WAVELET ANALYSIS OF CONSERVATIVE CASCADES 1 ...

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WAVELET ANALYSIS OF CONSERVATIVE CASCADES

SIDNEY RESNICK, GENNADY SAMORODNITSKY, ANNA GILBERT, AND WALTER WILLINGER

Abstract. A conservative cascade is an iterative process that fragments a given set into smallerand smaller pieces according to a rule which preserves the total mass of the initial set at eachstage of the construction almost surely and not just in expectation. Motivated by the importanceof conservative cascades in analyzing multifractal behavior of measured Internet traffic traces,we consider wavelet based statistical techniques for inference about the cascade generator , therandom mechanism determining the re-distribution of the set’s mass at each iteration. We providetwo estimators of the structure function, one asymptotically biased and one not, prove consistencyand asymptotic normality in a range of values of the argument of the structure function less thana critical value. Simulation experiments illustrate the asymptotic properties of these estimatorsfor values of the argument both below and above the critical value. Beyond the critical value,the estimators are shown to not be asymptotically consistent.

1. Introduction

A multiplicative cascade is an iterative process that fragments a given set into smaller andsmaller pieces according to some geometric rule and, at the same time, distributes the total massof the given set according to another rule. The limiting object generated by such a proceduregenerally gives rise to a singular measure or multifractal – a mathematical construct that is able tocapture the highly irregular and intermittent behavior associated with many naturally occurringphenomena, e.g., fully developed turbulence (see [10, 12, 4, 15] and references therein); spatialrainfall [6]; the movements of stock prices [14]; and Internet traffic dynamics [19, 3].

The generator of a cascade determines the re-distribution of the set’s total mass at everyiteration; it can be deterministic or random. Cascade processes with the property that thegenerator preserves the total mass of the initial set at each stage of the construction almost surelyand not just in expectation are called conservative cascades and are the main focus of this paper.Originally introduced by Mandelbrot [13] (also in the turbulence context), conservative cascadeshave recently been considered in [3] for use in describing the observed multifractal behavior ofmeasured Internet traffic traces. In particular, Feldmann et al. [3] build on empirical evidencethat measured Internet traffic is consistent with multifractal behavior by illustrating that “...data networks appear to act as conservative cascades!” They demonstrate that

• multiplicative and measure-preserving structure becomes most apparent when analyzingmeasured Internet traces at a particular layer within the well-defined protocol hierarchy

The visits of Sidney Resnick to AT&T Labs–Research were supported by AT&T Labs–Research and a NationalScience Foundation Grant from the Cooperative Research Program in the Mathematical Sciences. S. Resnick andG Samorodnitsky were also partially supported by NSF grant DMS-97-04982 and NSA grant MDA904-98-1-0041at Cornell University.

c©0000 American Mathematical Society0000-0000/00 $1.00 + $.25 per page

1

2 S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER

of today’s Internet Protocol (IP)-based networks, namely the Transport Control Protocol(TCP) layer and at the level of individual TCP connections,

• this structure is recovered at the aggregate level (i.e., when considering the superpositionprocess consisting of all IP packets generated by all active TCP connections) and causesaggregate Internet traffic to exhibit multifractal behavior.

Well short of providing a physical explanation to the all-important networking question of “Whydo packets within individual TCP connections conform to a conservative cascade?” the work in [3]is empirical in nature and relies on a number of heuristics for inferring multifractal behavior fromtraces of measured Internet traffic. However, to provide a more solid statistical basis for empiricalstudies of multifractal phenomena, progress in the area of statistical inference for multiplicativelygenerated multifractals is crucial.

In this paper we contribute to the effort of providing rigorous techniques for multifractalanalysis by investigating wavelet-based estimators for conservative cascades (i.e., for the classof multifractal processes generated by conservative cascades) and studying their large sampleproperties. In essence, the inference problem for conservative cascades consists of deducing froma single realization of the cascade process the distribution of the cascade generator that waspresumably used to generate the sample or signal at hand. Intuitively, the generator’s distributioncan be inferred from the degree of variability and intermittency exhibited locally in time by thesignal under consideration. It can be expressed mathematically in terms of the local Holderexponents which in turn characterize the singularity behavior of a signal locally in time. Moreover,since the local Holder exponent at a point in time t0 describes the local scaling behavior of thesignal as we look at smaller and smaller neighborhoods around t0, a wavelet-based analysis thatfully exploits the time- and scale-localization ability of wavelets proves convenient and is tailor-made for our purpose. On the one hand, we exploit here the fact that the singularity behavior ofa process can (under certain assumptions) be fully recovered by studying the singularity behaviorin the wavelet domain; i.e., by investigating the (possibly) time-dependent scaling properties ofthe wavelet coefficients associated with the underlying process in the fine-time scale limit. On theother hand, using Haar wavelets, the discrete wavelet transform of a conservative cascade can beexplicitly expressed in terms of the cascade’s generator (see for example [5]) and hence providesa promising setting for relating the local scaling behavior of the sample to the distribution ofthe underlying conservative cascade generator. In particular, we relate the distribution of thegenerator to an invariant of the cascade, namely the structure function or modified cumulantgenerating function (also known as Mandelbrot-Kahane-Peyriere (MKP) function [7]) and studythe statistical properties (i.e., asymptotic consistency, asymptotic normality, confidence intervals)of two wavelet-based estimators of this function.

Although the results in this paper have been largely motivated by our empirical investigationsinto the multifractal nature of measured Internet traffic [3, 5], we have clearly benefited fromthe recent random cascade work of Ossiander and Waymire [17]. Compared to the conservativecascades considered in this paper, random cascades are multiplicative processes with generatorsthat preserve the total mass of the initial set only in expectation and not almost surely. Thisapparently minor difference ensures independence within and across the different stages of a ran-dom cascade construction but gives rise to subtle dependencies inherent in conservative cascades.Ossiander and Waymire [17] study the large sample asymptotics of estimators that are defined inthe time-domain rather than in the wavelet-domain and allow for a rigorous statistical analysisof the scaling behavior exhibited by random cascades (for related work, see also [20]). While

WAVELET ANALYSIS OF CONSERVATIVE CASCADES 3

the large-sample properties of the time domain-based estimators considered in [17] and of thewavelet-based estimators studied in this paper are very similar, their potential advantages, dis-advantages and pitfalls when implementing and using them in practice require further studies.However, in combination, these different estimators provide a set of statistically rigorous tech-niques for multifractal analysis of highly irregular and intermittent data that are assumed to begenerated by certain types of multiplicative processes or cascades.

The rest of the paper is organized as follows. Sections 2–4 contain the basic facts about conser-vative cascades, their wavelet transforms, and some related quantities that are needed later in thepaper. Section 5 discusses the critical constants and Section 6 is concerned with certain martin-gales and leads into Section 7 where subcritical asymptotics (that is, asymptotics for values of theargument below the critical value) and strong consistency of our two wavelet-based estimatorsis established. Asymptotic normality of the estimators is explained and illustrated with somesimulated data in Section 8. We conclude in Section 9 with some supercritical asymptotics whenthe value of the arguement exceeds the critical value. The values of the estimators at large valuesof the argument of the structure function are uninformative and misleading, thus providing somepractical guidance for properly interpreting the plots associated with the estimation procedure.

2. The Conservative Cascade.

We now summarize the basic facts about the conservative cascade.Consider the binary tree. Nodes of the tree at depth l will be indicated by (j1, . . . , jl) ∈ 0, 1l.

Alternatively we consider successive subdivisions of the unit interval [0, 1]. After subdividing ltimes we have equal subintervals of length 2−l indicated by

I(j1, . . . , jl) =[

l∑

k=1

jk2k,

l∑

k=1

jk2k

+1

2l

)

, (j1, . . . , jl) ∈ 0, 1l.(2.1)

An infinite path through the tree is denoted by

j = (j1, j2, . . . ) ∈ 0, 1∞

and the first l entries of j are denoted by

j|l = (j1, . . . , jl).

We will sometimes write when convenient

j|l, jl+1 = (j1, . . . , jl, jl+1).

The conservative cascade is a random measure on the Borel subsets of [0, 1] which may beconstructed in the following manner. Suppose we are given a random variable W , called the

cascade generator , which has range [0, 1] and which is symmetric about 1/2 so that Wd= 1 −W.

