C. R. Acad. Sci. Paris, t. 326, Sikie I, p. 365-370, 1998

ProbabilitCs/Probabi/ify Theory

Two-point statistics of statistically self-similar cascades

Robert VOJAK

CEREMADE: universitt Paris IX, Dauphine. place du MarCchal-de-Lattre-de-Tassigny, 75775 F’aris

crdex 16, France

E-mail: vojak@ceremade.dauphine.fr URL : http://www.reremade.dauphine.fr/-vojak

(Reyu lr 20 juillet 1997, accepti apr&s &vision le 5 janvier 1998)

Abstract. The classical multifractal analysis is extented to higher orders in order to study the second order correlations existing between local dimensions of a class of random self-similar measures (in the sense of Arbeiter and Patzschke).

Statistiques de dew&me ordre sur une classe de mesures

alhatoires auto-similaires

R&urn& L’analyse multifractale classique est hendue aux ordres suptfrieurs ajin d’e’tudier les corrP1ation.s de second ordre entre les dimensions locales d’une classe de mesures al&atoires auto-similaires (au sens de Arbeiter et Patzschke).

Version frangaise abr&6e

On considiire une mesure ,U (dkterministe ou non) positive et dCfinie sur [0, 1). Soient b, n deux entiers, b > 2. On considkre l’ensemble C, = (0,. . . : b - l}” et C, = U,, C,, l’ensemble des mots de longueur finie. Pour tout 0 5 i < b, on note CT * 1: la concatCnation du mot 0 et de la lettre i (dans cet ordre).

Pour 71 don+, l’intervalle [0, 1) est partitionne en b” intervalles b-adiques I(n), u E C,, de longueur b-‘” :

I(u) = &qb-“,~~ib-i+~-” (1) i=l i=l

Si t E [0, l), on note I”(t) l’intervalle de la n-i6me gCnCration contenant t.

Note prCsentCe par Paul DEHEUVEIS.

0764-4442/98/03260365 0 AcadCmie des Sciences/Elsevier, Paris 365

R. Vojak

Now nous proposons de raffiner l’analyse multifractale classique de mesures en generalisant la notion de spectre multifractal. Par exemple, si o(t) designe la dimension locale de IL au point t, nous Ctudions la dimension de Hausdorff des points t pour lesquels les couples (n(t), cu(t + S)) (S > 0) coincident. De possibles applications de ces nouveaux spectres du second ordre concernent la prediction de dimensions locales et la caracterisation de mesures par ces nouveaux spectres (voir [lo]).

On s’interesse ici aux correlations d’ordre deux entre les dimensions locales de p calculees en des points de l’intervalle [O, I), dont les positions sont regulees par une suite (T~)~, donnee. Le terme general de cette suite est donne par T,, = b-‘“,l, oti (u,,). est une suite d’entiers, croissante avec u,, 5 n et telle que /j = lim 71.-l ‘uu,, existe. Ces correlations sont mesurees par les spectres de

Hausdorff Fi*’ et de Legendre .f, ‘;*) (le chiffre 2 designant l’ordre de la correlation) :

n(*)(t) = (

li;n logp(lTZ(t)) log- ,r,~(t), , limsup

log p(P(f + r,,))

n log jI”(1: + 7.,,)) 1 ’

K(*)(aJk’) = t E [I), 1); J*)(t) = (a,w’) { >

,

Fi*‘(n, a’) = dimK(*)(,, (2’) si K(*)(ry: (Y’) # Cn,

Fj2)(tr: a’) = -cm si Kc2)(o, a’) = c?.

p(c”, 0’) = jn; (trq + n’q’ - Tyq. q/q) 1

~(~~(4; q’) = - limsupn-l log, E C cL(I(0))lr /L(l(c))y’ r, ( UEC,,

Le mot ii E C, est donne par la relation t E I(o) e I: + T,, E I(5).

(2)

(3)

(4)

1. Cascades albatoires

Nous nous interessons ici a une classe de mesures aleatoires : les cascades statistiquement auto- similaires. On suppose donnees b variables aleatoires pa! . , p&r definies sur un espace de probabilite

b-l

(bZ,.T>Pr), et telles que E ( >

C p; = 1. Posons z)(0) = p(I(0)) pour tout IT E C,. On suppose par 7.=0

ailleurs que 0 < p(E) < cc presque sfirement. La mesure /L est une cascude statistiquement auto-

similaire si, pour tout cr E C,, le vecteur aleatoire I

p(r * 1:) ___ : 0 5 I < b

P(O) I est distribue selon la loi

du vecteur {pi ; 0 5 i < b}. On suppose Cgalement que les vecteurs aleatoires i

p(CJ * 1:) --;o:g%<b

Pl)(fl) sont independants pour differentes valeurs de (T E C,.

I

Cette classe de cascades aleatoires est un cas particulier de mesures Ctudiees par Arbeiter et Patzschke [I], et ont Cte considerees par by F. Ben Nasr [2], generalisant la construction de la martingale de Mandelbrot [6], [7], [8], [I?], [4].

Les proprietes classiques du premier ordre de ces mesures peuvent etre trouvees dans [3], pp. 684-689, et sont rappelees dans la proposition 1.

2. Rikltats

POsOns (X0 = -E(lo&pa) et ab-1 = -~(lO&p&l). Dans le cas b = 2, