A general decomposition theory for random cascades

a r X i v :m a t h /9410222v 1 [m a t h .P R ] 1 O c t 1994APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31,Number 2,October 1994,Pages 216-222

A GENERAL DECOMPOSITION

THEORY FOR RANDOM CASCADES

Edward C.Waymire and Stanley C.Williams Abstract.This announcement describes a probabilistic approach to cascades which,in addition to providing an entirely probabilistic proof of the Kahane-Peyri`e re theo-rem for independent cascades,readily applies to general dependent cascades.More-over,this uni?es various seemingly disparate cascade decompositions,including Ka-hane’s T-martingale decomposition and dimension disintegration.1.Brief history of the problem A theory of positive T-martingales was introduced in [K3]as the general frame-work for independent multiplicative cascades and random coverings.This includes spatial distributions of interest in both probability theory and in the physical sciences,e.g.[CCD,CK,DE,DG,DF,DM,F,GW,MS,MW,TLS,PW,S].For basic de?nitions,let T be a locally compact metric space with Borel sig-ma?eld B ,and let (?,F ,P )be a probability space together with an increasing sequence F n ,n =1,2,...,of sub-sigma?elds of F .A positive T-martingale is a sequence {Q n }of non-negative functions on T ×?such that:(i)for each t ∈T,{Q n (t,·):n =1,2,...}is a martingale adapted to F n ,n =1,2,...;(ii)for P -a.s.ω∈?,{Q n (·,ω):n =1,2,...}is a sequence of Borel measurable non-negative real-valued functions on T .Let M +(T )denote the space of positive Borel measures on T ,and suppose that {Q n (t )}is a positive T -martingale.For σ∈M +(T )let Q n σdenote the random measure de?ned by Q n σ<<σand dQ n σ

1991Mathematics Subject Classi?cation .Primary 60G57,60G30,60G42;Secondary 60K35,60D05,60J10,60G09.

Key words and phrases.Martingale,Hausdor?dimension,tree,random measure.

Received by the editors June 14,1993

1994American Mathematical Society 0273-0979/94$1.00+$.25per page

2 E.C.WAYMIRE AND S.C.WILLIAMS

Independent cascades.Let b≥2be a natural number,referred to as a branching number;and let T={0,1,...,b?1}N be the metric space for the ultrametric ρ(s,t)=b?a(s,t)where a(s,t)=inf{n:s n=t n},s=(s1,s2,...),t=(t1,t2,...). The countable set T?≡T?(∞):=∪∞n=0T?(n),where T?(n):=∪n k=1{0,1,...,b?1}k,provides a convenient labelling of the vertices of the in?nite b-ary tree.It is sometimes convenient to adjoin a root vertex denoted?to T?.For t∈T,n≥1,write t|n:=(t1,t2,...,t n)∈T?,t|0:=?.Forγ=(γ1,...,γn)∈T?,j∈{0,1,...,b?1}, de?ne|γ|:=n andγ?j:=(t1,...,t n,j).Also denote the n th generation partition by?γ:={t∈T:t||γ|=γ}.In this context the cascade generators are furnished by a(denumerable)family of i.i.d.nonnegative mean-one random variables{Wγ:γ∈T?}indexed by the tree and de?ned on a probability space(?,F,P).With this,let

(1.2)P n(t):= |γ|=n Wγ1?γ(t).

Now the homogeneous independent cascade is the positive T-martingale de?ned by Q n=P1P2···P n with respect to F n:=σ{Wγ:|γ|≤n}.De?neλ∞:=Q∞λ, where here and throughoutλdenotes the Haar measure on T.It may happen that λ∞=0a.s.,referred to as a degeneracy.The onset of nondegeneracy may be viewed as a critical phenomenon associated with the cascade.The fundamental theorem of [KP]on the structure of homogeneous independent cascades provides necessary and su?cient conditions on the distribution of the generators for(i)nondegeneracy and, (ii)divergence of moments ofλ∞[0,1],and provides in all cases of nondegeneracy (iii)the Hausdor?dimension of the support.Replacement of the generators at each generation by i.i.d.nonnegative random vectors W:=(W0,...,W b?1),E1