The symmetry implies that E(W ) = 1/2. We assume the random variable is not almost surelyequal to 1/2. There is a family of identically distributed random variables

W (j|l), j ∈ 0, 1∞, l ≥ 1each of which is identically distributed as W . These random variables satisfy the conservativeproperty

W (j|l, 1) = 1 −W (j|l, 0).(2.2)


Random variables associated with different depths of the tree are independent and random vari-ables of the same depth which have different antecedents in the tree are likewise independent.Dependence of random variables having the same depth is expressed by (2.2). The conservativecascade is the random measure µ∞ defined by

µ∞(I(j|l)) =l∏

i=1

W (j|i).(2.3)

Note the conservative property entails that

µ∞(I(j|l, 0)) + µ∞(I(j|l, 1)) = µ∞(I(j|l)),(2.4)

so that the weight of two offspring equals the weight of the parent. This implies∑

j|l

µ∞(

I(j|l))

= 1.(2.5)

3. Wavelet Coefficients.

We compute the wavelet transform

d−l,n =

∫ 1

0ψ−l,n(x)µ∞(dx), n = 0, . . . , 2l − 1; l ≥ 1,(3.1)

using the Haar wavelets

ψ−l,n(x) :=

2l/2, if 2n2l+1 ≤ x < 2n+1

2l+1 ,

−2l/2, if 2n+12l+1 ≤ x < 2n+2

2l+1 .(3.2)

We have by examining where the Haar wavelet is constant that

d−l,n = 2l/2

(

µ∞

(

[2n

2l+1,2n+ 1

2l+1))

− µ∞

(

[2n+ 1

2l+1,2n+ 2

2l+1))

)

.

Now suppose that∑l

k=1 jk/2k = n/2l. Then we have from the definition (2.3)

d−l,n =2l/2[

l∏

i=1

W (j|i)W (j|l, 0) −l∏

i=1

W (j|i)W (j|l, 1)]

=2l/2l∏

i=1

W (j|i)[

W (j|l, 0) −W (j|l, 1)]

and using the conservative property (2.2), this is

d−l,n =2l/2l∏

i=1

W (j|i)[

2W j|l, 0) − 1]

,(3.3)

for n = 0, 1, . . . , 2l − 1. Sometimes where convenient, we will also write

d−l,n = d(−l, j|l) = 2l/2l∏

i=1

W (j|i)[

2W j|l, 0) − 1]

.(3.4)


4. Notation Glossary.

Before continuing the analysis, we collect some notation in one place for easy reference. Weseek to estimate the distribution of the cascade generator W and this will be accomplished if weestimate

c(q) := 2E(W q), q > 0,(4.1)

or equivalently we could estimate the structure function

τ(q) = 1 + log2E(W q) = log2 c(q).(4.2)

The structure function will be estimated using estimators constructed from the process

Z(q, l) =∑

j|l

l∏

i=1

W (j|i)q|2W (j|l, 0) − 1|q(4.3)

and note from (3.3) that

Z(q, l) =1

2ql/2

2l−1∑

n=0

|d−l,n|q.(4.4)

Our analysis rests on the process M(q, l), which we will show to be a martingale and which isdefined as

M(q, l) =1

c(q)l

∑

j|l

l∏

i=1

W (j|i)q, q > 0, l ≥ 1,(4.5)

and note the normalization makes

E(M(q, l)) = 1.

There are further constant functions needed:

b(q) =E|2W − 1|q,(4.6)

a(q) =c(2q)

c2(q)=

E(W 2q)

2(

E(W q))2 ,(4.7)

ar(q) =c(rq)

cr(q)=

21−rE(W rq)(

E(W q))r .(4.8)

Note that a(q) = a2(q). Finally we need three variances

σ21(q) :=

1

c2Var(W q + (1 −W )q),(4.9)

σ22(q) :=

1

b2Var(|2W − 1|q).(4.10)

σ23(q) :=

1

b2Var(W q

1

c|2W2 − 1|q +

(1 −W1)q

c|2W3 − 1|q − |2W1 − 1|q

)

,(4.11)

where Wi, i = 1, 2, 3 are iid with the distribution of the cascade generator.It is convenient to define W = e−Y so that the Laplace transform of Y is

φ(q) := Ee−qY = E(W q)(4.12)


and

ar(q) =21−rφ(rq)

φr(q).

5. Critical Constants

We now define the quantity

q∗ := supq > 0 : a(q) < 1(5.1)

so that for q < q∗ we have a(q) < 1. It will turn out that when q < q∗, the sequence M(q, l), l ≥ 1is an L2-bounded and uniformly integrable martingale and this is the easiest case to analyze. Itis always the case that q∗ ≥ 1, which follows from the fact that

a(1) =E(W 2)

2(E(W ))2=E(W 2)

2(12 )2

= 2E(W 2)

so that

a(1) = 2(

Var(W ) +1

4

)

= 2E(W − 1

2)2 +

1

2≤ 2 · |1 − 1

2|2 +

1

2= 1.

The Mandelbrot-Kahane-Peyriere (MKP) Condition: Let W be the cascade generatorand define

Xq =W q

EW q, q > 0,

so that EXq = 1. The MKP Condition is satisfied for q if

E(Xq log2Xq) < 1(5.2)

iff

q

E(W q)E(W q logW ) − logE(W q) < log 2(5.3)

iff

q(log φ)′(q) − log φ < log 2 .(5.4)

Define

Λ∗ := q : E(Xq log2Xq) < 1.Then Λ∗ is an interval and we define the second critical constant

q∗ := supΛ∗.(5.5)

Why is q∗ considered a critical quantity? It turns out that the martingale M(q, l), l ≥ 1converges as l → ∞ to M(q,∞) where

M(q,∞) =

0, if q ≥ q∗,

something non-degenerate, if q < q∗.

It turns out that the associated martingale is uninformative asymptotically when q > q∗.The two critical constants are related numerically by the inequality

max(1, q∗/2) ≤ q∗ ≤ q∗.(5.6)


To see why the inequality q∗ ≤ q∗ in (5.6) is true, it suffices to show that if q > 0 satisfies a(q) < 1,then the MKP condition is satisfied for this q. However, if a(q) < 1, then

log 2 > log φ(2q) − 2 log φ(q) = log φ(2q) − log φ(q) − log φ(q)

=

∫ 2q

q(log φ)′(s)ds− log φ(q)

and since log φ is convex, (log φ)′ is increasing and the forgoing is bounded below by

≥(log φ)′(q)

∫ 2q

qds − log φ(q)

=q(log φ)′(q) − log φ(q).

We conclude

log 2 > q(log φ)′(q) − log φ(q)

which is equivalent to the MKP condition holding by (5.4).On the other hand, suppose that q∗ <∞. Since a(q∗) = 1, we have in the same way as above

log 2 = log φ(2q∗) − 2 log φ(q∗) = 2(

log φ(2q∗) − log φ(q∗))

− log φ(2q∗)

=2

∫ 2q∗

q∗

(log φ)′(s)ds − log φ(2q∗)

≤2(log φ)′(2q∗)

∫ 2q∗

q∗

ds− log φ(2q∗)

=2q(log φ)′(2q∗) − log φ(2q∗).

Therefore, the MKP condition does not hold for 2q∗, and so q∗ ≤ 2q∗.

Example 1. Suppose W is uniformly distributed on [0, 1]. In this case E(W q) = 1/(1 + q) andso

a(q) =1

2

(

1 +q2

2q + 1

)

and

q∗ = 1 +√

2 ≈ 2.4.

Likewise, q∗ satisfies the equation

log(1 + q) − q

1 + q= log 2

and so q∗ ≈ 3.311.

Example 2. Suppose more generally that W has the beta distribution with mean 1/2 (i.e., theshape parameters α and β are equal). Then

E(W q) =Γ(2α)Γ(α + q)

Γ(α)Γ(2α + q)(5.7)

and q∗ satisfies

4−α√πΓ(α+ 2q∗)Γ(2α + q∗)2 − Γ(α+ 1/2)Γ(2α + 2q∗)Γ(α+ q∗)

2 = 0.


Example 3. Suppose W has the two point distribution concentrating mass 1/2 at ±p for some0 ≤ p < 1/2. Then

E(W q) =1

2(pq + (1 − p)q)(5.8)

and

a(q) = 1 − 2pq(1 − p)q

(pq + (1 − p)q)2.

Note in this case that a(q) ↑ 1 as q ↑ ∞ so q∗ = q∗ = ∞.

Example 4. If W does not have a two point distribution but nevertheless has an atom of sizep1 < 1/2 at 1 (and hence by symmetry there is an atom of the same size at 0) we have q∗ < ∞(and, hence, also q∗ < ∞). To see this we express the condition (5.3), when q > 1, in theequivalent form

E(W q log2Wq)

E(W q) log2

(

2E(W q)) > 1 .(5.9)

Note that if W does not have a two point distribution, then for q > 1, we have P [W q < W ] > 0and E(W q) < E(W ) = 1/2, so log2(2E(wq)) < 0, which explains the sign reversal in (5.9)compared with (5.3).

By the dominated convergence theorem the numerator in (5.9) converges to 0 as q → ∞, whilethe denominator converges to P [W = 1] log2

(

2P [W = 1])

6= 0. Hence, (5.9) fails for large q.

Based on the experience provided in Examples 3 and 4, it natural to wonder how common itcan be that q∗ = ∞. This is discussed in the next proposition.

Proposition 5.1. Unless W has a two point distribution, it must be the case that q∗ <∞.

