SELF-SIMILAR RANDOM FRACTAL MEASURES USING CONTRACTION METHOD IN PROBABILISTIC

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SELF-SIMILAR RANDOM FRACTAL MEASURES USING CONTRACTION METHOD IN PROBABILISTIC

METRIC SPACES

JÓZSEF KOLUMBÁN, ANNA SOÓS, and IBOLYA VARGA Received 2 January 2003

Self-similar random fractal measures were studied by Hutchinson and Rüschen- dorf. Working with probability metric in complete metric spaces, they need the first moment condition for the existence and uniqueness of these measures. In this paper, we use contraction method in probabilistic metric spaces to prove the existence and uniqueness of self-similar random fractal measures replacing the first moment condition.

2000 Mathematics Subject Classification: 60G57, 28A80.

1. Introduction. Contraction methods for proving the existence and unique- ness of nonrandom self-similar fractal sets and measures were first applied by Hutchinson [7]. Further results and applications to image compression were obtained by Barnsley and Demko [3] and Barnsley [2]. At the same time, Fal- coner [5], Graf [6], and Mauldin and Williams [13] randomized each step in the approximation process to obtain self-similar random fractal sets. Arbeiter [1] and Olsen [15] studied self-similar random fractal measures applying non- random metrics. More recently, Hutchinson and Rüschendorf [8,9,10] intro- duced probability metrics defined by expectation for random measure and es- tablished existence, uniqueness, and approximation properties of self-similar random fractal measures. In these works a finite first moment condition is essential.

In this paper, we show that, using probabilistic metric spaces techniques, we can weaken the first moment condition for the existence and uniqueness of self-similar measures.

The theory of probabilistic metric spaces, introduced in 1942 by Menger [14], was developed by numerous authors, as it can be realized upon consult- ing [4,18] and the references therein. The study of contraction mappings for probabilistic metric spaces was initiated by Sehgal [19] and Sherwood [20].

2. Self-similar random fractal measures. Based on contraction properties of random scaling operators with respect tol^∗_p andl^∗∗_p , for 0< p <∞, on a space of random measures and their distributions, respectively, defined below, Hutchinson and Rüschendorf [8,9,10] gave a simple proof for the existence and uniqueness of invariant random measures. The underlying probability

space for the iteration procedure is also generated by selecting independent and identically distributed (i.i.d.) scaling laws for measures.

Let(X,d)be a complete separable metric space.

Definition2.1. Ascaling law with weightsis a 2N-tuple S:=

p1,S1,...,pN,SN

, N≥1, (2.1)

of positive real numberspi such that_N

i=1pi=1 and of Lipschitz mapsSi: X→X.

Let ri =LipSi, i∈ {1,...,N}. Denote by M=M(X)the set of finite-mass Radon measures onXwith weak topology. Ifµ∈M, then the measureSµ is defined by

Sµ= N i=1

piSiµ, (2.2)

whereSiµis the usual push-forward measure, that is, Siµ(A)=µ

S_i⁻¹(A)

forA⊆X. (2.3)

Definition2.2. The measureµsatisfies the scaling lawSoris a self-similar fractal measureifSµ=µ.

LetMqdenote the set of unit mass Radon measuresµonXwith finiteqth moment; that is,

Mq=

µ∈M|µ(X)=1,

Xd^q(x,a)dµ(x) <∞

, (2.4)

for some (and hence any)a∈X. Note that, ifp≥q, thenMp⊂Mq. Definition2.3. Theminimal metriclqonMqis defined by

lq(µ,ν)=inf

Xd^q(x,y)dγ(x,y) 1/q∧1

|π1γ=µ, π2γ=ν

, (2.5)

where∧denotes the minimum of the relevant numbers andπiγdenotes the ith marginal ofγ, that is, projection of the measureγ onX×Xonto theith component.

Thelqmetric has the following properties (see [16]).

