Dilated Fractional Stable Motions

51

0894-9840/04/0100-0051/0 © 2004 Plenum Publishing Corporation

Journal of Theoretical Probability, Vol. 17, No. 1, January 2004 (© 2004)

Dilated Fractional Stable Motions

Vladas Pipiras1 and Murad S. Taqqu2 , 3

1Department of Statistics and Operations Research, University of North Carolina atChapel Hill, New West, CB #3260, Chapel Hill, North Carolina 27599. E-mail: [email protected] of Mathematics, Boston University, 111 Cummington St., Boston, Massachusetts02215. E-mail: [email protected] To whom correspondence should be addressed.

Received May 6, 2002; revised November 24, 2003

Dilated fractional stable motions are stable, self-similar, stationary incrementsrandom processes which are associated with dissipative flows. Self-similarityimplies that their finite-dimensional distributions are invariant under scaling. Inthe Gaussian case, when the stability exponent equals 2, dilated fractional stablemotions reduce to fractional Brownian motion. We suppose here that the sta-bility exponent is less than 2. This implies that the dilated fractional stablemotions have infinite variance and hence they cannot be characterised by acovariance function. These dilated fractional stable motions are defined throughan integral representation involving a nonrandom kernel. This kernel plays afundamental role. In this work, we study the space of kernels for which thedilated processes are well-defined, indicate connections to Sobolev spaces,discuss uniqueness questions and relate dilated fractional stable motions toother self-similar processes. We show that a number of processes that have beenobtained in the literature, are in fact dilated fractional stable motions, forexample, the telecom process obtained as limit of renewal reward processes, theTakenaka processes and the so-called ‘‘random wavelet expansion’’ processes.

KEY WORDS: Stable, self-similar processes with stationary increments; dilatedfractional stable motions; uniqueness; flows; the telecom process.

1. INTRODUCTION

We study a class of processes {Xa(t)}t ¥ R, called dilated fractionalstable motions, which are symmetric a-stable, self-similar and havestationary increments. These processes appear in the decomposition of

stable, self-similar mixed moving averages obtained by the authors inPipiras and Taqqu. (7, 8)

We start by recalling the definitions of symmetric a-stable processesand of self-similarity. Dilated fractional stable motions are defined belowin Definition 1.1. A random variable t is symmetric a-stable (SaS, in short)with a ¥ (0, 2] if its characteristic function has the form E exp{iht}=exp{−sa |h|a}, where h ¥ R and s > 0 is a scale coefficient. The case a=2corresponds to the Gaussian distribution. We will be interested in the non-Gaussian case a ¥ (0, 2). A random process {Xa(t)}t ¥ R is called SaS if itslinear combinations ;n

j=1 hjXa(tj) are SaS random variables for h1, t1,...,hn, tn ¥ R.The most common way to define a SaS process is through an integral

with respect to a SaS random measure M(ds) on some space S. One maythink of M(ds), s ¥ S, as a sequence of independent SaS random variableswith the scale coefficients m(ds). Moreover, one assumes that m(ds) is ameasure on S and one calls it a control measure of Ma. An integral repre-sentation (also called a spectral representation) of a SaS process Xa is thenwritten as

{Xa(t)}t ¥ R=d 3F

Sft(s) Ma(ds)4

t ¥ R

, (1.1)

where=d stands for the equality in the sense of the finite-dimensional dis-tributions and {ft}t ¥ R is a collection of deterministic functions. The repre-sentation (1.1) means that the characteristic function of the process Xa canbe expressed as

E exp 3 i Cn

k=1hkXa(tk)4=exp 3 −F

S

: Cn

k=1hkftk (s): a m(ds)4 , (1.2)

where h1, t1,..., hn, tn ¥ R. One must therefore require that {ft}t ¥ R … La(S, m).In particular, (1.2) shows that the process Xa is indeed SaS. For moreinformation on stable processes, see Samorodnitsky and Taqqu. (13)

A process {Xa(t)} is called self-similar with index H> 0 if, for anyc > 0,

{Xa(ct)}t ¥ R=d {cHXa(t)}t ¥ R (1.3)

and it is called stationary increments if, for any h ¥ R,

{Xa(t+h)−Xa(h)}t ¥ R=d {Xa(t)−Xa(0)}t ¥ R. (1.4)

Perhaps the best known and simplest examples of SaS, H-self-similarstationary increments processes are the linear fractional stable motions

52 Pipiras and Taqqu

(LFSM, in short) defined for H ] 1/a through the integral representation

FR{a((t+u)H−

1a

+ −uH−1a

+ )+b((t+u)H−1a

− −uH−1a

− )} Ma(du), (1.5)

where a, b ¥ R, a ¥ (0, 2), H ¥ (0, 1), u+=max{u, 0}, u−=max{−u, 0},and the SaS random measure Ma has the Lebesgue control measure on R.To see that LFSM Xa is self-similar, that is, (1.3) holds, replace t by ctand u by cu in (1.5) and then use the heuristic relation c−1/aMa(d(cu))=d

Ma(du) valid for a SaS random measure on R with the Lebesgue controlmeasure. Self-similarity can be proved rigorously by using characteristicfunctions. To see that LFSM Xa has also stationary increments, considerthe integral representation of the increment Xa(t+h)−Xa(h), make thechange of variables uQ u−h and use the heuristic relationMa(d(u−h))=d

Ma(du). When H=1/a, the simplest examples of SaS, 1/a-self-similarstationary increments processes are

FR{c(1(0,.)(t+u)−1(0,.)(u))+d(ln |t+u|− ln |u|)} Ma(du), (1.6)

where c, d ¥ R, a ¥ (0, 2) and the SaS random measure Ma has theLebesgue control measure on R. When c ] 0, d=0, the process (1.6) is theusual SaS Lévy motion, which has independent increments. When c=0,d ] 0, the process has dependent increments and is called log-fractionalstable motion. Since a log-fractional stable motion is defined only fora ¥ (1, 2) (see Samorodnitsky and Taqqu (13)), we assume that d=0 in (1.6)when a ¥ (0, 1]. We will also call the processes (1.6) LFSM because (1.6)can be viewed as limiting cases of (1.5) when HQ 1/a (see Pipiras andTaqqu (8) for more information).It is important to note that different values of a and b in (1.5) lead

to different processes (for an exact formulation, see Theorem 7.4.5 inSamorodnitsky and Taqqu (13)). In fact, when a ¥ (0, 2), there are infinitelymany different SaS, self-similar stationary increments processes all differentfrom LFSM. For additional examples, see Samorodnitsky and Taqqu. (13)

We study here a new class of SaS, self-similar stationary incrementsprocesses defined as follows.

Definition 1.1. Let a ¥ (0, 2) and H> 0. A random process Xa iscalled a dilated fractional stable motion (DFSM, in short) if it has therepresentation

{Xa(t)}t ¥ R=d 3F

YFRFRe−os(F(y, e s(t+u))−F(y, e su)) Ma(dy, ds, du)4

t ¥ R

,

(1.7)

Dilated Fractional Stable Motions 53

where

o=H−1/a, (1.8)

F is a deterministic function and Ma is a SaS random measure onY×R×R with the control measure

ma(dy, ds, du)=n(dy) ds du.

The term ‘‘dilated’’ in the name DFSM refers to the dilation factor e s

in the representation (1.7). It is easy to see that DFSM are self-similar andhave stationary increments. They are special cases of the so-called stableself-similar mixed moving averages, that is, self-similar processes with amixed moving average representation, namely,

{Xa(t)}t ¥ R=d 3F

XFR(G(x, t+u)−G(x, u)) Ma(dx, du)4

t ¥ R

, (1.9)

where (X, X, m) is a measure space, G: X×RW R is a function andM is aSaS random measure on X×R with the control measure

ma(dx, du)=m(dx) du.

(To go from (1.7) to (1.9), set X=Y×R, x=(y, s), G(x, u)=e−osF(y, e su), and m(dx)=n(dy) ds.) In Pipiras and Taqqu, (7, 8) we pro-posed a decomposition of SaS self-similar mixed moving averages in threeparts: those determined by dissipative flows, mixed linear fractional stablemotions and processes of the ‘‘third kind.’’ It was shown that DFSM arecanonical processes for the dissipative class of processes.An example of DFSM is the telecom process, obtained as a limit of

renewal reward processes (see Pipiras and Taqqu (6)) and represented as

FRFR{(tN z−v)+−(0N z−v)+}(z−v)H−

2a−1

+ Ma(dz, dv), (1.10)

where Ma is a SaS random measure on R2 with the Lebesgue controlmeasure

ma(dz, dv)=dz dv

and aNb=min{a, b}. ( It can be written in the form (1.7) after a change ofvariables.) Another example is the so-called random wavelet expansionintroduced by Chi. (3) These and other examples of DFSM are given inSection 5 later.