Proof. Because of Example 3, we may assume that W does not have atoms at 0 and 1. Letp ∈ [0, 1/2) be the leftmost point of the support of the distribution of W . Then 1 − p is therightmost point of the support of the distribution of W . For 0 < ρ < 1, we have

θ(ρ) := P [W ≥ ρ(1 − p)] > 0.

Since the distribution of W is not a two point distribution, limρ→1 θ(ρ) <12 . Thus, we can find

and fix a value of 0 < ρ < 1 such that

θ(ρ) < 1/2, 0 < ρ < 1.

For this value of ρ, it is convenient to set

δ(ρ) := δ = ρ(1 − p).

Apply Jensen’s inequality with the convex function g(x) = x log x, x > 0, to get

E

(

W q

c(q)log2

(W q

c(q)

)

· 1[W≥δ]

)

=θ(ρ)E

(

W q

c(q)log2

W q

c(q)

∣

∣

∣W ≥ δ

)

≥θ(ρ)E(

W q

c(q)

∣

∣

∣W ≥ δ

)

log2

(

E

(

W q

c(q)

∣

∣

∣W ≥ δ

))

=E

(

W q

2E(W q)1[W≥δ]

)

· log2E

(

W q

2E(W q)

∣

∣

∣W ≥ δ

)

.(5.10)


Also we have

limq→∞

E

(

W q

c(q)1[W≥δ]

)

=1

2.(5.11)

To verify (5.11), note that

1

2= E

(W q

c(q)

)

= E(W q)

c(q)1[W≥δ]

)

+ E(W q

c(q)1[W<δ]

)

,

and

0 ≤ limq→∞

E(W q1[W<δ])

E(W q1[W≥δ])≤ lim

q→∞

1

P [W ≥ δ]E(W

δ

)q1[W/δ<1] = 0,

by dominated convergence. Thus from (5.10) and (5.11), we conclude

lim infq→∞

E(W q

c(q)log2

W q

c(q)· 1[W≥δ]

)

≥ 1

2· log2

( 1

2θ(ρ)

)

=: h > 0,(5.12)

because θ(ρ) < 1/2.We also claim

limq→∞

E(∣

∣

∣

W q

c(q)log

W q

c(q)

∣

∣

∣1[W<δ]

)

= 0.(5.13)

To verify (5.13), note the expectation is the same as

E(∣

∣

∣

W q

c(q)log

W q

c(q)

∣

∣

∣1[ Wq

c(q)< δq

c(q)]

)

.

Provided that

limq→∞

δq/c(q) = 0,(5.14)

we get for any ǫ < e−1, by the monotonicity of |x log x| in (0, e−1), that the expectation is boundedby |ǫ log2 ǫ| for q so large that δq/c(q) < ǫ. So it remains to check (5.14), or equivalently to check

limq→∞

E(W q

δq

)

= 0.

However, by Fatou’s lemma

lim infq→∞

E(W q

δq

)

≥ E(

lim infq→∞

(W

δ

)q1[W≥δ]

)

= ∞,

since P [W > δ] > 0 (otherwise, the definition of p would be contradicted).Our conclusion from (5.12) and (5.13) is that for all large q,

E

(

W q

c(q)log2

(W q

c(q)

)

)

= E

(

W q

c(q)log2

(W q

c(q)

)

)

1[W≥δ] + E

(

W q

c(q)log2

(W q

c(q)

)

)

1[W<δ] > 0.

Thus

EW q

2EW q

(

log2W q

E(W q)− log2 2

)

=1

2EXq log2Xq −

1

2> 0

and so the MKP condition (5.3) fails for all large q. Therefore, q∗ <∞.


5.1. Properties of the function ar(q). We need the following properties of the function ar(q).

Proposition 5.2. (i) For any fixed r > 1, the function ar(q) (and therefore a(q)) is strictlyincreasing in q > 0.

(ii) For any fixed q > 0, the function log ar(q) is strictly convex in the region r > 1.(iii) If q satisfies the MKP condition, then

d

drlog ar(q)

∣

∣

∣

r=1< 0,

and there exists r0 ∈ (1, 2) such that

ar0(q) < a1(q) = 1.(5.15)

(iv) If the MKP condition fails for q and the inequalities in (5.2), (5.3) or (5.4) are reversed tobecome strictly greater than, we have

d

drlog ar(q)

∣

∣

∣

r=1> 0,(5.16)

and there exists 0 < r1 < 1 and ar1(q) < a1(q) = 1.

Proof. (i) Recall the definition of φ from (4.12). For fixed r > 1, if we differentiate with respectto q, we get

(

φ(rq)

φr(q)

)′

=φr(q)rφ′(rq) − φ(rq)rφr−1(q)φ′(q)

φ2r(q).

This is positive iff

φ(q)φ′(rq) > φ(rq)φ′(q)

orφ′(rq)

φ(rq)>φ′(q)

φ(q).

Since r > 1, it suffices to show φ′/φ is strictly increasing which is true if its derivative is strictlypositive. The derivative is

φ(q)φ′′(q) − (φ′(q))2

φ2(q)

and this is strictly positive iff

φ(q)φ′′(q) > (φ′(q))2,(5.17)

that is iff

E(e−qY )E(Y 2e−qY ) >(

E(Y e−q/2Y · e−q/2Y ))2

which follows from the Cauchy–Schwartz inequality.(ii) Fix q > 0 and check that

d2

dr2(log ar(q)) =

q2

φ2(rq)

[

φ′′(rq)φ(rq) − (φ′(rq))2]

which is positive by (5.17).(iii) For fixed q > 0,

d

drlog ar(q) = q(log φ)′(qr) − log 2 − log φ(q),


so thatd

drlog ar(q)

∣

∣

∣

r=1=

ddrar(q)

ar(q)

∣

∣

∣

∣

∣

r=1

= q(log φ)′(q) − log φ(q) − log 2 < 0.

Since a1(q) = 1, we haved

drar(q)

∣

∣

∣

r=1< 0, a1(q) = 1.

Hence there exists r0 ∈ (1, 2) such that

ar0(q) < a1(q) = 1.

(iv) If

d

drlog ar(q)

∣

∣

∣

r=1=

ddrar(q)

ar(q)

∣

∣

∣

∣

∣

r=1

= q(log φ)′(q) − log φ(q) − log 2 > 0,

then since log a1(q) = 0, there exists r1 < 1 such that

log ar1(q) < 0 or ar1(q) < 1.

6. The Associated Martingale.

In this section we study the properties of the process M(q, l), l ≥ 1 defined in (4.5) for eachfixed q > 0. We define the increasing family of σ-fields

Fl := σW (j|l), j|l ∈ 0, 1lgenerated by the weights up to and including depth l.

Proposition 6.1. For each q > 0, the family

(M(q, l),Fl), l ≥ 1is a non-negative martingale with constant mean 1 such that M(q, l) converges almost surely toa limiting random variable M(q,∞):

M(q, l)a.s.→ M(q,∞), E(M(q,∞) ≤ 1.

If the MKP condition fails for q, then

P [M(q,∞) = 0] = 1,

and if q satisfies the MKP condition, then E(M(q,∞)) = 1 so that

P [M(q,∞) > 0] = 1.

Proof. The martingale property is easily established:

E(M(q, l + 1)|Fl) =∑

j|l,jl+1

E(

∏li=1W

q(j|i)cl

W q(j|l, jl+1)

c|Fl

)

=∑

j|l

l∏

i=1

W q(j|i)cl

∑

jl+1

E(

W q(j|l, jl+1)/c)

=M(q, l)2 ·E(W q)/c = M(q, l).


By the martingale convergence theorem (eg, [18], [16]) a non-negative martingale always convergesalmost surely. The last statements follow by the methods of Kahane and Peyriere ([8]). See alsoPropositions 6.2 and 6.3.

Example 5. Recall the example of the two point distribution of Example 3 of Section 4. In thiscase we have M(q, l) = 1 for all q > 0 and l ≥ 1. For verifying this, the key observation is that

W q + (1 −W )q = pq + (1 − p)q.(6.1)

Recall (5.8) and then observe for l > 1

M(q, l) =∑

j|l

l∏

i=1

(

W q(j|i)c

)

=∑

j|l−1

l−1∏

i=1

(

W q(j|i)c

)

∑

jl

W q(j|l − 1, jl)

c

=∑

j|l−1

l−1∏

i=1

(

W q(j|i)c

)

[

W q(j|l − 1, 0) +W q(j|l − 1, 1)]

/c

and since W (j|l − 1, 1) = 1 −W (j|l − 1, 0) we apply (6.1) to get

=∑

j|l−1

l−1∏

i=1

(

W q(j|i)c

)

[

pq + (1 − p)q]

/c

=∑

j|l−1

l−1∏

i=1

(

W q(j|i)c

)

= M(q, l − 1).

One can easily see that M(q, 1) = 1 and the assertion is shown.

Define M(q, 0) = 1 and let the martingale differences be

d(q, l) := M(q, l) −M(q, l − 1), l ≥ 1.