(a) Supposeαis a positive real,S:X→X is Lipschitz, and∨ denotes the maximum of the relevant numbers. Then, forq >0 and for measuresµandν,

...

we have the following properties:

l^q∨q ¹(αµ,αν)=αl^q∨q ¹(µ,ν), (2.6) l^q∨1q

µ1+µ2,ν1+ν2

≤l^q∨1q µ1,ν1

+l^q∨1q µ2,ν2

, (2.7)

lq(Sµ,Sν)≤(LipS)^q∧1lq(µ,ν). (2.8) The first property follows from the definition by settingγ =cγ, whereγ is optimal for(µ,ν), and the third follows by settingγ=Sγ. The second follows by settingγ=γ1+γ2, whereγiis optimal for(µi,νi), and also by noting that (a+b)^q≤a^q+b^qifa,b≥0 and 0< q <1.

(b) The pair(Mq,lq)is a complete separable metric space andlq(µn,µ)→0 if and only if

(i) µn→µ(weak convergence), (ii)

Xd^q(x,a)dµn(x)→

d^q(x,a)dµ(x)(convergence ofqth moments).

(c) Ifδa is the Dirac measure ata∈X, then

lq

µ,µ(X)δa

=

Xd^q(x,a)dµ(x) _1/q∧1

, lq

δa,δb

=d^1∧q(a,b).

(2.9)

Let Mdenote the set of all random measures µ with value inM, that is, random variables µ:Ω→M. LetM_q denote the space of random measures µ:Ω→Mqwith finite expectedqth moment. That is,

M_q:=

µ∈M|µ^ω(X)=1a.s., Eω

Xd^q(x,a)dµ^ω(x) <∞

. (2.10)

The notationEω indicates that the expectation is with respect to the vari- ableω. It follows from (2.10) thatµ^ω∈Mqa.s. Note thatM_p⊂M_qifq≤p. Moreover, sinceE^1/q|f|^q→exp(Elog|f|)asq→0,

M0:= ∪q>0M_q=

µ∈M|µω(X)=1a.s., Eω

X

logd(x,a)dµ^ω(x) <∞

. (2.11) For random measuresµ,ν∈M_q, define

l^∗q(µ,ν):=





 Eω^1/ql^qq

µ^ω,ν^ω

, q≥1, Eωlq

µ^ω,ν^ω

, 0< q <1. (2.12) One can check, as in [16], that(M_q,l^∗_q)is a complete separable metric space.

Note thatl^∗q(µ,ν)=lq(µ,ν)ifµandνare constant random measures.

Letᏹdenote the class of probability distributions onM, that is,

ᏹ= {Ᏸ=distµ|µ∈M}. (2.13) Letᏹqbe the set of probability distributions of random measuresµ∈M_q. For q≤p, it is to be noticed thatᏹp⊂ᏹq. Let

ᏹ0:= ∪q>0ᏹq. (2.14)

The minimal metric onᏹqis defined by l^∗∗_q

Ᏸ1,Ᏸ2

=inf

l^∗_q(µ,ν)|µ=^dD1, ν=^dD2

. (2.15)

It follows that(ᏹq,l^∗∗_q ) is a complete separable metric space with the next properties (see [16]):

(a) l^∗∗q (αᏰ1,αᏰ2)=αl^∗∗q (Ᏸ1,Ᏸ2), (b) l^∗∗q (Ᏸ1+Ᏸ2,Ᏸ3+Ᏸ4)≤l^∗∗q q

(Ᏸ1,Ᏸ3)+l^∗∗q q

(Ᏸ2,Ᏸ4), forᏰi∈ᏹq,i=1,2,3,4.

Definition2.4. A random scaling law with weightsor arandom scaling law for measureS=(p1,S1,p2,S2,...,pN,SN)is a random variable whose values are scaling laws, withN

i=1pi=1 a.s.

We write᏿=distSfor the probability distribution determined byS. Ifµ is a random measure, then the random measure Sµ is defined (up to probability distribution) by

Sµ:= N i=1

piSiµ⁽ⁱ⁾, (2.16)

whereS, µ⁽¹⁾,...,µ^(N) are independent of one another, and µ^{(i) d}=µ. If Ᏸ= distµ, we define᏿Ᏸ=distSµ.

Definition2.5. The measureµsatisfies the scaling lawSor is aself-similar random fractal measureifSµ=^dµ, or equivalently᏿Ᏸ=Ᏸ, whereᏰis called a self-similar random fractal distribution.

To generate a random self-similar fractal measure, we use theiterative pro- ceduredescribed as follows. Fixq >0. Beginning with a nonrandom measure µ0∈Mq (or, more generally, a random measureµ0∈M_q), one iteratively ap- plies i.i.d. scaling laws with distribution᏿to obtain a sequenceµnof random measures inM_qand a corresponding sequence Ᏸn of distributions inᏹqas follows.

(i) Select a scaling lawSvia the distribution᏿and define µ1=Sµ0=

n i=1

piSiµ0, (2.17)

...

that is,

µ1(ω)=Sµ0= n i=1

pi(ω)Si(ω)µ0, Ᏸ1d

=µ1. (2.18)

(ii) SelectS¹,...,S^N via᏿^with Sⁱ=(pⁱ1,S1ⁱ,...,pⁱN,SNⁱ), i∈ {1,2,...,N}, in- dependent of each other and ofS, and define

µ2:=S²µ0=

i,j

pip_jⁱSi◦S_jⁱµ0, Ᏸ2d

=µ0. (2.19)

(iii) SelectS^ij=(pⁱ1,S1^ij,...,pNⁱ,SN^i,j)via᏿, independent of one another and ofS¹,...,S^N,S, and define

µ3=S³µ0=

i,j,k

pip_jⁱp_k^ijSi◦S_jⁱ◦S_k^ijµ0, Ᏸ3d

=µ3, (2.20)

and so forth.

Thusµn+1=_N

i=1piSiµ⁽ⁱ⁾n , whereµn⁽ⁱ⁾ d

=µn d

=Ᏸn, S=^d᏿^{, and}µ⁽ⁱ⁾n andSare independent. It follows thatᏰn=᏿ᏰN−1=᏿ⁿᏰ0, whereᏰ0is the distribution ofµ0. In the caseµ0∈Mq,Ᏸ0is constant.

In the following, we define the underlying probability space for a.s. conver- gence (see [10]).

Aconstruction tree(or a construction process) is a mapω:{1,...,N}^∗→Γ, whereΓ is the set of (nonrandom) scaling laws. A construction tree specifies, at each node of the scaling law used for constructive definition, a recursive sequence of random measures. Denote the scaling law ofωat the nodeσ by the 2N-tuple

S^σ(ω)=ω(σ )=

p^σ1(ω),S1^σ(ω),...,pN^σ(ω),SN^σ(ω)

, (2.21) wherepi^σ are weights andSi^σ Lipschitz maps. The sample space of all con- struction trees is denoted by ˜Ω. The underlying probability space(Ω,˜ ᏷˜,P )˜ for the iteration procedure is generated by selecting i.i.d. scaling lawsω(σ )=^d S for eachσ∈ {1,...,N}^∗. We use the notation

p^σ=pσ1p^σσ¹2pσ^σ3^1σ2···pσ^σ1n^{···σn−1},

S^σ=Sσ1Sσ^σ2¹pσ^σ3^1σ2···Sσ^σn^{1···σn−1}, (2.22) where|σ| =nand wherep^σ_i andS_i^σdenote theith components of scaling law.

For a fixed measureµ0∈Mq, define µn=µn(ω)=

|σ|=n

p^σ(ω)S^σ(ω)µ0 (2.23)

forn≥1. This is identical to the sequence defined in an iterative procedure with an underlying spaceΩ=Ω. To see this, for ω∈Ωand 1≤i≤N, let ω⁽ⁱ⁾∈Ωbe defined by

ω⁽ⁱ⁾(σ )=ω(i∗σ ) (2.24)

forσ∈ {1,...,N}^∗. Then

p^i∗σ=pi(ω)p^σ ω⁽ⁱ⁾

, S^i∗σ=Si(ω)◦p^σ

ω⁽ⁱ⁾

. (2.25)

By construction,ω⁽ⁱ⁾are i.i.d. with the same distribution asω, and are inde- pendent of(p¹(ω),S¹(ω),...,pN(ω),SN(ω)). More precisely, for anyP mea- surable setsE,F⊂ΩandB⊂Γ,

P

{ω|ω∈E}

=P

ω|ω⁽ⁱ⁾∈E

, (2.26)

where{ω|ω⁽ⁱ⁾∈E}and {ω|ω^(j)∈E}are independent ifi=j, and{ω| (p1ω,S1(ω),...,pN(ω),SN(ω))∈B} and {ω|ω⁽ⁱ⁾∈E} are independent. It follows that

µn+1(ω)= N i=1

|σ|=n

p^i∗σS^i∗σ(ω)µ0= N i=1

pi(ω)Si(ω)µn

ω⁽ⁱ⁾

=Sµn(ω).

