψ c (x),u) = (b c (x)) 1 ψc ) 1/α ( ) c κ G(x, c(u g c (x))) J(x, c) = G 1,c (x)(u + g 1,c (x)) κ + + G 2,c (x)(u + g 1,c (x)) κ + G 3,c (x) a.e. u, where G 1,c,G 2,c,G 3,c an g 1,c are some functions. This shows that ψ c (x) E for a.e. x E or that E ψc 1 (E) a.e. Since ψc 1 (E) =ψ 1/c (E) an we have that E ψc 1 (E) a.e. implies ψ 1/c (E) E a.e., we get ψc 1 (E) E a.e. an hence ψc 1 (E) =E a.e. 18

19 7 Ientification of the mixe LFSM component We suppose throughout this section that α (1, 2). Our goal is to show that the conservative part process Xα C in (1.7) can be further ecompose uniquely into two inepenent processes X α (2) an X α (3). The process X α (2) is a mixe LFSM of Definition 5.1, whereas the process X α (3) has no mixe LFSM component, that is, it cannot be ecompose into two inepenent processes one of which is a mixe LFSM. To obtain our ecomposition, we will use the mixe LFSM component set E in Definition 6.1. Another approach, namely that base on the structure of the unerlying flow, can be foun in Section 10 below. Let now X α be a SαS self-similar process having representation (1.4) with the kernel function G. By Theorem 5.1 in Pipiras an Taqqu (2001a), we may assume without loss of generality that G satisfies the conitions of Definition 2.1 an hence that X α is generate by a multiplicative flow {ψ c c>0 in the sense of that efinition. ecall from Pipiras an Taqqu (2001a) that the processes Xα D an Xα C in the ecomposition (1.7) are then efine as Xα D (t) = (G(x, t + u) G(x, u))m α (x, u), D Xα C (t) = (G(x, t + u) G(x, u))m α (x, u), C where X = D C is the Hopf s ecomposition of the flow {ψ c c>0 into its issipative an conservative parts, respectively. Since the mixe LFSM component set E is a subset of the conservative part C by Proposition 6.1, one can ecompose the process Xα C into two processes as where X (2) α (t) = X (3) α (t) = E C\E X C α = X (2) α + X (3) α, (7.1) (G(x, t + u) G(x, u))m α (x, u), (7.2) (G(x, t + u) G(x, u))m α (x, u). (7.3) We will say that the ecomposition (7.1) is unique, if it oes not epen on the representation (1.4) of X α. Proposition 7.1 The ecomposition (7.1) is unique. Proof: We consier only the case κ 0. By Theorems 4.2 an 4.1 in Pipiras an Taqqu (2001a), there is a new space X an function G( x, u), x X, u, such that { G( x, t + u) G( x, u) t L α ( X, µ( x)u), which also satisfies the conitions (2.3) an (2.4) in Definition 2.1. This function G correspons to the so-calle minimal spectral representation of the process X α. (For more etails, see Pipiras an Taqqu (2001a).) Let { ψ c c>0 be the flow associate to G by Definition 2.1, C be the conservative part of the flow { ψ c c>0, an Ẽ be the set an G i, i =1, 2, 3, g be the functions efine for G by Definition 6.1. Since G correspons to a minimal spectral representation of the process X α, by Corollary 5.1 in Pipiras an Taqqu (2001a), there are measurable functions Φ 1 : X X, h : X \{0 an Φ 2, Φ 3 : X such that G(x, u) =h(x) G(Φ 1 (x),u+φ 2 (x))+φ 3 (x) (7.4) 19

20 a.e. µ(x)u an µ = µ h Φ 1 1, where µ h(x) = h(x) α µ(x). We want to show first that Φ 1 1 (Ẽ) =E an that Φ 1 1 ( C \ Ẽ) =C \ E a.e. µ(x). If x E, then a.e. µ(x), G 1 (x)(u + g(x)) κ + + G 2 (x)(u + g(x)) κ + G 3 (x) =h(x) G(Φ 1 (x),u+φ 2 (x))+φ 3 (x) or G(Φ 1 (x),u)= G 1(x) h(x) (u Φ 2(x)+g(x)) κ + + G 2(x) h(x) (u Φ 2(x)+g(x)) κ + G 3(x) Φ 3 (x) h(x) a.e. u, which shows that Φ 1 (x) Ẽ an hence E Φ 1 1 (Ẽ) a.e. µ(x). Conversely, if Φ 1(x) Ẽ, then a.e. µ(x), ( G(x, u) =h(x) G 1 (x)(u +Φ 2 (x)+ g(x)) κ + + G 2 (x)(u +Φ 2 (x)+ g(x)) κ + G ) 3 (x) +Φ 3 (x) a.e. u, which, by performing multiplications on the right-han sie, implies x E an hence Φ 1 1 (Ẽ) E a.e. µ(x). Therefore, Φ 1 1 (Ẽ) =E a.e. µ(x). elation (5.17) in the proof of Theorem 5.3 in Pipiras an Taqqu (2001a) shows that Φ 1 1 ( C 0 )=C 0 a.e. µ(x), where C 0 = C a.e. µ(x) an C 0 = C a.e. µ( x), an hence Φ 1 1 ( C) =C a.e. µ(x), since by using µ = µ h Φ 1 1, µ(ñ) =0forÑ X implies µ(φ 1 (Ñ)) = 0. Together with Φ 1 1 (Ẽ) =E a.e. µ(x), this implies that Φ 1 1 ( C \ Ẽ) =C \ E a.e. µ(x). (2) (3) Let now X α an X α be the processes in the ecomposition (7.1) obtaine from replacing G, E, C by G, Ẽ, C in (7.2) an (7.3), respectively. Then, for every a 1,,a n an t 1,,t n, n 1, we have, by (7.4), n ln E exp{i a k X α (2) n α (t k ) = a E k (G(x, t k + u) G(x, u)) µ(x)u = = n E k=1 Ẽ k=1 k=1 a k ( G(Φ 1 (x),t k + u +Φ 2 (x)) G(Φ α 1 (x),u+φ 2 (x))) h(x) α µ(x)u n = a Φ 1 1 (Ẽ) k ( G(Φ 1 (x),t k + u) G(Φ α 1 (x),u)) µ h (x)u k=1 n a k ( G( x, t k + u) G( x, α n (2) u)) µ( x)u = ln E exp{i a k X α (t k ), k=1 which shows that X (2) α = X (2) α. Similarly, X (3) α = k=1 X (3) α an hence ecomposition (7.1) is unique. We will say that a process oes not have a mixe LFSM component, if it cannot be ecompose into two inepenent SαS processes one of which is a mixe LFSM. We will also say that two processes X an X are essentially ifferent if there is no multiplicative constant c such that X(t) an c X(t) have the same finite-imensional istributions. Proposition 7.2 The process X α (2) in (7.2) is a mixe LFSM, while the process X α (3) in (7.3) is a self-similar process with stationary increments, generate by a conservative flow, which oes not have a mixe LFSM component. These processes are inepenent an essentially ifferent. 20