For l > 1 we have from the definition of M(q, l) that

d(q, l) =∑

j|l−1

l−1∏

i=1

(

W q(j|i)c

)

[W q(j|l − 1, 0) + (1 −W (j|l − 1, 0)q

c− 1]

=∑

j|l−1

l−1∏

i=1

(

W q(j|i)c

)

[

ξ(j|l)]

.(6.2)

We now easily see E(d(q, l)|Fl−1) = 0. For the conditional variance, note E(ξ(j|l)) = 0 and recallthe notation from (4.9)

σ21(q) = Var(ξ(j|l) =

1

c2Var(W q + (1 −W )q).

So the conditional variance of d(q, l) is

E(d2(q, l)|Fl−1) =E

(

(

∑

j|l−1

l−1∏

i=1

W q(j|i)c

ξ(j|l))2

|Fl−1

)


=∑

j|l−1p|l−1

l−1∏

i=1

(

W q(j|i)c

) l−1∏

i=1

(

W q(p|i)c

)

E(ξ(j|l)ξ(p|l)).

Since ξ(p|l)) ⊥ ξ(j|l)) if p|l 6= j|l we have

E(d2(q, l)|Fl−1) =∑

j|l−1

(

l−1∏

i=1

(

W q(j|i)c

)

)2σ2

1(q)

=∑

j|l−1

l−1∏

i=1

(

W 2q(j|i)c(2q)

)

al−1(q)σ21(q)

=M(2q, l − 1)al−1(q)σ21(q).

Thus the conditional variance of M(q, l) is

l∑

i=1

E(d2(q, i)|Fi−1) =l∑

i=1

M(2q, i − 1)ai−1(q)σ21(q).(6.3)

Furthermore,

E(d2(q, l)) = E(E(d2(q, l)|Fl−1)) = EM(2q, l − 1)al−1(q)σ21(q) = al−1(q)σ2

1(q)

and thus

Var(M(q, l)) =l∑

i=1

E(d2(q, i)) = σ21(q)

l∑

i=1

ai−1(q).(6.4)

This leads to the following facts.

Proposition 6.2. If q < q∗ so that a(q) < 1, the martingale (M(q, l),Fl), l ≥ 0 is L2-boundedand hence uniformly integrable. It follows that

E(M(q,∞) = 1, M(q, l) = E(M(q,∞)|Fl) ,(6.5)

and M(q, l) → M(q,∞) almost surely and in L2. Moreover, if q∗ ≤ q < q∗, then the martingale(M(q, l),Fl), l ≥ 0 is Lp-bounded for some 1 < p < 2 and, hence, still uniformly integrable,(6.5) still holds and M(q, l) →M(q,∞) almost surely and in Lp.

Remark. The proof will show that when q∗ ≤ q < q∗, we may take p = r0, where r0 is given inProposition 5.2 (iii). See (5.15).

Proof. Suppose first that q < q∗. We have from (6.4) that

supl≥0

E(M(q, l) − 1)2 = supl≥0

Var(M(q, l))

= liml→∞

↑l∑

i=1

E(d2(q, i)

=∞∑

i=1

ai−1(q)σ21(q) <∞.


The rest follows from standard martingale theory (eg, [16, page 68], [18]).Now let q < q∗ and we consider uniform integrability without L2 boundedness. Suppose

1 < p ≤ 2 and for two paths j1 and j2 denote by mj1,j2the largest i ≤ l such that j1|i = j2|i.

We have

E(M(q, l))p =1

c(q)lpE

(

∑

j1|l

∑

j2|l

l∏

i=1

W q(j1|i)W q(j2|i))p/2

≤ 1

c(q)lp

l∑

k=0

E

(

∑

j1|lj2|l

mj1,j2=k

l∏

i=1

W q(j1|i)W q(j2|i))p/2

≤ 1

clp(q)

l∑

k=0

E

(

∑

j|k

k∏

i=1

W 2q(j|i) ·

∑

j(1)k+1,...,j

(1)l

j(2)k+1,...,j

(2)l

j(1)k+1 6=j

(2)k+1

l∏

i=k+1

W q(j1, . . . , jk, j(1)k+1, . . . , j

(1)l |i)W q(j1, . . . , jk, j

(2)k+1, . . . , j

(2)l |i)

)p/2

≤ 1

clp(q)

l∑

k=0

∑

j|k

E(

W pq)k

·

E

(

∑

j(1)k+1,...,j

(1)l

j(2)k+1,...,j

(2)l

j(1)k+1 6=j

(2)k+1

l∏

i=k+1

W q(j1, . . . , jk, j(1)k+1, . . . , j

(1)l |i)W q(j1, . . . , jk, j

(2)k+1, . . . , j

(2)l |i)

)p/2

≤l∑

k=0

(c(pq)

cp(q)

)k 1

c(l−k)p(q)·

(

E∑

j(1)k+1,...,j

(1)l

j(2)k+1,...,j

(2)l

j(1)k+1 6=j

(2)k+1

l∏

i=k+1

W q(j1, . . . , jk, j(1)k+1, . . . , j

(1)l |i)W q(j1, . . . , jk, j

(2)k+1, . . . , j

(2)l |i)

)p/2

=

l∑

k=0

ap(q)k 1

c(l−k)p(q)

(

E(W (1 −W ))q(EW q)2(l−k−1)22(l−k−1) · 2)p/2

=

(

(

E(W (1 −W ))q)p/2 2p/2

c(q)

) l∑

k=0

ap(q)k.


Here a product over the empty set is equal to 1. By Proposition 5.2 (iii) there is a p ∈ (1, 2)such that ap(q) < 1. For this p the martingale (M(q, l),Fl), l ≥ 0 is Lp-bounded, and the restfollows, once again, from standard martingale theory.

6.1. The distribution of M(q,∞). The distribution of M(q,∞) satisfies a simple recursionwhich can be used to derive additional information.

Proposition 6.3. Suppose M(q,∞),M1(q,∞),M2(q,∞) are iid with the same distribution asM(q,∞), the martingale limit. Let W have the distribution of the cascade generator and supposeW and M(q,∞),M1(q,∞),M2(q,∞) are independent. Then

M(q,∞)d= W qM1(q,∞)

c(q)+ (1 −W )q

M2(q,∞)

c(q)(6.6)

and for any q > 0,

P [M(q,∞) = 0] = 0 or 1,(6.7)

so that E(M(q,∞)) = 1 implies P [M(q,∞) = 0] = 0.

Proof. We write

M(q,∞) = liml→∞

∑

j|l

l∏

i=1

W q(j|i)cl

= liml→∞

(

∑

j2,...,jl

l∏

i=1

W q(0, j2, . . . , jl)

cl+∑

j2,...,jl

l∏

i=1

W q(1, j2, . . . , jl)

cl

)

= liml→∞

(

W q(0)∑

j2,...,jl

l∏

i=2

W q(0, j2, . . . , jl)

cl+ (1 −W (0))q

∑

j2,...,jl

l∏

i=2

W q(1, j2, . . . , jl)

cl

)

d=W q(0)

M1(q,∞)

c+ (1 −W (0))q

M2(q,∞)

c.

Now we verify (6.7). Define

p0 =P [M(q,∞) = 0]

pW (0) =P [W = 0] = P [W = 1].

Then since c(q) 6= 0

p0 =P [M(q,∞) = 0] = P [W qM1(q,∞) + (1 −W )qM2(q,∞) = 0]

=P [′′,W = 0] + P [′′,W = 1] + P [′′, 0 < W < 1].

¿From this we conclude

p0 = 2pW (0)p0 + (1 − 2pW (0))p20

so that

p0(1 − 2pW (0)) = p20(1 − 2pW (0)).

If 0 < pW (0) < 1/2, then p0 = p20 and p0 = 0 or 1. If pW (0) = 1/2, then P [W = 0] = P [W =

1] = 1/2 and W has a two point distribution and hence from Example 5 we know M(q, l) = 1which implies M(q,∞) = 1.


7. Estimation: Subcritical Consistency

We propose two estimators of the structure function which depend on scaled summed powersof the wavelet coefficients Z(q, l), l ≥ 1. These are

τ1(q) = τ1(q, l) =log2 Z(q, l)

l=

log2

∑2l−1n=0 |d−l,n|q − ql/2

l(7.1)

τ2(q) = τ2(q, l) = log2

(

Z(q, l + 1)

Z(q, l)

)

= log2

(

∑2l+1−1n=0 |d−(l+1),n|q

2q/2∑2l−1

n=0 |d−l,n|q

)

.(7.2)

Analysis depends on showing that scaled versions of Z(q, l) are well-approximated by the mar-tingale and this is discussed next. Recall notational definitions (4.1), (4.2), (4.3), (4.6).

Proposition 7.1. For q > 0,Z(q, l)

clb−M(q, l)

P→ 0.

If q 6= q∗, the convergence is almost sure and if q < q∗, the convergence is in L2. Thus

Z(q, l)

clb→M(q,∞)(7.3)

in the appropriate sense, depending on the case.