(2.27) In [8], Hutchinson and Rüschendorf proved the following theorem.

Theorem2.6. LetS=(p1,S1,p2S2,...,pN,SN)be a random scaling law with _N

i=1pi=1a.s. Assumeλq:=Eω(_N

i=1pir_i^q) <1and Eω



^N

i=1

pid^q

Sia,a

<∞ for someq >0,and fora∈Y . (2.28)

Then the following facts hold.

(a)The operatorS:Mq→Mqis a contraction map with respect tol^∗q. (b)There exists a self-similar random measureµ^∗.

(c)Ifµ0∈Mp(or, more generally,M_q), then Eω^1/ql^qq

µk,µ^∗

≤ λ^k/qq

1−λ^1/qq

E^1/qω l^qq µ1,Sµ0

→0, q≥1,

Eωlq µk,µ^∗

≤ λ^kq

1−λqEωlq µ¹,Sµ⁰

→0, 0< q <1,

(2.29)

ask→ ∞. In particularµn→µ^∗a.s. in the sense of weak convergence of mea- sures.

Moreover, up to probability distribution,µ^∗is the unique unit mass random measure withEω

lnd(x,a)dµ^ω<∞, which satisfiesS.

...

Using contraction method in probabilistic metric spaces, instead of condi- tion (2.28), we can give a weaker condition for the existence and uniqueness of invariant measure. More precisely, we prove the following theorem.

Theorem 2.7. LetS= (p1,S1,p2,S2,...,pN,SN) be a random scaling law which satisfies_N

i=1pi=1a.s., and supposeλq:=ess sup(_N

i=1pir_i^q) <1for someq >0. If there existα∈Mqand a positive numberγ such that

P

ω∈Ω|lq

α(ω),Sα(ω)

≥t

≤γ

t ∀t >0, (2.30) then there existsµ^∗such thatSµ^∗=µ^∗a.s.

Moreover, up to probability distribution,µ^∗is the unique unit mass random measure which satisfiesS.

Remark2.8. If condition (2.28) is satisfied, then condition (2.30) also holds.

To see this, leta∈Xandα(ω):=δafor allω∈Ω. We have P

ω∈Ω|lq δa(ω

,Sδa(ω)

≥t

=P







ω∈Ω|lq



^N

i=1

piδa(ω), N i=1

piSiδa(ω)



≥t









≤P







ω∈Ω| N i=1

pilq

δa(ω),Siδa(ω)

≥t









=P







ω∈Ω| N i=1

pid^q Sia,a

≥t









≤1 tEω



^N

i=1

pid^q

Sia,a

=γ t.

(2.31)

However, condition (2.30) can also be satisfied if

Eω



^N

i=1

pid^q

Sia,a

= ∞ ∀q >0. (2.32)

LetΩ=]0,1]with the Lebesque measure, letXbe the interval[0,∞[, and let N=1. DefineS:X→XbyS^ω(x)=x/2+e^1/ω. This map is a contraction with ratio 1/2. Forq >0, the expectationEωd^q(S0,0)= ∞, however

P

ω∈Ω|lq(S0,0)≥t

=1

t (2.33)

for allt >0.

3. Invariant sets inE-spaces

3.1. Menger spaces. LetRdenote the set of real numbers andR+:= {x∈ R:x≥0}. A mapping F :R→[0,1] is called a distribution functionif it is

nondecreasing, left continuous with inf_t∈RF(t)=0 and sup_t∈RF(t)=1 (see [4]). By∆we will denote the set of all distribution functionsF. Let∆be ordered by the relation “≤”, that is,F≤Gif and only ifF(t)≤G(t)for all realt. Also F < Gif and only ifF≤GbutF=G. We set∆⁺:= {F∈∆:F(0)=0}.