The rest of the paper is organized as follows. In Section 2, we provideequivalent representations of DFSM and describe some connections toother processes. Section 3 concerns the space of integrands F for whichDFSM are well-defined and Section 4 points to connections to Sobolevspaces. Examples of DFSM processes are given in Section 5. Uniquenessquestions related to DFSM are investigated in Sections 6 and 7. Section 8contains the proofs of the results of Section 7.

2. EQUIVALENT REPRESENTATIONS AND CONNECTIONS TOOTHER PROCESSES

There are many other ways to represent DFSM in (1.7) as indicated inthe following lemma.

Lemma 2.1. Other representations for the process Xa in (1.7) are asfollows:

Xa(t)=d FYF.

0FRz−H(F(y, z(t+u))−F(y, zu)) Ma(dy, dz, du) (2.1)

=d FYF.

0FRzH−

2a(F(y, z−1(t+u))−F(y, z−1u)) Ma(dy, dz, du) (2.2)

=d FYFRFRe−Hs(F(y, e st+v)−F(y, v)) Ma(dy, ds, dv) (2.3)

=d FYF.

0FRz−H−

1a(F(y, zt+v)−F(y, v)) Ma(dy, dz, dv) (2.4)

=d FYF.

0FRzH−

1a(F(y, z−1t+v)−F(y, v)) Ma(dy, dz, dv), (2.5)

where=d denotes the equality in the sense of the finite-dimensional distri-butions. The control measure ofMa is n(dy) dz du.

Proof. The lemma follows by making proper changes of variablesin (1.7). i

The equivalent representations given in Lemma 2.1 point to connec-tions of DFSM to other types of processes that have been discussed in theliterature. Indeed:

1. The representation (2.5) can be expressed as

Xa(t)=d FY2F.

0zH−

1aF2 1 y, t

z2Ma(dy, dz) (2.6)


where Y2=Y×R, y=(y, v) ¥ Y×R=Y2 and

F2(y, s)=F2(y, v, s)=F(y, s+v)−F(y, v).

In other words, DFSM is a mixed fractional motion in the senseof Burnecki et al. (2)

2. Consider the representation (2.4). Let S=Y×R, m(ds)=m(dy, dv)=n(dy) dv, and let

Tt(s)=Tt(y, v)=(y, t+v) (2.7)

be a measure preserving flow on S (that is, {Tt}t ¥ R is a collectionof maps such that Tt1+t2 (s)=Tt1 (Tt2 (s)) for all s ¥ S and t1, t2 ¥ R,and m(T−1t A)=m(A) for all t ¥ R and measurable sets A). Let Ma

be a SaS random measure on S×R+ with the control measure

ma(s, z)=m(ds) z−aH−1 dz.

Then, the representation (2.4) can be expressed in distribution as

FSFR+

(F(Ttz(s))−F(s)) Ma(ds, dz). (2.8)

When S is any measure space and {Tt}t ¥ R is any measure preserv-ing flow on S, not necessarily of the form (2.7), the processes (2.8)(if well-defined) are also SaS, self-similar with exponent H andhave stationary increments. In other words, the class of DFSMprocesses is a subset of a larger class of processes (2.8), introducedand studied by Mori and Sato. (5)

The following result shows that DFSM are different from the usualLFSM defined in (1.5), at least when (Y, Y, n) is a so-called standardLebesgue spaces and a ¥ (1, 2). A standard Lebesgue space is made of astandard Borel space (Y,Y) and a s-finite measure n. A standard Borelspace (Y,Y) is a measurable space isomorphic (that is, there is a one-to-one, onto and bimeasurable map) to a subset of a complete separablemetric space equipped with the s-algebra of its Borel sets. A typicalexample of a standard Lebesgue space is (Rn, B(Rn)) with the Lebesguemeasure and discrete point masses. Standard Lebesgue spaces are conve-nient and have been used in the context of stable processes by Rosinski, (9)

Pipiras and Taqqu, (7, 8) and others.


Proposition 2.1. Let (Y, Y, n) be a standard Lebesgue space anda ¥ (1, 2). Then, DFSM in (1.7) and LFSM in (1.5) have different finite-dimensional distributions.

Proof. Theorems 4.1 and 5.1 in Pipiras and Taqqu (8) show thatDFSM is generated by a so-called dissipative flow (Theorem 4.1) andLFSM is generated by a so-called conservative flow. Theorem 5.3 (and theremark following it) in Pipiras and Taqqu (7) show that this implies thatthese two processes cannot have the same finite-dimensional distributions.

i

3. THE SPACE OF INTEGRANDS

Recall that the DFSM process (1.7) is well-defined if and only if itskernel

ft(y, s, u)=e−os(F(y, e s(t+u))−F(y, e su))

belongs to

La(Y×R×R, n(dy) ds du)

for all t ¥ R. We now study the set of functions F for which this is the case.Let a ¥ (0, 2), H> 0, and (Y, Y, n) be a measure space as before. Set

Cn, a, H={F: Y×RW R such that ||F||Cn, a, H <.}, (3.1)

where

||F||aCn, a, H=FYFRFR

|F(y, z1)−F(y, z2)|a

|z1−z2 |aH+1n(dy) dz1 dz2 (3.2)

=2 FYFRF.

0h−aH−1 |F(y, u+h)−F(y, u)|a n(dy) du dh (3.3)

=2 FY

1F.0h−aH−1 ||ghF(y, · )||

aLa(R) dh2 n(dy) (3.4)

with the notation

gh g( · )=g( ·+h)−g( · )

and ||g||aLa(R)=>R |g(x)|a dx. Observe that the relation (3.3) follows bymaking the change of variables z1=u+h, z2=u on the set {z1 \ z2} and


z1=u, z2=u+h on the set {z1 < z2} in (3.2). The relation (3.4) followsobviously from (3.3).

Proposition 3.1. Let a ¥ (0, 2), H> 0, and o=H−1/a. The DFSMprocess (1.7) is well-defined if and only if the function F belongs to theclass Cn, a, H, defined in (3.1).

Proof. The process Xa in (1.7) is well defined if and only if, for allt ¥ R,

I=FYFRFRe−oas |F(y, e s(t+u))−F(y, e su)|a n(dy) ds du <..

The change of variables e s(t+u)=z1 and e su=z2 leads to e s=t−1(z1−z2),u=tz2/(z2−z1) and the Jacobian

: “s“z1 “s“z2

“u“z1

“u“z2

:= :1

z1 −z2− 1z1 −z2

− tz2(z1 −z2)

2

tz1(z1 −z2)

2

:= t(z1−z2)2

.

Observe that while t positive (say) implies z1 > z2, the integral over theregion z1 < z2 yields the same value. Therefore, for all t ¥ R,

I=12|t|Ha F

YFRFR

|F(y, z1)−F(y, z2)|a

|z1−z2 |aH+1n(dy) dz1 dz2

=12|t|Ha ||F||aCn, a, H <.. i

WhenY={1} and n(dy)=d{1}(dy), we will use the notationCa, H=Cn, a, Hand also suppress the variable y=1 in the norms (3.2)–(3.4). Hence

||F||aCa, H=2 FRF.

0h−aH−1 |F(u+h)−F(u)|a du dh.

4. CONNECTION TO SOBOLEV SPACES

The space Ca, H is related to the well-known Sobolev type spaces. Moreprecisely, when a ¥ [1, 2) and H ¥ (0, 1), we have

Ca, H 5 La(R)=WH, a(R), (4.1)

where WH, a(R) is the Sobolev space of fractional order H (see, forexample, Theorem 7.48 in Adams, (1) p. 214). The space WH, a(R)=Ca, H 5 La(R) is also called the Slobodeckij space (see Triebel (1983), p. 36,


and Runst and Sickel (1996), p. 12). The space Ca, H is strictly larger thanWH, a(R) because there are functions in Ca, H which are not in La(R) andhence not inWH, a(R) (e.g., see Example 5.1 later).While fractional order Sobolev spaces WH, a(R) have been extensively

studied, it would be interesting to know whether results obtained for thesespaces can be extended to bigger spaces Ca, H. We are particularly interestedin embedding results, that is results of the type CaŒ, HŒ … Ca, H, since onewould be able to show that a given function F belongs to Ca, H by showingthat it belongs to CaŒ, HŒ. We do not have a simple answer to the abovequestion because we believe that the study of spaces Ca, H is more complexthan that ofWH, a(R). For example, it follows immediately from (3.4) that

WHŒ, a(R) …WH, a(R)

for all

0 < H <HŒ.