Proof. Begin by writing

Z(q, l)

clb−M(q, l) =

∑

j|l

l∏

i=1

W q(j|i)c

[ |2W (j|l, 0) − 1|qb

− 1]

=∑

j|l

l∏

i=1

W q(j|i)c

ξ(j|l, 0)(7.4)

where ξ(j|l, 0) ⊥ ξ(p|l, 0) if j|l 6= p|l. Also Eξ(j|l, 0) = 0 and recall the notation from (4.10)

σ22(q) := Eξ2(j|l, 0) =

1

b2Var(|2W − 1|q).

If q < q∗, so a(q) < 1, then similar to the calculations leading to (6.3) and (6.4) we find

E(Z(q, l)

clb−M(q, l)

)2=∑

j|l

∑

p|l

E

(

l∏

i=1

W q(j|i)c

l∏

i=1

W q(p|i)c

ξ(j|l, 0)ξ(p|l, 0))

=σ22(q)

(2EW 2q)l

c2l(q)= σ2

2(q)al(q)

→0

as l → ∞ since a(q) < 1. This shows the L2–convergence.For q > 0, the same method shows

E(

(Z(q, l)

clb−M(q, l)

)2|Fl

)

=σ22(q)M(2q, l)al(q)


=σ22(q)

∑

j|l

l∏

i=1

W 2q(j|i)c2(q)

=: σ22(q)V (q, l),(7.5)

and we need to show V (q, l) → 0, almost surely as l → ∞. If the MKP condition fails, thenM(q, l) → 0 as l → ∞ and

V (q, l) ≤(

M(q, l))2 → 0.

If the MKP condition holds, then from Proposition 5.2 (iii), there exists r0 ∈ (1, 2) such thatar0(q) < 1, and for p = r0/2 ∈ (1/2, 1) we have by the triangle inequality

0 ≤ V (q, l)p ≤∑

j|l

l∏

i=1

W 2pq(j|i)c2p(q)

= (ar0(q))lM(r0q, l)

a.s.→ 0,(7.6)

as l → ∞, since M(r0q,∞) <∞ almost surely.So in all cases V (q, l) → 0. For any δ > 0, ǫ > 0 we have

P [∣

∣

∣

Z(q, l)

clb−M(q, l)

∣

∣

∣> ǫ|Fl] =P [

∣

∣

∣

Z(q, l)

clb−M(q, l)

∣

∣

∣> ǫ|Fl]1[V (q,l)σ2

2(q)>δ]

+ P [∣

∣

∣

Z(q, l)

clb−M(q, l)

∣

∣

∣> ǫ|Fl]1[V (q,l)σ2

2(q)≤δ]

≤1[V (q,l)σ22(q)>δ] + ǫ−2E

(

(Z(q, l)

clb−M(q, l)

)2|Fl

)

1[V (q,l)σ22(q)≤δ]

=1[V (q,l)σ22(q)>δ] + ǫ−2V (q, l)σ2

2(q)1[V (q,l)σ22(q)≤δ]

≤1[V (q,l)σ22(q)>δ] +

δ2

ǫ2.

Take expectations and use V (q, l)a.s.→ 0 and the arbitrariness of δ to conclude

Z(q, l)

clb−M(q, l)

P→ 0,

as l → ∞.For almost sure convergence, when q < q∗, we get from (7.6) that

V (q, l) ≤(

ar0(q)1/p)lM1/p(r0q, l)

and so∑

l V (q, l) <∞ almost surely. Thus, for any ǫ > 0,

∑

l

P [|Z(q, l)

clb−M(q, l)| > ǫ|Fl] ≤ ǫ−2

∑

E

(

(Z(q, l)

clb−M(q, l)

)2|Fl

)

= (const)∑

l

V (q, l) <∞,

and by a generalization of the Borel-Cantelli lemma ([16, page 152]) we have

Z(q, l)

clb−M(q, l)

a.s.→ 0.

For q > q∗, we prove almost sure convergence from Proposition 5.2 (iv) in a similar way.

We use this comparison result Proposition 7.1 to get consistent estimators of the structurefunction τ(q) in the subcritical case, by which we mean the case where the MKP condition holds.


Proposition 7.2. Define τi(q) for i = 1, 2 by (7.1) and (7.2). Provided q < q∗, so that the MKPcondition holds, both estimators are almost surely consistent for τ(q):

τi(q)a.s.→ τ(q), i = 1, 2,

as l → ∞.

Proof. In (7.3), take logarithms to the base 2 to get

log2 Z(q, l) − l log2 c(q) − log2 b→ log2M(q,∞),(7.7)

almost surely as l → ∞. Divide through by l to get consistency of τ1(q). To get the consistencyof τ2(q), note from (7.7) that

log2 Z(q, l + 1) − log2 Z(q, l) − (l + 1 − l)τ(q) → 0

almost surely which proves consistency of τ2(q).

8. Subcritical Asymptotic Normality of Estimators.

In this section we discuss second order properties of the estimators τi(q), i = 1, 2, definedin (7.1) and (7.2). The asymptotic normality for τ1(q) requires a bias term which cannot beeliminated. This drawback, is eliminated by using τ2(q), whose definition in terms of differencingremoves the bias term. However, take note of the suggestive remarks at the end of this Section8 about mean squared error.

For this section it is convenient to write EFl and PFl for the conditional expectation andconditional probability with respect to the σ-field Fl.

8.1. Asymptotic Normality of τ1(q). Begin by writing

Z(q, l)

clb−M(q, l) =

∑

j|l

l∏

i=1

W q(j|i)c

[ |2W (j|l, 0) − 1|qb

− 1]

(8.1)

=:∑

j|l

Z(j|l)(8.2)

where

EFl(Z(j|l)) =0

EFl(Z(j|l))2 =

( l∏

i=1

W 2q(j|i)c(2q)

)

al(q)σ22(q),

and recall σ22(q) is defined in (4.10). Therefore,

∑

j|l

EFl(Z(j|l))2 = M(2q, l)al(q)σ22(q).(8.3)

Our strategy for the central limit theorem is to regard Z(q,l)clb

−M(q, l) as a sum of random variableswhich are conditionally independent given Fl and then apply the Liapunov condition ([18]) forasymptotic normality in a triangular array.


Proposition 8.1. If 2q < q∗, then as l → ∞

PFl

[

Z(q,l)clb

−M(q, l)√

M(2q, l)al(q)σ22(q)

≤ x

]

→ P [N(0, 1) ≤ x] a.s.(8.4)

where N(0, 1) is a standard normal random variable with mean 0 and variance 1. Taking expec-tations in (8.4) yields

P

[

Z(q,l)clb

−M(q, l)√


≤ x

]

→ P [N(0, 1) ≤ x].(8.5)

Proof. By Proposition 5.2 (iii) there is δ > 0 such that both 2q + δ < q∗ and

a1+δ/2(2q) < 1.(8.6)

Asymptotic normality in (8.4) will be shown if we establish the Liapunov condition∑

j|lEFl |Z(j|l)|2+δ

(M(2q, l)al(q))(2+δ)/2→ 0 a.s.,(8.7)

where the denominator comes from (8.3). The numerator in the left side of (8.7) is boundedabove by

EFl

∑

j|l

∣

∣

∣

∣

∣

l∏

i=1

W q(j|i)c

∣

∣

∣

∣

∣

2+δ∣

∣

∣

|2W (j|l, 0) − 1|qb

− 1∣

∣

∣

2+δ

=c1∑

j|l

( l∏

i=1

W q(2+δ)(j|i)c((2 + δ)q)

)

cl((2 + δ)q)

cl(2+δ)(q)

=c1M((2 + δ)q, l)(a2+δ(q))l,

where

c1 = E∣

∣

∣

|2W (j|l, 0) − 1|qb

− 1∣

∣

∣

2+δ.

So the ratio in (8.7), apart from constants, is bounded by

M((2 + δ)q, l)(a2+δ(q))l

M(2q, l)1+δ/2(a2(q))(1+δ/2)l∼ M((2 + δ)q,∞)(a2+δ(q))

l

M(2q,∞)1+δ/2(a2(q))(1+δ/2)l.

Note that the two random variables M((2 + δ)q,∞) and M(2q,∞) are non zero with probability1 by Proposition 6.1. Check that

a2+δ(q)

(a2(q))1+δ/2= a1+δ/2(2q) < 1.

So the Liapunov ratio is asymptotic to a finite nonzero random variable times (a1+δ/2(2q))l where

a1+δ/2(2q) < 1 and the result is proven.

Remark 8.1. In the denominator of (8.5) we may replace M(2q, l) by its limit M(2q,∞). Thisfollows since almost surely 0 < M(2q,∞) <∞ for 2q < q∗ and thus

(

Z(q,l)clb

−M(q, l)√


,

√

M(2q, l)

M(2q,∞)

)

⇒ (N(0, 1), 1)


by [2]. The desired result is obtained by multiplying components.