A GENERAL DECOMPOSITION THEORY FOR RANDOM CASCADES3 independence.The point of focus in[KP]is a distributional?xed-point equation for the total mass of the cascade.Extensions to a limited class of dependent cas-

cades,namely,?nite-state Markov generators,have been found along similar lines

in[WW1].However,even in the countably in?nite-state Markov context,the?xed-point analysis does not readily extend to dependent generators.It is interesting to

note that the?xed-point equation also arises in certain interacting particle context,

[HL,DL].The approach described in this note may be of independent interest when applied to the?xed-point problem under dependence.

To de?ne the general dependent random cascade,we begin with a probabil-ity measure p0(dx)and a collection of mean-one transition probability kernels

q n(dx|x0,...,x n?1),on B[0,∞),x i≥0,n≥https://www.360docs.net/doc/0012099993.html,ing the Kolmogorov exten-

sion theorem,one may construct a unique probability measure P on the product space?:=[0,∞)T?(∞),:=:=B T?(∞)[0,∞)together with the coordinate projec-

tion random variables Wγ(ω):=ωγsuch that(i)W?has distribution p0(dx);

(ii)for any t∈T,the conditional distribution of W t|n+1given Wγ,|γ|≤n,is

q n+1(dx|W t|0,W t|1,...,W t|n);and(iii)for s∈T,s|n+1=t|n+1,W s|n+1is con-

ditionally independent of W t|n+1given Wγ,|γ|≤n[WW2].We refer to this model as the cascade generator corresponding to the given transition kernels q n and initial

distribution p0.By relabeling the states,the two-state“Markov chains on trees”

constructed in[Pr,Sp]may be adapted as a special case to illustrate this general setting.

Given a cascade generator,one can de?ne a positive T-martingale by applying the formula(1.2).The resulting random measureλ∞:=Q∞λwill be referred to

as the dependent cascade.Our approach to the study of cascades in this generality

is based on three elements,namely,(i)a weight system perturbation,(ii)a size-bias transform,and(iii)a general percolation method.

De?nition2.1.A weight system is a family F of real-valued functions Fγ:?→

[0,∞)indexed by the tree,where for eachγ∈T?,Fγisσ{Wγ|j:j≤|γ|}measur-

able,such that Q F,n(t):= |γ|=n j≤n(Wγ|j)Fγ1?γis a positive T-martingale, referred to as a weighted cascade.A weight system F is called a weight decomposi-

tion in the case F c:={1?Fγ:γ∈T?}is also a weight system.

Note that a weight system is a weight decomposition if and only if the weights are bounded between0and1.

Assuming that one already has a nondegenerate limit cascade,Peyri`e re[P]de-?nes a probability Q on?×T,named the Peyri`e re probability in[K1],for the joint distribution of a randomly selected path and the cascade generators along the path.This probability plays an important role in the analysis of the structure of independent cascades provided nondegeneracy has been established.The following probability is a useful extension of this notion in two directions:namely,(i)it does not require an a priori nondegeneracy condition;and(ii)it permits perturbations by a weight system.For a given weight system F,let Q F,n(t):=b?n j≤n(W t|j)F t|n, t∈T,n∈N.De?ne a sequence Q F,n of measures on?×T by

(2.1) ?×T f(ω,t)Q F,n(dω×dt):=E P T f(ω,t)Q F,n(t)λ(dt),

for bounded measurable functions f.Then one normalizes the masses of the Q F,n by a factor Z?:=EW?F?and extends to a probability Q F using the Kolmogorov extension theorem.One requires Z?>0here and throughout.