21 Proof: The processes X α (2) an X α (3) are inepenent because their kernels have isjoint supports, that is, E (C \ E) = (see Theorem in Samoronitsky an Taqqu (1994)). The process X α (2) is a mixe LFSM by construction. The process X α (3) has representation (1.4) an hence has stationary increments. It is also self-similar. Inee, Xα C = X α (2) + X α (3), X α (2) an X α (3) are inepenent an Xα C an X α (2) are self-similar. Therefore, X α (3) is self-similar because its characteristic function is the ratio of the characteristic functions of two self-similar processes. Let us show now that X α (3) has no mixe LFSM component when κ 0. Suppose there exist two inepenent SαS processes Y α an Z α such that = Y α + Z α, (7.5) X (3) α where Y α is a mixe LFSM, efine by (5.1) with some functions Ĝ1, Ĝ2 L α (Y, µ(y)) an supp{ĝ1, Ĝ2 = Y. The structure of the process Z α plays no role in the proof. Observe only that, by (7.5), Z α has a stochastic integral representation. Inee, since SαS processes Y α + Z α (= X (3) α ) an Y α have stochastic integral representations, these processes are necessarily separable in probability an hence so is their sum Z α (see Theorem an Exercise 3.20 in Samoronitsky an Taqqu (1994)). Then, by the same Theorem , we have Z α (t) = 1 0 g t(z)m α (z) for some {g t t L α (0, 1), where M α has the Lebesgue control measure z. Since Y α an Z α are inepenent, one can represent the sum Y α + Z α as V f t(v)m α (v), where V is mae up of two isjoint parts Y an (0, 1). Formally, V =(Y ) (0, 1) = {v : v =(y, u) Y or v = z (0, 1), f t (v) =g t (z), if v = z, an f t (v) =Ĝ1(y)((t + u) κ + u κ +)+Ĝ2(y)((t + u) κ u κ ), if v =(y, u), an M α has the control measure m(v) equal to z on (0, 1) an µ(y)u on Y. We may suppose without loss of generality that supp{f t,t = V. Then, one can relate the representations of Y α + Z α an X (3) α in (7.5): by Theorem 1.1 in osiński (1995), there are functions Φ 1 : V C \ E, Φ 2 : V an h : V \{0 such that f t (v) =h(v)(g(φ 1 (v),t+φ 2 (v)) G(Φ 1 (v), Φ 2 (v))) a.e. m(v)t. Consiering only those v =(y, u) Y, wehave Ĝ 1 (y)((t+u) κ + u κ +)+Ĝ2(y)((t+u) κ u κ )=h(y, u)(g(φ 1 (y, u),t+φ 2 (y, u)) G(Φ 1 (y, u), Φ 2 (y, u))) a.e. µ(y)ut. Then fixing u = u 0 for which the relation hols a.e. µ(y)t an making the change of variables t + u 0 = w, we get that there are new measurable functions Φ 1 : Y C \ E, Φ 2, Φ 3 : Y an h : Y \{0 such that Ĝ 1 (y)w κ + + Ĝ1(y)w κ = h(y)g(φ 1 (y),w+φ 2 (y)) + Φ 3 (y) (7.6) a.e. µ(y)w. Hence, Φ 1 (y) E a.e. µ(y). If µ is not a zero measure, then relation (7.6) contraicts the fact that Φ 1 (y) C \ E. The case of κ = 0 may be prove in a similar way. Finally, there is no constant c 0 such that cx α (2) = X α (3) essentially ifferent), since X (3) α has no mixe LFSM component. 8 Example of a process of the thir kin (in other wors, the processes are In this section, we provie an example of a mixe moving average process generate by a conservative flow which has only process X α (3) in its ecomposition (7.1). (We refer to such processes as processes of the thir kin.) Existence of such a process turne out to be a nontrivial problem. 21