Remark 8.2. Set

Nl :=

Z(q,l)clb

−M(q, l)√


.(8.8)

Then in R2, as l → ∞

(Nl, Nl+1) ⇒(

N1(0, 1), N2(0, 1))

,

where Ni(0, 1), i = 1, 2 are iid standard normal random variables.To see this, write for any x, y ∈ R

P [Nl ≤ x,Nl+1 ≤ y] =EPFl+1 [Nl ≤ x,Nl+1 ≤ y]

=E1[Nl≤x]PFl+1 [Nl+1 ≤ y].

By Proposition 8.1,PFl+1 [Nl+1 ≤ y] = Φ(y) + ǫl(y) a.s.

where Φ(y) is the standard normal cdf and where ǫl(y)L1→ 0 and |ǫl(y)| ≤ 2. So

P [Nl ≤ x,Nl+1 ≤ y] =E1[Nl≤x]

(

Φ(y) + ǫl(y))

=E1[Nl≤x]Φ(y) + o(1)

from the dominated convergence theorem, and hence we get convergence to

→Φ(x)Φ(y).

We now describe how this central limit behavior transfers to τ1(q).

Corollary 8.1. Under the assumptions in force in Proposition 8.1, we have(

τ1(q) − τ(q))

− l−1 log2 bM(q, l)√M(2q,∞)al(q)σ2

2 (q)

l log 2·M(q,l)

⇒ N(0, 1).(8.9)

Remark. The bias term l−1 log2

(

bM(q, l))

cannot be neglected.

Proof. For brevity, write

d(q) := M(2q,∞)al(q)σ22(q),(8.10)

and using the notation of (8.8) we have

Z(q, l) = clb(

Nl

√

d(q) +M(q, l))

.

Since

τ1(q) =1

llog2 Z(q, l),

we havelτ1(q) = l log2 c+ log2 b+ log2

(

Nl

√

d(q) +M(q, l))

and thus

l(

τ1(q) − τ(q))

= log2 bM(q, l) + log2

(

1 +Nl

√

d(q)

M(q, l)

)

.


Since by (5.6) and assumption 2q < q∗ we have q < q∗, we know that d(q) → 0. Therefore,

Nl

√

d(q)/M(q, l) → 0, and the desired result follows by using the relation log(1 + x) ∼ x forx ↓ 0.

The bias term in (8.9) is an unpleasant feature and thus we consider how to remove it bydifferencing.

8.2. Asymptotic Normality of τ2(q). We now consider the asymptotic normality of τ2(q). Itis possible to proceed from Proposition 8.1 but it turns out to be simpler to proceed with a directproof.

Proposition 8.2. Suppose 2q < q∗. Then

τ2(q) − τ(q)√M(2q,∞)al(q)σ2

3(q)

log 2·M(q,∞)

⇒ N(0, 1),(8.11)

where σ23(q) is defined in (4.11).

Proof. Begin by observing that

Z(q, l + 1)

cl+1b− Z(q, l)

clb

=∑

j|l

( l∏

i=1

W q(j|i)c

)

[

W q(j|l, 0)c

|2W (j|l, 0, 0) − 1|qb

+W q(j|l, 1)

c

|2W (j|l, 1, 0) − 1|qb

− |2W (j|l, 0) − 1|qb

]

=:∑

j|l

( l∏

i=1

W q(j|i)c

)

H(j|l)

where we have set

H(j|l) =W q(j|l, 0)

c

|2W (j|l, 0, 0) − 1|qb

+W q(j|l, 1)

c

|2W (j|l, 1, 0) − 1|qb

− |2W (j|l, 0) − 1|qb

.

(8.12)

Check that

EFlH(j|l) =1

2+

1

2− 1 = 0

and

EFlH2(j|l) =σ23(q).

It follows that conditionally on Fl we may treat

Z(q, l + 1)

cl+1b− Z(q, l)

clb


as a sum of iid random variables with (conditional) variance

EFl

(Z(q, l + 1)

cl+1b− Z(q, l)

clb

)2=∑

j|l

EFl

(

(

l∏

i=1

W q(j|i)c(q)

)

H(j|l))2

=∑

j|l

(

l∏

i=1

W 2q(j|i)c(2q)

)

al(q)EH2(j|l)

=M(2q, l)al(q)σ23(q).

As in the proof of Proposition 8.1 we may check the Liapunov condition and conclude

Z(q,l+1)cl+1b

− Z(q,l)clb

√

M(2q,∞)al(q)σ23(q)

⇒ N(0, 1),

or equivalently(

c−1 Z(q,l+1)Z(q,l) − 1

)

Z(q,l)clb

√

M(2q,∞)al(q)σ23(q)

⇒ N(0, 1),

and since Z(q, l)/(clb) →M(q,∞) we have(

c−1 Z(q,l+1)Z(q,l) − 1

)

M(q,∞)√

M(2q,∞)al(q)σ23(q)

⇒ N(0, 1).(8.13)

Since

c−1Z(q, l + 1)

Z(q, l)− 1

P→ 0,

it follows that

τ2(q) − τ(q) = log2

(

c−1Z(q, l + 1)

Z(q, l)

)

=log(

1 +(

c−1 Z(q,l+1)Z(q,l) − 1

)

)

log 2

∼c−1 Z(q,l+1)

Z(q,l) − 1

log 2

in probability. Combine this with (8.13) to complete the proof.

For statistical purposes, the result (8.11) contains unobservables so as in [20, 17], considerationneeds to be given to replacing quantities which are not observed by observable estimators. Weassume that the random measure µ∞ is observed, or equivalently that the wavelet coefficientsd−l,n are known. This means we have the quantities Z(q, l).

We define the following useful observable quantity

D2(q, l) =∑

j|l

l∏

i=1

W 2q(j|i)[

W q(j|l, 0)|2W (j|l, 0, 0) − 1|qZ(q, l + 1)

+W q(j|l, 1)|2W (j|l, 1, 0) − 1|q

Z(q, l + 1)


− |2W (j|l, 0) − 1|qZ(q, l)

]2

(8.14)

=:∑

j|l

l∏

i=1

W 2q(j|i)V 2(j|l)

=:∑

j|l

l∏

i=1

W 2q(j|i)[

A+B

Z(q, l + 1)− C

Z(q, l)

]2

.

Note that in this notation,

H(j|l) =A+B

cb− C

b,

where H(j|l) is defined in (8.12). Recall also that EH2(j|l) = σ23(q). In terms of the wavelet

coefficients we have

D2(q, l) =∑

j|l

[

|d(−l, (j|l, 0))|q2−q(l+1)/2

Z(q, l + 1)+

|d(−l, (j|l, 1))|q2−q(l+1)/2

Z(q, l + 1)− |d(−l, (j|l)|q2−ql/2

Z(q, l)

]2

,

(8.15)

showing that D2(q, l) is an observable statistic.

Corollary 8.2. Suppose 2q < q∗. Then

τ2(q) − τ(q)

D(q, l)/ log 2⇒ N(0, 1)(8.16)

as l → ∞.

Proof. Because of (8.11), it suffices to show

D2(q, l)

M(2q,∞)al(q)σ23(q)/M2(q,∞)

P→ 1,

as l → ∞. This is equivalent to showing

Z2(q,l)c2l(q)b2(q)

D2(q, l)


P→ 1.

After some simple algebra, this ratio is the same as∑

j|l

∏li=1

W 2q(j|i)c(2q)

M(2q, l)σ23(q)

[

A+B

bc

(

Z(q, l)/clb

Z(q, l + 1)/cl+1b

)

− C

b

]2

.

Since M(2q, l) → M(2q,∞), it suffices to show that the numerator converges in probability toM(2q,∞)σ2

3(q). Due to (7.3), we write the numerator as

∑

j|l

l∏

i=1

W 2q(j|i)c(2q)

(

[A+B

bc− C

b

]2+ op(1)2

(A+B

bc− C

b

)

+ op(1)2(A+B

bc

)2)

=I + II + III.


Quantiles of standard normal

tau1

-2 -1 0 1 20.0

40.0

60.0

80.1

00.1

20.1

4

10121416

q = 0.75, shape = 1

Quantiles of standard normalta

u2

-2 -1 0 1 2-0

.05

0.0

0.0

50.1

0

10121416

q = 0.25, shape = 1

Figure 1. Normal QQ-plots of τ1(q) − τ(q) (left) and τ2(q) − τ(q) (right).

As in Theorem 3.5 of [17],

I →M(2q,∞)E(A+B

bc− C

b

)2= M(2q,∞)σ2

3(q),

as desired. The terms I and II can readily be shown to go to 0.

Figure 1 shows normal QQ plots of τi(q) − τ(q), i = 1, 2, from simulated cascade data withbeta distributed cascade generator with shape parameter 1 (this makes the distribution uniform).The left plot is for q = 0.75 and the right is for q = 0.25. Each plot presents 4 graphs as thedepth l increases to 16. Note the better agreement of τ2(q) to normality compared with τ1(q).