Throughout this paper, H will denote the heaviside distribution function defined by

H(x)=





0, x≤0,

1, x >0. (3.1)

LetXbe a nonempty set. For a mappingᏲ:X×X→∆⁺ andx,y∈X, we will denoteᏲ(x,y)byFx,y, and the value ofFx,yatt∈RbyFx,y(t), respec- tively. The pair (X,Ᏺ) is a probabilistic metric space(briefly PM space) ifX is a nonempty set andᏲ:X×X→∆⁺ is a mapping satisfying the following conditions:

(1) Fx,y(t)=Fy,x(t)for allx,y∈Xandt∈R; (2) Fx,y(t)=1, for everyt >0, if and only ifx=y; (3) ifFx,y(s)=1 andFy,z(t)=1, thenFx,z(s+t)=1.

A mappingT :[0,1]×[0,1]→[0,1]is called at-normif the following con- ditions are satisfied:

(4) T (a,1)=afor everya∈[0,1]; (5) T (a,b)=T (b,a)for everya,b∈[0,1]; (6) ifa≥candb≥d, thenT (a,b)≥T (c,d);

(7) T (a,T (b,c))=T (T (a,b),c)for everya,b,c∈[0,1].

AMenger spaceis a triplet(X,Ᏺ,T ), where(X,Ᏺ)is a PM space,T is at- norm, and instead of condition (3), we have the stronger condition

(8) Fx,y(s+t)≥T (Fx,z(s),Fz,y(t))for allx,y,z∈Xands,t∈R+.

The(t, )-topology in a Menger space was introduced in 1960 by Schweizer and Sklar [17]. The base for the neighbourhoods of an elementx∈Xis given by

Ux(t, )⊆X:t >0, ∈]0,1[

, (3.2)

where

Ux(t, ):=

y∈X:Fx,y(t) >1−

. (3.3)

In 1969, Sehgal [19] introduced the notion of a contraction mapping in PM spaces. The mappingf:X→Xis said to be acontractionif there existsr∈ ]0,1[such that

Ff (x),f (y)(r t)≥Fx,y(t) (3.4) for everyx,y∈Xandt∈R+.

...

A sequence(xn)n∈NfromXis said to befundamentalif

n,m→∞lim Fx_m,x_n(t)=1 (3.5)

for allt >0. The elementx∈Xis calledlimit of the sequence(xn)n∈N, and we write lim_n→∞xn=x orxn→x if lim_n→∞Fx,x_n(t)=1 for allt >0. A PM (Menger) space is said to becompleteif every fundamental sequence in that space is convergent.

LetAandBbe nonempty subsets ofX. Theprobabilistic Hausdorff-Pompeiu distancebetweenAandBis the functionFA,B:R→[0,1]defined by

FA,B(t):=sup

s<tT

inf

x∈A

sup

y∈BFx,y(s),inf

y∈B

sup

x∈AFx,y(s)

. (3.6)

In the following, we recall some properties proved in [11,12].

Proposition3.1. If Ꮿis a nonempty collection of nonempty closed bounded sets in a Menger space(X,Ᏺ,T )withTcontinuous, then(Ꮿ,ᏲᏯ,T )is also Menger space, whereᏲᏯis defined byᏲᏯ(A,B):=FA,Bfor allA,B∈Ꮿ^.

Proposition3.2. LetTm(a,b):=max{a+b−1,0}. If(X,Ᏺ,Tm)is a com- plete Menger space andᏯis the collection of all nonempty closed bounded sub- sets ofXin(t, )-topology, then(Ꮿ,ᏲᏯ,Tm)is also a complete Menger space.