However, this does not imply that Ca, HŒ … Ca, H for all 0 < H <HŒ, as thefollowing example shows.

Example 4.1. Let a ¥ (1, 2), H ¥ (0, 1) and consider the function Fthat appears in the telecom process (1.10), namely,

F(z)=(zN0+1)+=˛0, if z [ −1,

1+z, if −1 < z [ 0,

1, if z > 0.

We will show that F ¥ Ca, H for H ¥ (1/a, 1) but F ¨ Ca, H for H ¥ (0, 1/a].Observe that

||F||aCa, H=2 F−1

−.dz1 F

0

−1dz2

(1+z2)a

(z2−z1)aH+1+2 F

−1

−.dz1 F

.

0dz2

1(z2−z1)aH+1

+2 FF−1 < z1 < z2 < 1

dz1 dz2(z2−z1)a

(z2−z1)aH+1

+2 F0

−1dz1 F

.

0dz2

(−z1)a

(z2−z1)aH+1

=: I1+I2+I3+I4.


If H ¥ (1/a, 1), then ||F||Ca, H <., since

I1=2aH

F0

−1dz2(1+z2)a−aH <.

by using a−aH=a(1−H) > 0,

I2=2aH

F.

0dz2(1+z2)−aH <.,

because aH> 1,

I3=2

a−aHF0

−1dz1(−z1)a−aH <.,

and

I4=2aH

F0

−1dz1(−z1)a−aH <..

If H ¥ (0, 1/a], then ||F||Ca, H=. since I2=.. Hence, we do not haveCa, HŒ … Ca, H when

0 < H [ 1/a <HŒ < 1.

Observe that in this example F ¨ La(R).

Although the complete structure of spaces Ca, H is still to be deter-mined, we can nevertheless provide some insight on these spaces. Observethat the norm in (3.4) can be expressed as

||F||aCa, H=2 Fc

0h−aH−1 ||ghF||

aLa(R) dh+2 F

.

ch−aH−1 ||ghF||

aLa(R) dh

=: ||F||a1, Ca, H+||F||a2, Ca, H

, (4.2)

where ||F||a1, Ca, H and ||F||a2, Ca, H

denote the corresponding integrals in (4.2).The following lemma can be used to conclude that ||F||1, Ca, H <. or||F||2, Ca, H <. and thus that ||F||Ca, H <..

Lemma 4.1. Let a ¥ (1, 2) and H ¥ (0, 1). Then,

(i) if F is an absolutely continuous function with the derivativeFŒ ¥ La(R), then

||F||1, Ca, H <.;


(ii) if F ¥ La(R), then

||F||2, Ca, H <..

In particular, if F ¥ La(R) and FŒ ¥ La(R), then

||F||Ca, H <..

Proof. We first prove (i). By writing

ghF(u)=FR1[0, h)(z) FŒ(u+z) dz

and by using the generalized Minkowski’s inequality

>FR|g(x, · )| dx>

La(R)[ F

R||g(x, · )||La(R) dx

valid for a > 1, we obtain that

||ghF||aLa(R) [ h

a ||FŒ||aLa(R).

The conclusion ||F||1, Ca, H <. follows since −aH−1+a+1=a(1−H) > 0.Part (ii) follows by using ||ghF||

aLa(R) [ C ||F||

aLa(R) and the integrability

condition −aH−1+1=−aH< 0. i

5. EXAMPLES OF DILATED FRACTIONAL STABLE MOTIONS

There are interesting examples given below of processes that havethe representation (1.7) with Y={1} and n(dy)=d{1}(dy). Example 5.4involves Y={1, 2}. These processes are well-defined under the conditionson a and H that are specified (this can be verified by showing that thecorresponding kernel F belongs to the space Cn, a, H ).

Example 5.1. If a ¥ (1, 2) and H ¥ (1/a, 1), then the process

Xa(t)=F.

0FR(((t+u)N0+x)+−(uN0+x)+) xH−

2a−1Ma(dx, du), t ¥ R,

where SaS random measure Ma has the Lebesgue control measure, is well-defined. It is the telecom process mentioned in Section 1. We can derive thecanonical representation (1.7) for the process Xa by using (2.2) or, directly,as follows:


Xa(t)=d F

RFR((t+u)N0+e−s)+−(uN0+e−s)+) e−(H−

1a−1) sMa(ds, du)

=FRFR(e s(t+u)N0+1)+−(e suN0+1)+) e−(H−

1a) sMa(ds, du)

=FRFRe−os(F(e s(t+u))−F(e su)) Ma(ds, du),

where o=H−1/a and

F(z)=(zN0+1)+, z ¥ R.

This is the function F considered in Example 4.1.

Example 5.2. Let a ¥ (0, 2), H ¥ (0, 1), and a ¥ R be such that eitherH< 1/a <H−a (here, a < 0) or H−a < 1/a <H (here, a > 0). Then, theprocess

Xa(t)=F.

0FR((t+u)a+Nxa−ua+Nxa) xH−

2a−aMa(dx, du), t ¥ R,

where Ma has the Lebesgue control measure, is called a mixed truncatedfractional stable motion. To derive the canonical representation (1.7), write

Xa(t)=d F

RFR(e sa(t+u)a+N1−e saua+N1) e−(H−

1a) sMa(ds, du)

=FRFRe−os(F(e s(t+u))−F(e su)) Ma(ds, du),

where o=H−1/a and

F(z)=za+N1, z ¥ R.

(Note that the shape of the function F is quite different for a > 0 anda < 0.) For more information on the mixed truncated fractional stablemotion see Surgailis et al. (14)

Example 5.3. If a ¥ (0, 2) and H ¥ (0, 1/a), it is easy to verify thatthe function

F(z)=1{|z| [ 1}(z), z ¥ R,


belongs to the space Ca, H. It corresponds to the process

Xa(t)=FRFR(1{es |t+u| [ 1}(s, u)−1{es |u| [ 1}(s, u)) e−(H−

1a) sMa(ds, du)

=d F.

0FR(1{|u−t| < x}(x, u)−1{|u| < x}(x, u)) xH−

2aMa(dx, du). (5.1)

This last expression resembles the H-self-similar process with stationaryincrements constructed by means of integral geometry in Takenaka. (15) If,for t ¥ R, we let

At={(x, u) ¥ R+×R : |u− t| < x}

and define

Bt=A0 g At=(A0 0At) 2 (At 0A0),

then the process constructed by Takenaka (15) for a ¥ (0, 2) and H ¥ (0, 1/a)is given by

X2a(t)=F.

0FR1Bt (x, u) x

H− 2aM2 a(dx, du)

=F.

0FR|1At (x, u)−1A0 (x, u)| x

H− 2aM2 a(dx, du)

=F.

0FR|1{|u−t| < x}(x, u)−1{|u| < x}(x, u)| xH−

2aM2 a(dx, du),

where M2 a has the Lebesgue control measure (see also Samorodnitsky andTaqqu, (13) Chapter 8). The difference between Xa and X2a is that the kernelin the definition of X2a is taken in absolute value. Nevertheless, the pro-cesses Xa and X2a have the same finite-dimensional distributions. This is aconsequence of the following simple observation,

1{|u−t| < x}(x, u)−1{|u| < x}(x, u)=h(x, u) |1{|u−t| < x}(x, u)−1{|u| < x}(x, u)|,

for all t, u, x, where h(x, u)=−1, if |u| < x, and h(x, u)=1, if |u| \ x (drawa picture in the (u, x) plane). Since h does not depend on t and since|h(x, u)|=1, one can use characteristic functions to prove the equality ofthe finite-dimensional distributions. Therefore, the processes (5.1) areTakenaka processes.