Concluding remarks on mean squared error: Examining Corollary 8.1 and Proposition 8.2yields that in the region 2q < q∗, the conditional mean squared error of τ1(q) is of the form

O(1)p (al(q))2

l2+O

(2)p (1)

l2

while that of τ2(q) is O(3)p

(

al(q))2

.

9. Supercritical Asymptotics; Lack of Consistency

A critical issue with both the wavelet based estimator and the moment based ones used in [17],is that the asymptotic properties of the estimators are only valid in a certain range of q-values.For the wavelet estimators, we require q < q∗ for consistency and for the asymptotic normalityresults we require 2q < q∗. We now show that the range q > q∗ is uninformative for our estimatorsand in fact our estimators are misleading when extended to inference for values beyond q∗. Areliable estimate of q∗ would be valuable information. In place of such an estimate it is likelythat a graphical procedure is possible based on the following.


2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

q = 8 Z/max.Z shape = 10

1012141618

Figure 2. Density plots of Z(q, l)/Z∨(q, l).

Let τ∨i (q) (i = 1, 2) have the same definition as τi(q) except that sum is replaced by max. Thuswe can define by analogy with (4.3)

Z∨(q, l) =∨

j|l

l∏

i=1

W (j|i)q|2W (j|l, 0) − 1|q.

Note that

Z∨(q, l) =(

Z∨(1, l))q.

For large values of q, namely for q ≥ q∗, Z(q, l) is sufficiently well approximated by its largestsummand Z∨(q, l). Figure 2 presents a density plot of simulated values of Z(q, l)/Z∨(q, l) as thedepth l increases from 10 to 18; note the densities concentrate most mass around the point 1.The cascade generator is a beta distribution with shape parameter 10. Based on the idea ofapproximating Z(q, l) by Z∨(q, l), since logZ∨(q, l) = q logZ∨(1, l) is linear in q, we anticipatethat τ1(q) should also be linear in q rendering τ1(q) largely uninformative for inference purposesin the q ≥ q∗–region. A rough estimate of q∗ would be provided by the q-value where the plotsof τ1(q) starts to look linear.

Computer simulations offer strong support for these remarks. Figure 3 shows overlaid simulatedvalues for τi(q), τ

∨i (q), i = 1, 2 for large values of q. In the range of q–values beyond q∗ ≈ 3.3, it is

remarkable how linear the plots for τ1(q) and τ∨1 (q) look and also how closely τ∨1 (q) approximatesτ1(q). Note the values in the plots have been multiplied by -1 to make the plots increasing andthe cascade generator is a beta distribution with shape parameter 1.

We now assume that q∗ <∞ and examine this supercritical phenomenon when q ≥ q∗ in moredetail. We will prove the asymptotic linearity of the estimator τ1(q) for q ≥ q∗. In particular, theestimator τ1(q) is not consistent when q > q∗, and neither is the estimator τ2(q).


q

tau(

q)

0 2 4 6 8 10

-10

12

34 tau.1

tau.2theory

max.1max.2

levels = 16, q.step = 0.5, shape = 1

Figure 3. Plots of τi(q), τ∨i (q) for q∗ ≈ 2.4.

We start by introducing new notation. Let

U(q, l) = c(q)lM(q, l) =∑

j|l

l∏

i=1

W (j|i)q, q > 0, l ≥ 1 ,(9.1)

U∗(l) = maxj|l

l∏

i=1

W (j|i), l ≥ 1 ,(9.2)

and define, for q > 0,

m(q) = lim supl→∞

1

llog2 U(q, l) ,(9.3)

m(q) = lim infl→∞

1

llog2 U(q, l)

as well as

m∗ = lim supl→∞

1

llog2 U

∗(l) ,(9.4)

m∗ = lim infl→∞

1

llog2 U

∗(l) .


It is immediate that for all q > 0 and 0 ≤ θ ≤ q

(U∗(l))q ≤ U(q, l) ≤ (U∗(l))θ U(q − θ, l) ≤ 2l (U∗(l))q .(9.5)

In particular, for every q > 0

m(q) − 1 ≤ qm∗ ≤ m(q) ,(9.6)

m(q) − 1 ≤ qm∗ ≤ m(q)

almost surely.Note that it follows from Proposition 6.1 that for 0 < q < q∗

m(q) = m(q) = τ(q) .(9.7)

Since by the triangle inequality for all 0 < ρ < 1 and q > 0

(U(q, l))ρ ≤ U(ρq, l) ,(9.8)

we see that

m(ρq) ≥ ρm(q) .(9.9)

For a q ≥ q∗ and 0 < ρ < q∗/q we hence get

m(q) ≤ 1

ρm(ρq) =

1

ρτ(ρq) ,

and letting ρ ↑ q∗/q we conclude that for every q ≥ q∗

m(q) ≤ qτ(q∗)

q∗.(9.10)

On the other hand, it follows from (9.5) that for all q > 0 and 0 ≤ θ ≤ q

m(q) ≤ θm∗ +m(q − θ) .

Using (9.6) we obtain from here

m(q1) ≤ θm(q2)

q2+m(q1 − θ)

for all q1, q2 > 0 and 0 ≤ θ ≤ q1. In particular, if 0 < q1 < q∗, then for every 0 < q3 < q1 wechoose θ = q1 − q3 and conclude, using (9.7), that

m(q2)

q2≥ m(q1) −m(q3)

q1 − q3=τ(q1) − τ(q3)

q1 − q3.

Therefore, for all q > 0

m(q)

q≥ sup

0<p<q∗τ ′(p) .(9.11)

However,

sup0<p<q∗

τ ′(p) = sup0<p<q∗

E(W q log2W )

E(W q)≥ E(W q∗ log2W )

E(W q∗)

=1

q∗(1 + log2E(W q)) =

τ(q∗)

q∗


by the definition of q∗. Substituting into (9.11) immediately gives us

m(q) ≥ qτ(q∗)

q∗(9.12)

for all q > 0. Comparing (9.12) with (9.10), we see that

m(q) := liml→∞

1

llog2 U(q, l) = q

τ(q∗)

q∗(9.13)

for any q ≥ q∗. Moreover, using (9.6) with q → ∞ and (9.13) we immediately conclude that

m∗ := liml→∞

1

llog2 U

∗(l) =τ(q∗)

q∗.(9.14)

Remark 9.1. For the non–conservative cascades, for which the random variables

W (j|l), j ∈ 0, 1∞, l ≥ 1are iid, a statement analogous to (9.14) is equivalent to the so called first birth problem; see forinstance [9]. For the particular case of uniformly distributed W in the context of conservativecascades see also [11].

We are now ready to establish the asymptotic behavior of the estimator τ1(q) = τ1(q, l) in thesupercritical case.

Theorem 9.1. Let q ≥ q∗. Then, as l → ∞,

τ1(q, l) → qτ(q∗)

q∗a.s.(9.15)

In particular, the estimator τ1(q, l) is not a consistent estimator of τ(q) if q > q∗.

Proof. Denote

mZ(q) = lim supl→∞

1

llog2 Z(q, l)

and

mZ(q) = lim infl→∞

1

llog2 Z(q, l) .

Since Z(q, l) ≤ U(q, l) for all q and l, we immediately conclude by (9.13) that

mZ(q) ≤ m(q) = qτ(q∗)

q∗.(9.16)

For the corresponding lower bound on mZ(q), note that since P (W 6= 1/2) > 0 and P (W = 0) <1/2 (otherwise q∗ = ∞), there is a θ > 0 such that

p1 := P (|2W − 1| ≥ θ) > 0 and p2 := P (min(W, 1 −W ) ≥ θ) > 0 .

Let 0 < ǫ < 1. Note that it follows from (9.14) that for all l large enough,

P(

U∗(l) ≥ 2(1−ǫ)lτ(q∗)/q∗)

≥ 1

2.(9.17)

For l ≥ 1 let

Nl = card

j|l : W (j|i) ≥ θ for all i = 1, . . . , l

.


By definition N0 = 1. Observe that for all l ≥ 0

Nl+1 = Nl +Ml ,

where, given N0, N1, . . . , Nl, the distribution of Ml is Binomial with parameters Nl and p2.Therefore, (Nl) is a supercritical branching process with progeny mean m = 1 + p2 > 1 andextinction probability 0. By Theorem I.10.3 of [1], page 30,

liml→∞

Nl

(1 + p2)l= N > 0 a.s. .(9.18)

Let now 0 < δ < 1. It follows by the definition of (Nl) that for every l ≥ 1

Z(l, q) ≥ θ[δl]q maxk=1,... ,N[δl]

U∗k

(

l − [δl])q|2W (l)

k − 1|q ,(9.19)

where(

U∗k

(

l − [δl])

, k ≥ 1)

are iid with the law of U∗(

l − [δl])

and

(W(l)k , k ≥ 1) are iid with the law of W .

The two sequences are independent, and also independent of N[δl]. All the random variablesdefined above can be assumed to be defined, for all l and k, on the same probability space(Ω,F , P ).

We introduce several events. Let d = (1 + p2)1/2 > 1. Put

Ω1 =

Nl ≥ dl for all l large enough

.