4 E.C.WAYMIRE AND S.C.WILLIAMS

The third ingredient to this theory is a generalization of an idea considered without proof in[K1],which may be viewed as a percolation method;see[WW2] for proofs.By independently pruning the tree,one studies the critical parameters governing the survival of mass,i.e.nondegeneracy of the percolated cascade,to determine dimension estimates on the support of the cascade.This is similar to an idea explored in[L]but di?ers by the distinction between locations and amounts of positive mass.

The following results provide the foundations for the general theory being an-nounced here.

Theorem2.1(A Lebesgue decomposition).Letπ?denote the coordinate projec-tion map of?×T onto?.Then

d Q F?π?1?=Z?1

λF,∞(T)1(λF,∞(T)<∞)dP+1(λF,∞(T)=∞)d Q F?π?1?whereλF,∞=Q F,∞λ.

Theorem2.2(A size-biased disintegration).Given a weighting system F,

Q F(dω,dt)=P F,t(dω)λ(dt),

where

dP F,t

c j

)M n,

where

M n=n?1

j=0c j b?(n?j) |γ|=n?j

γ(1)=t(j+1)

n?j

i=1(W[t|j]?[γ|i])Fγ

is a nonnegative P F,t-submartingale with respect to F t,n,whose Doob decomposition has the predictable part A n=b?1

A GENERAL DECOMPOSITION THEORY FOR RANDOM CASCADES5 Theorem 2.5(Kahane decomposition).The Kahane decomposition[K2,K4],

Q n(t)=Q′n(t)+Q′′n(t),is equivalent to a weight decomposition with respect to the weights de?ned by F t|n=Q′n(t)

λn(?γ)

where

νγ([0,s]):= [0,s](λ∞)α(?γ)ν(dα),n=|γ|.

Remark1.Two interesting classes of dependent cascades are:(i)Markov cascades and,(ii)exchangeable cascades.One may compute the weighting decompositions for Markov cascades,for example,in terms of harmonic measures corresponding to hitting probabilities of the survival classes introduced in[WW1].Also for ex-changeable cascades one may compute these weighting decompositions in terms of conditioned de Finetti measures applied to a partition of M+(T).

Remark2.One may show that the maps s→F s,γare nondecreasing in s and the Lebesgue-Stieltjes measures associated with the maps s→ F s,t|nλn(dt)converge vaguely to the spectral measureν.In fact,if the map s→F s,γis absolutely continuous with respect to some measureσ,then t→ dF t,γ

c j

e E P W log W.

Now use Theorem2.3,with c j=c?j,to conclude thatλ∞(T)<∞,P1,t-a.s.,and therefore,by Theorem2.2,after integrating out t,λ∞(T)<∞,Q1-a.s.Now apply Theorem2.1to see that Eλ∞(T)=1.

A necessary condition for nondegeneracy.To see why the entropy condition E P W log b W<1is also necessary for nondegeneracy,suppose that E P W log b W≥

6 E.C.WAYMIRE AND S.C.WILLIAMS

1,P t(W=b)<1.Fix t∈T.If E P W log b W>1,then one obtainsλn(T)→∞P1,t-a.s.directly from Theorem2.3.If E P W log b W=1but P t(W=b)<1,then, since under the size-biasing W>0a.s.,one has that lim sup logλn(T)=∞,P1,t-a.s.by the Chung-Fuchs null-recurrence criterion for the random walk along the t-path.Integrate out t and use Theorems2.2and2.1(in that order)to conclude thatλ∞(T)=0,P-a.s.The special case of the degeneracy P t(W=b)=1,i.e. P(W=b)=1

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[WW1] E.Waymire and S.C.Williams,Markov cascades,preprint.

[WW2]

,A cascade decomposition theory with applications to Markov and exchangeable cascades,preprint.

Mathematics Department,Oregon State University,Corvallis,Oregon97330 E-mail address:waymire@https://www.360docs.net/doc/0012099993.html,

Mathematics Department,Utah State University,Logan,Utah84321

E-mail address:williams@https://www.360docs.net/doc/0012099993.html,