Example 5.4. When a ¥ (0, 2) and H ¥ (0, 1), Samorodnitsky andTaqqu (12) introduced a class of SaS, H-self-similar processes with station-ary increments given by

Xa(t)=FRn(||t1+u||H−

na−||u||H−

na) Ma(du), t ¥ R, (5.2)

where n \ 2, || · || is the usual Euclidean norm, u=(u1,..., un) ¥ Rn, and1=(1,..., 1) ¥ Rn, and a SaS random measureMa has the Lebesgue controlmeasure. Let us show that the processes (5.2) have the canonical represen-tation (1.7). Consider the case n=2 first. By making the change ofvariables u1=u and u2=u+v, we can first represent (5.2) with n=2 as themixed moving average

Xa(t)=d F

R2(((t+u)2+(t+u+v)2)

H2 −

1a−(u2+(u+v)2)

H2 −

1a) Ma(du, dv).

(5.3)

To see that (5.3) has the canonical representation (1.7), consider theintegral in (5.3) over the regions v > 0 and v < 0 separately and make thechanges of variables v=e−s and v=−e−s, respectively. Then, by setting

F(1, z)=(z2+(z+1)2)H2 −

1a, F(2, z)=(z2+(z−1)2)

H2 −

1a, z ¥ R,

(5.4)

and Y={1, 2}, we have

Xa(t)=d FYFRFRe−(H−

1a) s(F(y, e s(t+u))−F(y, e su)) Ma(dy, ds, du),

(5.5)

where Ma has the control measure d{1, 2}(dy) ds du, which is the represen-tation (1.7).In the case n \ 3, one can represent the process (5.2) through the

DFSM representation (5.5) with the space Y={1, 2}×Rn−2 and the corre-sponding points y=(y0, y1,..., yn−2), where y0 ¥ {1, 2} and (y1,..., yn−2)¥ Rn−2, the SaS random measure Ma(dy, ds, du) with the control measured{1, 2}(dy0) dy1 · · · dyn−2 ds du and the kernel function

F(1, y1,..., yn−2, z)=(z2+(z+1)2+(z+y1)2+·· ·+(z+yn−2)2)H2 −

n2a,

F(2, y1,..., yn−2, z)=(z2+(z−1)2+(z+y1)2+·· ·+(z+yn−2)2)H2 −

n2a,(5.6)

where z ¥ R.


Observe that the processes Xa provide examples of 1/a–self-similarprocesses with stationary increments by setting H=1/a and a ¥ (1, 2) (sothat H ¥ (0, 1)). Samorodnitsky and Taqqu showed that these process aredifferent from the usual log-fractional stable motion and the independentincrements stable Lévy motion. This, in fact, follows also directly fromProposition 2.1.

Example 5.5. Recently, Chi (3) constructed a class of stationary self-similar distributions on generalized functions. The construction was moti-vated by a probabilistic modeling of natural images and the distributionsobtained were called random wavelet expansions. They are defined by

Za(g)=FRFR

1FReHŒsg(e st+v) g(t) dt2Ma(ds, dv),

where a ¥ (0, 2), HŒ ¥ R, Ma is a SaS random measure with the Lebesguecontrol, g is a suitably chosen function, and g belongs to some space of testfunctions. (For more information, we refer the reader to Chi. (3) The set-upin Chi (3) is more general than here: t and v may belong to Rd, d \ 1, andMa may be replaced by other random measures like a Gaussian or infini-tely divisible ones.) To see the connection between the stationary processZa and the stationary increment process Xa given by (1.7), replace HŒ by−H+1, let F be a function such that its derivative F(t) — dF(t)/dt=g(t)and make the change of variables v=e su. We can then represent Za as

Za(g)=FRFR

1FRe−(H−

1a) se sF(e s(t+u)) g(t) dt2Ma(ds, du).

Hence, Za is in fact the derivative, viewed as a generalized process, of thestationary increment process Xa, having the representation (1.7) withY={1} and n=d{1}.

6. ON THE UNIQUENESS OF THE DFSM PROCESSES

One would like to know whether the DFSM processes, which aredefined by (1.7), are really different for different F’s, namely whether theyhave different finite-dimensional distributions. The following result canoften be used to conclude that the distributions of two given processes aredifferent. It uses the notion of standard Lebesgue spaces given in Section 2.

Theorem 6.1. Let a ¥ (0, 2) and H> 0. Suppose that the process Xahas two representations (1.7): one on the standard Lebesgue space (Y, n)


and with the function F: Y×RW R, and the other on the standardLebesgue space (Y2 , n) and with the function F2 : Y2 ×RW R. Then, thereexist measurable functions F1: YW Y2 and h, F2, F3, F4: YW R such that

F(y, z)=h(y) F2(F1(y), eF2(y)(z+F3(y)))+F4(y) (6.1)

a.e. n(dy) dz.

Proof. By Lemma 4.1(i), in Pipiras and Taqqu, (7) there exist measur-able functions F1: Y×R×RW Y2 and h, F2, F3: Y×R×RW R such that

e−os(F(y, e s(t+u))−F(y, e su))

=h(y, s, u) e−oF2(y, s, u)

×(F2(F1(y, s, u), eF2(y, s, u)(t+u+F3(y, s, u)))

−F2(F1(y, s, u), eF2(y, s, u)(u+F3(y, s, u))))

a.e. n(y) ds du dt. By making the change of variables t=e−sz−u, we getthat

F(y, z)=F(y, e su)+eosh(y, s, u) e−oF2(y, s, u)

×(F2(F1(y, s, u), eF2(y, s, u)(e−sz+F3(y, s, u)))

−F2(F1(y, s, u), eF2(y, s, u)(u+F3(y, s, u))))

a.e. n(y) ds du dz. By fixing s=s0 and u=u0, for which this equation holdsa.e. n(dy) dz and setting h(y)=eos0h(y, s0, u0) e−oF2(y, s0, u0), we obtain theresult. i

Observe that Theorem 6.1 does not provide an ‘‘if and only if ’’ con-dition. One can obtain, however, such a condition in the special caseY=Y2={1} and n=n=d{1}.

Corollary 6.1. Suppose that the processes Xa and X2a have the repre-sentation (1.7) with the spaces Y=Y2={1}, the measures n=n=d{1} andthe functions F and F2 , respectively. If Xa and X2a have the same finite-dimensional distributions, then

F(z)=aF2(bz+c)+d a.e. dz, for some constants a, c, d ¥ R and b > 0.(6.2)

Conversely, if condition (6.2) holds, then Xa and X2a have the samefinite-dimensional distributions up to a multiplicative constant. They haveidentical finite-dimensional distributions if |a| bo=1.


Proof. The first statement of the corollary follows from Theorem 6.1.To show the second statement, assume that the condition (6.2) holds andlet tk, hk ¥ R, k=1,..., n. Then, by making the change of variables u=u−b−1e−sc and s=s− ln b below, we have

− ln E exp 3 i Cn

k=1hkXa(tk)4

=FRFR

: Cn

k=1hke−os(F(e s(tk+u))−F(e su)) :

a

ds du

=FRFR

: Cn

k=1hke−osa(F2(be s(tk+u)+c)−F2(be su+c)) :

a

ds du

=(|a| bo)a FRFR

: Cn

k=1hke−os(F2(e s(tk+u))−F2(e su)) :

a

ds du

=−ln E exp 3 i Cn

k=1hk |a| boX2a(tk)4 .

This show that the processes Xa and |a| boX2a have the same finite-dimen-sional distributions. i

In practice, one may not want to distinguish between processes whichdiffer by a multiplicative constant. Hence, we say that two processes X andX2 are essentially identical if there exists a multiplicative constant c suchthat X(t) and cX2 (t) have the same finite-dimensional distributions. If theseprocesses are not essentially identical, we say that they are essentially dif-ferent. It is easy to see that the following holds.

Corollary 6.2. (i) Suppose that Xa has the representation (1.7) withthe standard Lebesgue space (Y, n) and with the function F: Y×RW R,and X2a has the representation (1.7) with the standard Lebesgue space(Y2 , n) and with the function F2 : Y2 ×RW R. If condition (6.1) does nothold, then the processes Xa and X2a are essentially different.

(ii) Suppose that the assumptions of Corollary 6.1 hold. Then theprocesses Xa and X2a are essentially identical if and only if condition (6.2)holds.