It follows from (9.18) that P (Ω1) = 1. Let, further,

Ω(l)2 =

d[δl]⋃

k=1

|2W (l)k − 1| ≥ θ and U∗

k

(

l − [δl])

≥ 2(1−ǫ)(l−[δl])τ(q∗)/q∗

,

l ≥ 1. Note that by (9.17) we have P (Ω(l)2 ) ≥ 1− e−cδl for some c > 0 and all l ≥ 1, and so letting

Ω2 = lim infl→∞

Ω(l)2 ,

we see by Borel-Cantelli lemma that P (Ω2) = 1. Therefore, P (Ω1 ∩ Ω2) = 1 as well. However,for every ω ∈ Ω1 ∩ Ω2 we have by (9.19)

Z(l, q) ≥ θ[δl]q2q(1−ǫ)(l−[δl])τ(q∗)/q∗θq

for all l large enough, which implies that

mZ(q) ≥ qδ log2 θ + (1 − ǫ)(1 − δ)qτ(q∗)

q∗a.s. .

Letting δ → 0 and ǫ→ 0 we conclude that

mZ(q) ≥ qτ(q∗)

q∗.(9.20)

Now the statement (9.15) follows from (9.16) and (9.20).


Finally, it follows from part (iv) of Proposition 5.2 that

τ(q) > qτ(q∗)

q∗

for all q > q∗. Hence, the estimator τ1(q, l) is not a consistent estimator of τ(q) if q > q∗.

Here is an immediate corollary.

Corollary 9.1. The estimator τ2(q, l) is not a (strongly) consistent estimator of τ(q) if q > q∗.

Proof. Notice that for every l ≥ 1,

τ1(q, l) =1

l

l−1∑

j=0

τ2(q, j) ,

where Z(q, 0) = 1. Therefore, if for some q > q∗ τ2(q, l) → τ(q) a.s. as l → ∞, then so doesτ1(q, l), which contradicts Theorem 9.1.

An estimator related to τ2(q, l) is

τ3(q, l) = log2

(

U(q, l + 1)

U(q, l)

)

:= log2R(q, l), l ≥ 1 .

Since

1

llog2 U(q, l) =

1

l

l−1∑

j=0

τ3(q, j) ,(9.21)

where U(q, 0) = 1, (9.13) and the same argument as that of Corollary 9.1 shows that τ3(q, l)is not a strongly consistent estimator of τ(q) if q > q∗ (even though it is a strongly consistentestimator of τ(q) if q < q∗). We can say more, however. Note that 0 ≤ R(q, l) ≤ 2 for all q andl. Furthermore,

ER(q, l) = c(q) = 2τ(q) for all q and l .

Therefore, if for some q > q∗, τ3(q, l) converges a.s. to some limit τ3(q) as l → ∞, then the 2τ3(q)

must have a finite expectation equal to 2τ(q). On the other hand, by (9.13) and (9.21) we musthave τ3(q) equal to qτ(q∗)/q∗ a.s.. This contradiction shows that τ3(q, l) cannot converge a.s. asl → ∞ if q > q∗.

We conjecture that the same is true for τ2(q, l), in the sense that it does not converge a.s. asl → ∞ if q > q∗. A possibility is that τ2(q, l) converges in probability, and is weakly consistentfor q > q∗. Whether or not this is true remains an open question.

10. Concluding Remarks

While Ossiander and Waymire’s estimator for τ(q) is consistent for random cascades, we alsocheck empirically by simulation that it is an appropriate time domain method for conservativecascades. By time domain estimator we mean

τtime(q) =1

llog2

(

∑

j|l

|µ∞(I(j|l))|q)

.


q

tau(

q)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-1.0

-0.5

0.0

0.5

1.0 tau.1

tau.2theorytime.one

levels = 16, q.step = 0.1, instances = 50, shape = 1

Figure 4. Plots of the two wavelet estimators and the time domain estimator forτ(q) with q < q∗ ≈ 3.3.

We show in Figure 4 that the time domain estimator gives equally good results compared withthe two wavelet estimators. The cascade generator is a beta distribution with shape parameter1. The plot is for q values below q∗ ≈ 3.3. Note that the τ values are multiplied by -1.

One of the advantages of the wavelet method is its ability to filter deterministic trends becausedifferent wavelet families have different vanishing moments; i.e., they are orthogonal to low degreepolynomials. The Haar wavelets are “blind” to additive constants. Figure 5 illustrates the failureof the time domain method to cope with the presence of an additive constant. The cascade isgenerated with a beta distribution of shape parameter 1 then a fixed constant 0.1 is added to thecascade. The wavelet estimators give the same values regardless of the presence of the additiveconstant.

Do our wavelet methods work with other wavelet families. One reason for using other wavelets isthat the Haar wavelets have only one vanishing moment and can remove only an additive constant.Other wavelet families with higher vanishing moments can remove higher degree deterministictrends. Empirical simulated evidence (shown in Figure 6) suggests that other wavelets do indeedwork. Figure 6 shows that τ1 works quite well in the case of the D4 wavelet and the presenceof an additive linear trend. Note that the D4 wavelet has four vanishing moments (i.e., it isblind to cubic polynomials). The time domain method performs poorly as does τ2. Theoreticalinvestigations are necessary to confirm the validity of the wavelet method for wavelets other thanthe Haar.


q

tau(

q)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-1.0

-0.5

0.0

0.5

1.0

tau.1tau.2theorytime.trend.one

levels = 16, q.step = 0.1, instances = 50, shape = 1, trend added

Figure 5. Plots of the wavelet and time domain estimators for a cascade withan additive constant.

References

[1] K. Athreya and P. Ney. Branching Processes. Springer-Verlag, New York, 1972.[2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.[3] A. Feldmann, A. C. Gilbert, and W. Willinger. Data networks as cascades: Investigating the multifractal

nature of Internet WAN traffic. In Proc. of the ACM/SIGCOMM’98, pages 25–38, Vancouver, B.C., 1998.[4] U. Frisch and G. Parisi. Fully developed turbulence and intermittancy. In M. Ghil, editor, Turbulence and

Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland, Amsterdam, 1985.[5] A.C. Gilbert, W. Willinger, and A. Feldmann. Scaling analysis of conservative cascades, with applications to

network traffic. IEEE Transactions on Information Theory, 45(3):971–991, 1999.[6] V. K. Gupta and E. C. Waymire. A statistical analysis of mesoscale rainfall as a random cascade. Journal of

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random cascades. Ann. Appl. Probab., 2(4):819–845, 1992.[8] J.P. Kahane and J. Peyriere. Sur certaines martingales de b. mandelbrot. Advances in Mathematics, 22:131–

145, 1976.[9] J.F.C. Kingman. The first birth problem for an age–dependent branching process. The Annals of Probability,

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C.R. Doklady Acad. Sci. URSS (N.S.), 30:299–303, 1941.[11] H. Mahmood. Evolution of Random Search Trees. Wiley, New York, 1992.


q

tau(

q)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-10

12 tau.1

tau.2theorytime.trend.one

levels = 16, q.step = 0.1, instances = 50, shape = 1, trend added, D4

Figure 6. Plots of the wavelet and time domain estimators using D4 waveletsand in the presence of an additive linear trend.

[12] B. B. Mandelbrot. Intermittant turbulence in self-similar cascades: Divergence of high moments and dimensionof the carrier. Journal of Fluid Mechanics, 62:331–358, 1974.

[13] B. B. Mandelbrot. Limit Lognormal Multifractal Measures. In Gotsman, Ne’eman, and Voronel, editors, TheLandau Memorial Conference, pages 309–340, Tel Aviv, 1990.

[14] B.B. Mandelbrot. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer-Verlag, NewYork, 1998.

[15] C. Meneveau and K. R. Sreenivasan. Simple multifractal cascade model for fully developed turbulence.Phys. Rev. Lett., 59:1424–1427, 1987.

[16] J. Neveu. Discrete-Parameter Martingales, volume 10 of North-Holland Mathematical Library. North Holland,Amsterdam, 1975. Translated from the French original by T.P. Speed.

[17] M. Ossiander and E. Waymire. Statistical estimation for multiplicative cascades. To appear; available fromossiand,[email protected], 1999.

[18] S.I. Resnick. A Probability Path. Birkhauser, Boston, 1998.[19] R. H. Riedi and J. Levy-Vehel. Tcp traffic is multifractal: A numerical study. Preprint, 1997.[20] Brent M. Troutman and Aldo V. Vecchia. Estimation of Renyi exponents in random cascades. Bernoulli,

5(2):191–207, 1999.


School of Operations Research and Industrial Engineering and Department of Statistical Sci-

ence, Cornell University, Ithaca, NY 14853

E-mail address: [email protected], [email protected]

AT&T Labs–Research, 180 Park Avenue, Florham Park, NJ 07932

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Date post:	04-Jan-2017
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WAVELET ANALYSIS OF CONSERVATIVE CASCADES 1 ...

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