3.2.E-spaces. The notion ofE-space was introduced by Sherwood [20] in 1969. Next we recall this definition. Let(Ω,᏷,P)be a probability space and let (Y ,ρ)be a metric space. The ordered pair(Ᏹ,Ᏺ)is anE-space over the metric space(Y ,ρ)(briefly, anE-space) if the elements ofᏱare random variables from ΩintoY andᏲis the mapping fromᏱ×Ᏹ^into∆⁺defined viaᏲ(x,y)=Fx,y, where

Fx,y(t)=P

ω∈Ω|d

x(ω),y(ω)

< t

(3.7) for everyt∈R. Usually(Ω,᏷,P)is called the base and(Y ,ρ)the target space of theE-space. IfᏲsatisfies the condition

Ᏺ(x,y)=H forx=y, (3.8)

with H defined in Section 3.1, then (Ᏹ,Ᏺ)is said to be a canonicalE-space.

Sherwood [20] proved that every canonicalE-space is a Menger space under T=Tm, whereTm(a,b)=max{a+b−1,0}. In the following, we suppose that Ᏹis a canonicalE-space.

The convergence in anE-space is exactly the probability convergence. The E-space(Ᏹ,Ᏺ)is said to be complete if the Menger space(Ᏹ,Ᏺ,Tm)is complete.

Proposition 3.3. If (Y ,ρ) is a complete metric space, then the E-space (Ᏹ,Ᏺ)is also complete.

Proof. This property is well known for Y = R (see, e.g., [21, Theorem VII.4.2]). In the general case, the proof is analogous.

Let(xn)n∈Nbe a Cauchy sequence of elements ofᏱ^{, that is,}

n,m→∞lim Fx_n,x_n+m(t)=1 ∀t >0. (3.9) First we show that there exists a subsequence(xn_k)k∈Nof the given sequence which is convergent almost everywhere to a random variablex. We set positive numbers _iso that_∞

i=1 i<∞and putδp=_∞

i=p i, p=1,2,....For eachi, there is a natural numberkisuch that

P

ω∈Ω|ρ

xk(ω),xl(ω)

≥ i

< i fork,l≥ki. (3.10) We can assume thatk1< k2<···< ki<···. Then

P

ω∈Ω|ρ

xk_i+1(ω),xk_i(ω)

≥ i

< i fork,l≥ki. (3.11)

We put

Dp= ∪^∞_i=p

ω∈Ω|ρ

xk_i+1,xk_i

≥ i

. (3.12)

ThenP (Dp) < δp. Finally, for the intersectionD= ∩^∞_p=1Dp, we obviously haveP (D)=0 sinceδp→0. We will show that the sequence(xk_i(ω))has a finite limitx(ω)at every pointω∈ {ω∈Ω|ρ(xk(ω),xm(ω)) > t} \D. For somepwe havex∉Dp. Consequently,ρ(xk_i+1(ω),xk_i(ω)) < i, for alli≥p. It follows that for any two indicesiandjsuch thatj > i≥p, we have

ρ

xk_j(ω),xk_i(ω)

≤

j−1 m=i

ρ

xkm+1(ω),xkm(ω)

<

j−1 m=i

m<

∞ m=i

m=δi. (3.13) Thus lim_i,j→∞ρ(xk_j(ω),xk_i(ω))=0. This means that(xk(ω))k∈Nis a Cauchy sequence for everyωwhich implies the pointwise convergence of(xk_i)i∈Nto a finite-limit function. Now remains only to put

x(ω)=





limxk_i(ω) forω∉D,

0 forω∈D (3.14)

to obtain the desired limit random variable. By Lebesque theorem (see, e.g., [21, Theorem VI.5.2]),xk_i→xwith respect tod. Thus, every Cauchy sequence inᏱhas a limit, which means that the spaceᏱis complete.

The next result was proved in [12].

...

Theorem3.4. Let(Ᏹ,Ᏺ)be a completeE-space,N∈N^∗, and letf1,...,fN: Ᏹ→Ᏹbe contractions with ratiosr1,...,rN, respectively. Suppose that there exist an elementz∈Ᏹand a real numberγsuch that

P

ω∈Ω|ρ

z(ω),fi z(ω)

≥t

≤γ

t (3.15)

for alli∈ {1,...,N}and for all t >0. Then there exists a unique nonempty closed bounded and compact subsetKof Ᏹsuch that

f1(K)∪···∪fN(K)=K. (3.16) Corollary 3.5. Let (Ᏹ,Ᏺ) be a completeE-space and letf :Ᏹ→Ᏹ^{be a} contraction with ratior. Suppose there existz∈Ᏹand a real numberγ such that

P

ω∈Ω|ρ

z(ω),f (z)(ω)

≥t

≤γ

t ∀t >0. (3.17) Then there exists a uniquex⁰∈Ᏹsuch thatf (x⁰)=x⁰.