The second part of this corollary shows that the telecom process inExample 5.1, defined by the function

F1(z)=(zN0+1)+, z ¥ R,


the mixed truncated fractional stable motion in Example 5.2, defined by thefunction

F2(z)=za+N1, z ¥ R,

(if a ] 1), and the process in Example 5.3, defined by the function

F3(z)=1{|z| [ 1}, z ¥ R,

are essentially different.In the case a=1, F1(z)=F2(z+1) and therefore the telecom process

in Example 5.1 is essentially identical to a mixed truncated fractional stablemotion in Example 5.2. Since |a| bo=1, the two process have, in fact, thesame finite-dimensional distributions.By using Theorem 6.1, one may also show that the processes in

Example 5.4 are essentially different from the processes in Examples 5.1–5.3.Indeed, let

F(y, z)=F(i, y1,..., yn−2, z)

be the kernel function (5.6) of the process Xa in Example 5.4, where

y=(i, y1,..., yn−2) ¥ {1, 2}×Rn−2

and F2(z) be one of the kernel functions of processes X2a in Examples 5.1–5.3.If the processes Xa and X2a are essentially identical, then by Theorem 6.1,there are functions a, c, d: YW R and b: YW (0,.) such that

F(y, z)=a(y) F2(b(y) z+c(y))+d(y) (6.3)

a.e. n(dy) dz, and also by reversing the roles of the processes inTheorem 6.1, there are y ¥ Y, a, c, d ¥ R, and b > 0 such that

F2(z)=aF(y, bz+c)+d (6.4)

a.e. dz. It is easy to verify that neither the relation (6.3) nor the relation(6.4) holds. (In general, it is enough to verify that only one of the relations(6.3) and (6.4), whichever is easier to deal with, does not hold.) Forexample, if

F2(z)=za+N1


is the kernel function in Example 5.2, then for y=(1, y1,..., yn−2), therelation (6.3) becomes

(z2+(z+1)2+(z+y1)2+·· ·+(z+yn−2)2)H2 −

n2a

=a(1, y1,..., yn−2)((b(1, y1,..., yn−2) z

+c(1, y1,..., yn−2))a+N1)+d(1, y1,..., yn−2) (6.5)

a.e. dy1 · · · dyn−2 dz, and the relation (6.4) becomes

za+N1=a((bz+c)2+(bz+c+(−1) ı+1)2+·· ·+(bz+c+yn−2)2)H2 −

n2a+d(6.6)

a.e. dz, for some fixed y=(ı, y1,..., yn−2) where ı=1, 2. Neither the rela-tion (6.5) nor the relation (6.6) holds. For example, by fixing y1,..., yn−2 sothat the relation (6.5) holds a.e. dz and taking z negative large enough, theright-hand side of (6.5) becomes a constant while the left-hand side behaveslike a power of z. Hence, the processes Xa and X2a are essentially different.Since Samorodnitsky and Taqqu (12) have shown that the processes in

Example 5.4 differ from each other if they correspond to different valuesof n, we get:

Proposition 6.1. The processes in Examples 5.1, 5.2 (a ] 1), 5.3, and5.4 are essentially different.

7. FINER UNIQUENESS RESULTS WHEN Y IS FINITE ORCOUNTABLE

Let Xa and X2a be two DFSM processes defined through finite orcountable spaces Y and Y2 , respectively. Let |Y| denote the number of ele-ments of Y. We may suppose without loss of generality that Y … Z andY2 … Z. We will now introduce a scheme to determine whether the processesXa and X2a have different (or the same) finite-dimensional distributions.The scheme uses the notion of a minimal integral representation of a SaSprocess.Minimal integral representations, which were introduced by Hardin (4)

and developed by Rosinski, (10) are particularly useful in our context. (Forother applications, see Rosinski (9) and Pipiras and Taqqu. (7)) We first recalltheir definition. Let (S, BS, m) be a standard Lebesgue space where BS is as-algebra of Borel sets of S (see Section 2). Then, a spectral representation{ft}t ¥ R … La(S, BS, m) of a SaS process Xa in (1.1) is called minimal if


(M1) supp{ft: t ¥ R}=S a.e. and

(M2) s{ft/fs, s, t ¥ R}=BS (modulo m).

( The support supp{ft: t ¥ R} is defined as a minimal (m-a.e.) set A suchthat

m{s ¥ S : ft(s) ] 0, s ¨ A}=0 for all t ¥ R.

We say that two s-algebras are equal modulo m if their sets are equalm-a.e.)We will work with a condition equivalent to (M2), namely,

(M2Œ) for any non-singular f: SW S such that, for some k: SWR0{0} and every t ¥ R,

ft(f(s))=k(s) ft(s) a.e. m(ds), (7.1)

we have f(s)=s a.e. m(ds). (A map f: SW S is non-singular if m p f−1°m,that is, m(A)=0 for A ¥BS implies m(f−1(A))=0.) The equivalence of(M2) and (M2Œ) is proved in Rosinski. (10)

Recall that we study here DFSM processes defined through spacesY … Z, that is, processes

{Xa(t)}t ¥ R=d 3F

YFRFRe−os(F(n, e s(t+u))−F(n, e su)) Ma(dn, ds, du)4

t ¥ R

,

(7.2)

where a SaS random measure M(dn, ds, du) has the control measuredY(dn) ds du. We will suppose throughout the rest of this section that thereis no n ¥ Y such that F(n, z)=const a.e. dz since, otherwise, the integrandof (7.2) corresponding to that value of n will be zero. There is therefore nopoint in including it. The following result characterizes minimal represen-tations of Xa given by (7.2).

Theorem 7.1. Let Xa be a DFSM process defined through the spaceY … Z as in (7.2). Then,

(i) when |Y|=1, the representation (7.2) is always minimal,and

(ii) when |Y| \ 2, the representation (7.2) is minimal if and only ifthere is no n, m ¥ Y, n ] m, such that

F(n, z)=a(n, m) F(m, b(n, m) z+c(n, m))+d(n, m) (7.3)


holds a.e. dz, for some a(n, m) ] 0, b(n, m) > 0, and c(n, m),d(n, m) ¥ R.

The proof of this result can be found in the next section.

Example 7.1. Let a ¥ (1, 2) and H ¥ (0, 1/a). Let also

G(1, z)=z−1+ N1

denote the kernel in Example 5.2 with a=−1 (when a ¥ (1, 2) andH ¥ (0, 1/a), we have H< 1/a <H+1) and

G(2, z)=1{|z| [ 1}(z)

be the kernel in Example 5.3. Since the kernels G(1, z) and G(2, z) cannotbe related as in (7.3), the DFSM process

F{1, 2}

FRFRe−os(G(n, e s(t+u))−G(n, e su)) Ma(dn, ds, du), (7.4)

where the control measure of Ma(dn, ds, du) is d{1, 2}(dn) ds du, is definedthrough a minimal representation.

Theorem 7.1 states that, when |Y| \ 2, the kernels of non-minimalrepresentations are related by (7.3). This suggests that, if the representation(7.2) is not minimal, one can always modify it to a minimal representationas follows. Say that the relation n ’ m on Y holds if and only if (7.3) issatisfied a.e. dz, with a(n, m) ] 0, b(n, m) > 0, and c(n, m), d(n, m) ¥ R.Then, n ’ m is an equivalence relation. Let Y=;p ¥ P Cp denote the parti-tion of Y into the corresponding equivalence classes and let also np, p ¥ P,be representative elements from the equivalence classes Cp. The followingresult provides a minimal representation for the process Xa given by (7.2).It is proved in the next section.

Theorem 7.2. With the above notation, the process Xa in (7.2) hasthe minimal representation

F{np, p ¥ P}

FRFRe−os(F2(np, e s(t+u))−F2(np, e su)) Ma(dnp, ds, du), (7.5)

where

F2(np, z)=3 Cn ¥ Cp

|a(n, np)|a |b(n, np)|ao41/a

F(np, z). (7.6)


Example 7.2. Consider the process Xa in Example 5.4 with n=2,which a DFSM process with Y={1, 2} and the kernel functions F(i, z),i=1, 2, z ¥ R, in (5.4). The representation of Xa is not minimal byTheorem 7.1, since F(1, z)=F(2, z+1) for z ¥ R and hence (7.3) holdswith a(1, 2)=b(1, 2)=c(1, 2)=1 and d(1, 2)=0. There is in this caseonly one equivalence class C, with two elements n=1, 2 and representativeelement, say 2. By Theorem 7.2, the process Xa can be represented by theminimal representation (7.5), where (7.6) is given by

F2(1, z)={|a(1, 2)|a |b(1, 2)|ao+|a(2, 2)|a |b(2, 2)|ao} F(2, z)

=21/aF(2, z)

=21/a(z2+(z−1)2)H2 −

1a,

that is, Xa has the minimal representation

FRFR21/ae−os(((e s(t+u))2+(e s(t+u)−1)2)

H2 −

1a

−((e su)2+(e su−1)2)H2 −

1a) Ma(ds, du), (7.7)

whereMa(ds, du) has the control measure ds du.

The next result provides a simple way to find out if two DFSM pro-cesses (7.2) having minimal integral representations are essentially identicalor not. The proof of this result can be found in the next section.

Theorem 7.3. Let Xa and X2a be two DFSM processes definedthrough spaces Y … Z and Y2 … Z, respectively. Suppose that their integralrepresentations are minimal. Then,

(i) Xa and X2a are essentially different if |Y| ] |Y2 |,and

(ii) Xa and X2a are essentially identical if and only if |Y|=|Y2 | and iftheir kernels are related in the following way: there is a one-to-one, onto map p: YW Y2 such that, for n ¥ Y,

F(n, z)=a(n) F2(p(n), b(n) z+c(n))+d(n) (7.8)

holds a.e. dz, with a(n) ] 0, b(n) > 0, c(n), d(n) ¥ R, and a(n) b(n)o

¥ {−k, k}, k ] 0. The processes are identical in distribution if andonly if k=1.


In view of the previous three theorems, one can determine whether twoDFSM processes Xa and X2a defined through spaces Y … Z are essentiallyidentical by using the following

Methodology:

(1) Use Theorems 7.1 and 7.2 to find minimal representations for theprocesses Xa and X2a.

(2) Apply Theorem 7.3 to determine whether the processes Xa andX2a are essentially identical.

We illustrate this approach in the following examples. The first showshow one can use minimal representations to prove part of Proposition 6.1.

Example 7.3. Consider the DFSM Xa in Example 5.4 with n=2. ByExample 7.2, Xa has a minimal representation (7.7) with Y={1} and thekernel function

21/aF(2, z)=21/a(z2+(z−1)2)H2 −

1a.

Consider, on the other hand, any of the processes X2a in Examples 5.1–5.3,which are DFSM with Y2={1} and corresponding kernel functions F2 .By Theorem 7.1(i), their integral representations are minimal. Hence, byTheorem 7.3(ii), since the kernels 21/aF(2, z) and F2(z) cannot be related asin (7.8), the processes Xa and X2a are essentially different.

Example 7.4. Consider a sequence of functions {F(n, · )}n ¥ Z … Ca, Hand a sequence c={cn}n ¥ Z … R such that cn ] 0, n ¥ Z, and

Cn|cn |a ||F(n, · )||

aCa, H<..

Suppose that, for n ] m, F(n, · ) and F(m, · ) cannot be related as in (7.3) ofTheorem 7.1. Then, the DFSM process

Xca(t)=FZFRFRcne−os(F(n, e s(t+u))−F(n, e su)) Ma(dn, ds, du), (7.9)

where the control measure ofMa(dn, ds, du) is dZ(dn) ds du, is well-definedand its integral representation is minimal. Now choose two sequences a andb with the same properties as the sequence c. Consider the correspondingDFSM processes Xaa and X

ba. Since the difference of kernels is due to the


sequences a={an} and b={bn}, in view of Theorem 7.3, the processes Xaa

and Xba are essentially identical if and only if |an |/|bn |=const, n ¥ Z.

8. THE PROOFS OF THEOREMS 7.1–7.3

Proof of Theorem 7.1. Consider the case (i). We first show that thecondition (M1) of minimality, namely,

A0 :=supp{F(e s(t+u))−F(e su), t ¥ R}=R×R a.e. ds du,

is satisfied. The proof uses ideas of that of Lemma 4.2 in Pipiras andTaqqu. (7) As in that lemma,

A0=supp{F(e s(t1+u))−F(e s(t2+u)), t1, t2 ¥ R}.

By making the change of variables t1 Q ew(t1+h), t2 Q ew(t2+h), anduQ ewu, we obtain that

A0=supp{F(e s+w(t1+u+h))−F(e s+w(t2+u+h)), t1, t2 ¥ R}

and hence that, for all w, h ¥ R,

A0=A0+(w, h),

where by A0+(w, h) we mean the shifted set {(s+w, u+h): (s, u) ¥ A0}.Fubini’s theorem implies (see also Lemma 3.1 in Pipiras and Taqqu (7))that 1A0 (s+w, u+h)=1A0 (s, u) a.e. ds du dw dh. By making the changeof variables s+w=x and u+h=y, we get 1A0 (x, y)=1A0 (s, u) a.e.dx dy ds du. Since we have A0 ]” a.e. (otherwise, Xa — 0), we may fix(x0, y0) ¥ A0 such that 1=1A0 (x0, y0) and hence 1=1A0 (s, u) a.e. ds du.This implies that A0=R×R a.e.We will now show that the condition (M2Œ) of minimality holds (see

(7.1)). Suppose that

f(s, u)=(f1(s, u), f2(s, u)): R×RW R×R

is a map such that, for all t,

e−of1(s, u)(F(ef1(s, u)(t+f2(s, u)))−F(ef1(s, u)f2(s, u)))

=k(s, u) e−os(F(e s(t+u))−F(e su)) (8.1)


a.e. ds du. We need to show that f(s, u)=(s, u) a.e. By Fubini’s theorem,relation (8.1) holds a.e. ds du dt as well. By replacing t by t+v in (8.1), weget that

e−of1(s, u)(F(ef1(s, u)(t+v+f2(s, u)))−F(ef1(s, u)f2(s, u)))

=k(s, u) e−os(F(e s(t+v+u))−F(e su)) (8.2)

a.e. ds du dt dv and, by subtracting (8.1) from (8.2), we obtain that

e−of1(s, u)(F(ef1(s, u)(t+v+f2(s, u)))−F(ef1(s, u)(t+f2(s, u))))

=k(s, u) e−os(F(e s(t+v+u))−F(e s(t+u))) (8.3)

a.e. ds du dt dv. By making the change of variables t+u=z, we further getfrom (8.3) that

e−of1(s, u)(F(ef1(s, u)(z+v+f2(s, u)−u))−F(ef1(s, u)(z+f2(s, u)−u)))

=k(s, u) e−os(F(e s(z+v))−F(e sz)) (8.4)

a.e. ds du dz dv. By making a further change of variables vQ ew−st andzQ ew−sz, we obtain from (8.4) that, for fixed t,

(ef1(s, u)−s)−o (F(ef1(s, u)−sew(t+z)+ef1(s, u)(f2(s, u)−u))

−F(ef1(s, u)−sewz+ef1(s, u)(f2(s, u)−u)))

=k(s, u)(F(ew(t+z))−F(ewz)) (8.5)

a.e. ds du dw dz. If f(s, u)=(f1(s, u), f2(s, u)) ] (s, u) a.e. ds du, then, bychoosing specific values of s and u, there are 0 < a ] 1 or b ] 0 such that

a−oe−ow(F(aew(t+z)+b)−F(aewz+b))=ke−ow(F(ew(t+z))−F(ewz))(8.6)

a.e. dw dz. It is enough to show that (8.6) does not hold. Relation (8.6)implies that

FRFRa−oae−oaw |F(aew(t+z)+b)−F(aewz+b)| a dw dz

=|k|a FRFRe−oaw |F(ew(t+z))−F(ewz)| a dw dz. (8.7)


Since the two integrals in (8.7) are equal, it follows that either

|k|=1

or

FRFRe−oaw |F(ew(t+z))−F(ewz)|a dw dz=0.