4. Proof ofTheorem 2.7. Before the proof of the theorem, we give two lem- mas.

LetᏱqbe the set of random variables with values inMqand letᏱq(α)be the set

Ᏹq(α):=

β∈Ᏹq| ∃γ >0, P

ω∈Ω|lq

α(ω),β(ω)

≥t

≤γ t ∀t >0

. (4.1) Lemma4.1. For allα∈Mq,M_q⊂Ᏹq(α).

Proof. Forβ∈M_q, we have P

ω∈Ω|lq

α(ω),β(ω)

≥t

=

lq(α(ω),β(ω))≥tdP≤1 t

Ωlq

α(ω),β(ω) dP=1

tEωlq

α(ω),β(ω) . (4.2) Sinceβ∈ᏹq, we haveγ=Eωlq(α(ω),β(ω)) <∞for allt >0.

Lemma4.2. The pair(Ᏹq,Ᏺ)is a completeE-space.

Proof. The lemma follows by choosingY:=ᏱqandᏲµ,ν(t):=P ({ω∈Ω| lq(µ(ω),ν(ω)) < t})inProposition 3.3.

Proof ofTheorem2.7. LetSbe a random scaling law. Definef:Ᏹq→Ᏹq

byf (µ)=Sµ, that is,

Sµ(ω)=

i

p_i^ωS_i^ωµ ω⁽ⁱ⁾

. (4.3)

We first claim that ifµ∈Ᏹq, thenSµ∈Ᏹq. For this, choose i.i.d.µ(ω⁽ⁱ⁾)=^dµ(ω) and(p^ω1,S1^ω,...,p^ω_N,S^ω_N)=^d Sindependent ofµ(ω). Forq≥1 andbi=S_i⁻¹(a), using (2.8), we compute

Xd^q(x,a)d

Sµ^ω(x)

=l^qq



^N

i=1

p^ωi Si^ωµ ω⁽ⁱ⁾

,δa





=l^qq



^N

i=1

p^ω_i S_i^ωµ ω⁽ⁱ⁾

, N i=1

p_i^ωS_i^ωδb_i





≤ N i=1

p^ωi ri^ql^qq µ

ω⁽ⁱ⁾ ,δb_i

.

(4.4)

Sinceµ(ω⁽ⁱ⁾)∈Mq, we have

Xd^q(x,a)d Sµ(x)

<∞. (4.5)

We can deal with the case 0< q <1 similarly by replacingl^qqwithlq:

Xd^q(x,a)d

Sµ^ω(x)

=lq



^N

i=1

p^ω_i S_i^ωµ ω⁽ⁱ⁾

,δa





=lq



^N

i=1

p^ω_i S_i^ωµ ω⁽ⁱ⁾

, N i=1

p_i^ωS_i^ωδb_i





≤ N i=1

p^ω_i r_i^qlq

µ ω⁽ⁱ⁾

,δb_i

<∞.

(4.6)

To establish the contraction property, we considerµ,ν∈Ᏹq, µ

ω⁽ⁱ⁾=dµ(ω), ν

ω⁽ⁱ⁾=dν(ω), i∈ {1,2,...,N}, (4.7) andq≥1. We have

Ff (µ),f (ν)(t)=P

ω∈Ω|lq

f µ(ω)

,f ν(ω)

< t

=P







ω∈Ω|lq



^N

i=1

p^ω_i S_i^ωµ ω⁽ⁱ⁾

, N i=1

p_i^ωS_i^ων ω⁽ⁱ⁾

< t









≥P









ω∈Ω|



^N

i=1

p^ω_i ri

_q l^qq

µ ω⁽ⁱ⁾

,ν ω⁽ⁱ⁾



1/q

< t











≥P$

ω∈Ω|% λql^qq

µ(ω),ν(ω)&1/q

< t'

=Fµ,ν

t

λ^1/qq

(4.8) for allt >0.