In the latter case, we have F(ew(t+z))−F(ewz)=0 a.e. dw dz or, after achange of variables, F(v)=F(u) a.e. du dv or F(u)=const a.e. This impliesXa — 0 which is a contradiction.In the case |k|=1, we have from (8.6) that

a−oe−ow |F(aew(t+z)+b)−F(aewz+b)|=e−ow |F(ew(t+z))−F(ewz)|(8.8)

a.e. dw dz. Then, by setting t=1 and making the change of variableszQ z−a−1e−wb below, we obtain that

FRa−oae−oaw |F(aew(1+z))−F(aewz)| a dz

=FRe−oaw |F(ew(1+z))−F(ewz)| a dz

a.e. dw. Denoting the last integral by G(w), it follows that G(w)=G(w+ln a) a.e. dw or that G is periodic with period ln a. Since G ¥

La(R, dw) and Xa – 0, we obtain a contradiction to the assumption a ] 1.If a=1 but b ] 0 in (8.8), by setting t=1 and making the change ofvariables zQ ze−w, we get

e−Hw |F(ew+z+b)−F(z+b)|=e−Hw |F(ew+z)−F(z)|

a.e. dw dz. It follows that

FRe−Haw |F(ew+z+b)−F(z+b)| a dw

=FRe−Haw |F(ew+z)−F(z)| a dw=: J(z),

showing that J(z)=J(z+b) a.e. dz, namely that J is periodic with period b.Since J ¥ La(R, dz) and Xa – 0, we obtain a contradiction to the assumption


b ] 0. Hence, f(s, u)=(s, u) a.e. ds du and the condition (M2Œ) of mini-mality holds.Consider now part (ii) of the theorem. To prove the necessity, we

suppose that the representation (7.2) is minimal and we argue by contra-diction. Suppose there are n ] m such that (7.3) holds. The idea is to userelation (7.1) to construct a non-identity map f that essentially maps n intom and m into n. More precisely, the map

f(y, s, u)=˛ (m, s+ln b(n, m), u+e−sb(n, m)−1 c(n, m)), if y=n,

(n, s− ln b(n, m), u−e−sc(n, m)), if y=m,

(y, s, u), otherwise,

is non-singular, f(y, s, u) – (y, s, u) and there is a function k such that,with the notation

f(y, s, u)=(f0(y, s, u), f1(y, s, u), f2(y, s, u)),

we have

e−of1(y, s, u)(F(f0(y, s, u), ef1(y, s, u)(t+f2(y, s, u)))

−F(f0(y, s, u), ef1(y, s, u)f2(y, s, u)))

=k(y, s, u) e−os(F(y, e s(t+u))−F(y, e su)). (8.9)

To verify this, use the definition of f=(f0, f1, f2) and relation (7.3). Rela-tion (8.9) contradicts to the condition (M2Œ) of minimality.To show the sufficiency, suppose that (7.3) does not hold for any

n ] m. We need to show that the representation is minimal. One can showas in the case (i) that

supp{F(y, e s(t+u))−F(y, e su), t ¥ R}=Y×R×R a.e. dY(dy) ds du,

that is, the condition (M1) of minimality holds. In order to show thecondition (M2Œ), suppose that (8.9) holds for some non-singular f=(f0, f1, f2). We need to show that f is the identity map, that is, f(y, s, u)=(y, s, u) a.e. Arguing as in the case (i), we obtain (see (8.5)) that, for allt ¥ R,

e−o(f1(y, s, u)−s)e−ow(F(f0(y, s, u), ef1(y, s, u)−sew(t+z)+ef1(y, s, u)(f2(y, s, u)−u))

−F(f0(y, s, u), ef1(y, s, u)−sewz+ef1(y, s, u)(f2(y, s, u)−u)))

=k(y, s, u) e−ow(F(y, ew(t+z))−F(y, ewz)), (8.10)


a.e. dY(dy) ds du dw dz. Since we suppose that (7.3) does not hold forn ] m, we must have f0(y, s, u)=y a.e. ds du. Indeed, if f0(y, s, u) ] y a.e.ds du, then, by fixing s and u such that f0(n, s, u)=m and n ] m, it followsfrom (8.10) that

a(n, m)(F(m, b(n, m) ew(t+z)+c(n, m))−F(m, b(n, m) ewz+c(n, m)))

=F(n, ew(t+z))−F(n, ewz)

a.e. dw dz, for some a(n, m) ] 0, b(n, m) > 0 and c(n, m) ¥ R. By makingthe change of variables ew(t+z)=u, ewz=v and then fixing v, one obtainsthat (7.3) holds for n ] m, which contradicts minimality. Hence, f0(y, s, u)=y a.e. ds du, and it follows from (8.10) that, for all y ¥ Y and t ¥ R,

e−o(f1(y, s, u)−s)e−ow(F(y, ef1(y, s, u)−sew(t+z)+ef1(y, s, u)(f2(y, s, u)−u))

−F(y, ef1(y, s, u)−sewz+ef1(y, s, u)(f2(y, s, u)−u)))

=k(y, s, u) e−ow(F(y, ew(t+z))−F(y, ewz)) (8.11)

a.e. ds du dw dz. Arguing as in the case (i), one can conclude thatf1(y, s, u)=s and f2(y, s, u)=u a.e. ds du and hence that the condition(M2Œ) of minimality holds as well. i

Proof of Theorem 7.2. By the discussion preceding Theorem 7.2, wehave that, for n ¥ Cp,

F(n, z)=a(n, np) F(np, b(n, np) z+c(n, np))+d(n, np)

a.e. dz. It follows that, for n ¥ Cp and t ¥ R,

F(n, e s(t+u))−F(n, e su)

=a(n, np)(F(np, b(n, np) e s(t+u)+c(n, np))−F(np, b(n, np) e su+c(n, np)))

a.e. ds du or that

e−os(F(n, e s(t+u))−F(n, e su))

=a(n, np) b(n, np)o e−o(s+ln b(n, np))

· (F(np, e s+ln b(n, np)(t+u+e−(s+ln b(n, np))c(n, np))

−F(np, e s+ln b(n, np)(u+e−(s+ln b(n, np))c(n, np))))

a.e. ds du. By making the change of variables

s+ln b(n, np)=s


and

u+e−(s+ln b(n, np))c(n, np)=u

in the second integral below, we obtain that, for all hj, tj ¥ R, j=1,..., n,

FRFRe−oas : C

n

j=1hj(F(n, e s(tj+u))−F(n, e su)) :

a

ds du

=|a(n, np)|a |b(n, np)|ao

×FRFRe−oas : C

n

j=1hj(F(np, e s(tj+u))−F(np, e su)) :

a

ds du.

This implies that

Cn ¥ Y

FRFRe−oas : C

n

j=1hj(F(n, e s(tj+u))−F(n, e su)) :

a

ds du

=Cp ¥ P

3 Cn ¥ Cp

|a(n, np)|a |b(n, np)|ao4

×FRFRe−oas : C

n

j=1hj(F(np, e s(tj+u))−F(np, e su)) :

a

ds du,

which yields the representation (7.5). The representation (7.5) is minimal byTheorem 7.1 and the construction of equivalence classes Cp because, whenp1, p2 ¥ P and p1 ] p2, the kernels F(np1 , z) and F(np2 , z) cannot be relatedas in (7.3). i

Proof of Theorem 7.3. For part (i), we may suppose without loss ofgenerality that |Y| > |Y2 |. If Xa and X2a are essentially identical, then byTheorem 6.1, there are n1, n2 ¥ Y and m ¥ Y2 such that n1 ] n2 and

F(ni, z)=h(ni) F2(m, ea(ni)z+b(ni))+c(ni) (8.12)

a.e. dz, for i=1, 2. By expressing F2 in terms of F(n1, z) and F(n2, z), weget

h(n1)−1 F(n1, e−a(n1)(z−b(n1)))−h(n1)−1 c(n1)

=h(n2)−1 F(n2, e−a(n2)(z−b(n2)))−h(n2)−1 c(n2)


and

F(n1, z)=h(n1) h(n2)−1 F(n2, ea(n1)−a(n2)z+e−a(n2)(b(n1)−b(n2)))

+c(n1)−h(n1) h(n2)−1 c(n2)

a.e. dz. Hence F(n1, z) and F(n2, z) are related as in (7.3). ByTheorem 7.1(ii), this contradicts minimality.Consider now part (ii). We first prove the sufficiency. If there is a one-

to-one and onto map p: YQ Y2 such that qcondition-compare-minimal-finite holds with a(n) b(n)o ¥ {−k, k}, k ] 0, we obtain that

Xa(t)=d FYFRFRe−os(F(y, e s(t+u))−F(y, e su)) Ma(dy, ds, du)

=FYFRFRa(y) e−os(F2(p(y), b(y) e s(t+u)+c(y))

−F2(p(y), b(y) e su+c(y))) Ma(dy, ds, du)

=d |k| FYFRFRe−o(s+ln b(y))(F2(p(y), e s+ln b(y)(t+u+e−s− ln b(y)c(y)))

−F2(p(y), e s+ln b(y)(u+e−s− ln b(y)c(y)))) Ma(dy, ds, du)

=d |k| FY2FRFRe−ow(F2(y, ew(t+z))−F2(y, ewz)) Ma(dy, dw, dz)

=d X2a(t),

where in the step before last, we used the fact that Ma is symmetric andhence

a(y) b(y)oMa(dy, ds, du)=d |k| Ma(dy, ds, du).

In the last step, we made the change of variables p(y)=y, s+ln b(y)=w,and u+e−s− ln b(y)c(y)=z.We now turn to the necessity in part (ii) of the theorem. Suppose

without loss of generality that the processes Xa and X2a are identical indistribution. We will then have to show in particular that a(n) b(n)o ¥{−1, 1}. To simplify the notation, denote the kernels of Xa and X2a by

Ft(y, s, u)=e−os(F(y, e s(t+u))−F(y, e su)),

F2t(y, s, u)=e−os(F2(y, e s(t+u))−F2(y, e su)),(8.13)


respectively. By part (i), we have |Y|=|Y2 |. Since the representations of Xaand X2a are minimal, by using Theorem 2.2 in Rosinski, (9) there are twomaps, a one-to-one and onto non-singular map (with a non-singularinverse),

F(y, s, u)=(f0(y, s, u), f1(y, s, u), f2(y, s, u)): Y×R×RQ Y2 ×R×R

and a map E(y, s, u): Y×R×RQ {−1, 1}, such that

Ft(y, s, u)=E(y, s, u) 3d(dY2 ×L×L) p Fd(dY×L×L)

(y, s, u)41/a

F2t(F(y, s, u))(8.14)

a.e. dY(dy) ds du, where d is a counting measure and L stands for theLebesgue measure on R. The term in braces corresponds to the Jacobian ofthe transformation. Setting

k(y, s, u)−1=E(y, s, u) 3d(dY2 ×L×L) p Fd(dY×L×L)

(y, s, u)41/a

,

we get

F2t(F(y, s, u))=k(y, s, u) Ft(y, s, u)

a.e. dY(dy) ds du. Taking into account the notation (8.13), this relation isidentical to (8.9) with F in the left-hand side of (8.9) replaced by F2 . Thus,arguing as in (8.9) and (8.10) of the proof of Theorem 7.1, we get (8.10),which, rewritten in our context, becomes: for all t ¥ R,

Ft(y, w, z)

=k(y, s, u)−1 F2t(f0(y, s, u), w+f1(y, s, u)−s, z+e s−w(f2(y, s, u)−u))

a.e. dY(dy) dw dz ds du. By fixing s=s0 and u=u0 in the last relation, weobtain that, for all t ¥ R,

Ft(y, w, z)=h(y) F2t(F2(y, w, z)) (8.15)

a.e. dY(dy) dw dz, where h is some function and

F2(y, w, z)=(p(y), w+k1(y), z+e−wk2(y)) (8.16)

with some functions p: YW Y2 , k1, k2: YW R. By comparing (8.14) and(8.15), we obtain that

F2t(y, w, z)=g(y, w, z) F2t(F2(F−1(y, w, z))) (8.17)


a.e. dY2 (dy) dw dz, for some function g ] 0 a.e. Since F2 and F−1 are non-singular, the map F2 p F−1 is non-singular as well. Then, the minimality ofthe representation {F2t} (see (7.1)) and relation (8.17) imply that F2 p F−1 isan identity map a.e. or that

F2(y, w, z)=F(y, w, z) (8.18)

a.e. dY(dy) dw dz. By (8.18), since the map F is one-to-one and onto, themap p: YQ Y2 is one-to-one and onto as well. Then, by using (8.18) and(8.16), we have

d(dY2 ×L×L) p Fd(dY×L×L)

(y, s, u)=d(dY2 ×L×L) p F2

d(dY×L×L)(y, s, u)=1 (8.19)

a.e. dY(dy) ds du. To verify the second equality in (8.19), observe first that,for a fixed y ¥ Y, the map

(s, u)W (s+k1(y), u+e−sk2(y))

preserves the measure L×L since its Jacobian is 1. Moreover, the first entryin (8.16) does not involve the variables w, z and the map p: YW Y2 is one-to-one and onto (for example, if Y=Y2 finite, then p is a permutation). Ittherefore does not affect the counting measure and its contributions to theoverall Jacobian is 1.By using (8.18) and (8.19), and by comparing (8.14) and (8.15), we

conclude that h(y)=E(y, s, u) a.e. dY(dy) ds du or that h(y) ¥ {−1, 1} forall y ¥ Y, since E(y, s, u) ¥ {−1, 1}. Hence, relation (8.15) can be rewrittenas

e−ow(F(y, ew(t+z))−F(y, ewz))

=h(y) e−o(w+k1(y))(F2(p(y), ew+k1(y)(t+z+e−wk2(y)))

−F2(p(y), ew+k1(y)(z+e−wk2(y))))

a.e. dY(dy) dw dz, where h(y) ¥ {−1, 1} and p: YW Y2 is a one-to-one andonto map, and hence as

F(y, ew(t+z))−F(y, ewz)

=a(y)(F2(p(y), b(y) ew(t+z)+c(y))−F2(p(y), b(y) ewz+c(y)))(8.20)


a.e. dY(dy) dw dz, where

a(y)=h(y) e−ok1(y),

b(y)=ek1(y),

c(y)=ek1(y)k2(y).

By making the change of variables ew(t+z)=v, ewz=u in (8.20), we get

F(y, v)=a(y) F2(p(y), b(y) v+c(y))−a(y) F2(p(y), b(y) u+c(y))+F(y, u)

a.e. dY(dy) dv du. Then, by fixing u=u0, we obtain that, for all y ¥ Y,

F(y, v)=a(y) F2(p(y), b(y) v+c(y))+d(y)

a.e. dv, where d(y) ¥ R. This proves the necessity in part (ii) becausep: YW Y2 is one-to-one and onto, and

a(y) b(y)o=(h(y) e−ok1(y))(ek1(y))o=h(y) ¥ {−1, 1}. i

ACKNOWLEDGMENTS

This research was partially supported by the NSF Grants DMS-0102410 and ANI-9805623 at Boston University.

REFERENCES

1. Adams, R. A. (1975). Sobolev Spaces, Academic Press, New York.2. Burnecki, K., Rosinski, J., and Weron, A. (1998). Spectral representation and structure ofstable self-similar processes. In Karatzas, I., Rajput, B. S., and Taqqu, M. S. (eds.),Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943–1995,Trends in Mathematics, Birkhäuser, pp. 1–14.

3. Chi, Z. (2001). Construction of stationary self-similar generalized fields by randomwavelet expansion. Probab. Theory Related Fields 121(2), 269–300.

4. Hardin, Jr., C. D. (1982). On the spectral representation of symmetric stable processes.J. Multivariate Anal. 12, 385–401.

5. Mori, T., and Sato, Y. (1996). Construction and asymptotic behavior of a class of self-similar stable processes. Japanese J. Math. 22(2), 275–283.

6. Pipiras, V., and Taqqu, M. S. (2000). The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6(4), 607–614.

7. Pipiras, V., and Taqqu, M. S. (2002). Decomposition of self-similar stable mixed movingaverages. Probab. Theory Related Fields 123(3), 412–452.

8. Pipiras, V., and Taqqu, M. S. (2002). The structure of self-similar stable mixed movingaverages. Ann. Probab. 30(2), 898–932.

9. Rosinski, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23(3),1163–1187.


10. Rosinski, J. (1998). Minimal integral representations of stable processes. Preprint.11. Runst, T., and Sickel, W. (1996). Sobolev Spaces of Fractional Order, Nemytskij Opera-tors, and Nonlinear Partial Differential Equations, Walter de Gruyter & Co., Berlin.

12. Samorodnitsky, G., and Taqqu, M. S. (1990). (1/a)-self-similar processes with stationaryincrements. J. Multivariate Anal. 35, 308–313.

13. Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Processes: StochasticModels with Infinite Variance, Chapman & Hall, New York, London.

14. Surgailis, D., Rosinski, J., Mandrekar, V., and Cambanis, S. (1998). On the mixing struc-ture of stationary increment and self-similar SaS processes. Preprint.

15. Takenaka, S. (1991). Integral-geometric construction of self-similar stable processes.Nagoya Math. J. 123, 1–12.

16. Triebel, H. (1983). Theory of Function Spaces, Birkhäuser Verlag, Basel.


Dilated Fractional Stable Motions

Documents

Transcript of Dilated Fractional Stable Motions