...

In case 0< q <1, one replacesl^qqeverywhere bylq: Ff (µ),f (ν)(t)=P

ω∈Ω|lq

f µ(ω)

,f ν(ω)

< t

=P







ω∈Ω|lq



^N

i=1

p^ω_i S_i^ωµ ω⁽ⁱ⁾

, N i=1

p_i^ωS_i^ων ω⁽ⁱ⁾

< t









≥P









ω∈Ω|



^N

i=1

p^ωi

ri

q

lq

µ ω⁽ⁱ⁾

,ν ω⁽ⁱ⁾



1/q

< t











≥P

ω∈Ω|% λqlq

µ(ω),ν(ω)&

< t

=Fµ,ν

t λq

(4.9) for all t >0. Thus S is a contraction map with ratio λ^1/q∧1q . We can apply Corollary 3.5forr=λ^1/q∧q ¹. Ifµ^∗is the unique fixed point ofSandµ⁰∈Mq, then

FSⁿµ0,µ^∗(t)=P

ω∈Ω|lq

Sⁿµ0,µ^∗

< t

≥P

ω∈Ω| λ^n/qq

1−λ^1/qq

lq µ⁰,Sµ0

< t

=Fµ0,Sµ0

t

1−λ^1/qq λ^n/qq

,

n→∞limFSⁿµ0,µ^∗(t)=1 ∀t >0.

(4.10)

Fromµn+1(ω)=Sµn(ω), it follows thatµn→µ^∗ exponentially fast. More- over, forq≥1,

∞ i=1

P l^qq

Sⁿν0,µ^∗

≥

≤ ∞ i=1

el^qq

Sⁿµ⁰,µ^∗

≤c ∞ i=1

λⁿq

<∞. (4.11) This implies by Borel-Cantelli lemma thatlq(µn,µ^∗)→0 a.s.

For the uniqueness, letᏰbe the set of probability distribution of members ofᏱq. We define the probability metric onᏰ^by

FᏭ,Ꮾ(t)=sup

s<t

sup

Fµ,ν(s)|µ=^dᏭ, ν=^dᏮ. (4.12) To establish the contraction property of᏿, we considerᏭ,Ꮾ∈Ᏸ. Forq≥1, we get

F᏿Ꮽ,᏿Ꮾ(t)=sup

s<t

sup

FSµ,Sν(s)|µ=^dᏭ, ν=^dᏮ

≥sup

s<t

sup

Fµ,ν

s

λ^1/qq

|µ=^d Ꮽ, ν=^dᏮ

=FᏭ,Ꮾ

t

λ^1/qq

(4.13)

for allt >0. For 0< q <1, the demonstration is similar.

ConsiderᏰ1andᏰ2such that᏿Ᏸ1=Ᏸ1and᏿Ᏸ2=Ᏸ2. SinceᏰ1=᏿ⁿ(Ᏸ1)andᏰ2=᏿ⁿ(Ᏸ2), we have

FᏰ1,Ᏸ2(t)≥FᏰ1,Ᏸ2

t rⁿ

(4.14) for allt >0. Using lim_n→∞rⁿ=0, it follows that

FᏰ1,Ᏸ2(t)=1 (4.15)

for allt >0.

Remark4.3. Sinceλ^1/qq →max_iriasq→ ∞, we can regard [12, Theorem 4.2]

as a limit case ofTheorem 2.7. More precisely, if max_iri<1, then sprtµ^∗ is the unique compact set satisfying the random scaling law for sets(S¹,...,SN). Acknowledgment. This work was partially supported by Sapientia Foun- dation.

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József Kolumbán: Faculty of Mathematics and Computer Science, Babes-Bolyai Uni- versity, 3400 Cluj-Napoca, Romania

E-mail address:[email protected]

Anna Soós: Faculty of Mathematics and Computer Science, Babes-Bolyai University, 3400 Cluj-Napoca, Romania

E-mail address:[email protected]

Ibolya Varga: Faculty of Mathematics and Computer Science, Babes-Bolyai University, 3400 Cluj-Napoca, Romania

E-mail address:[email protected]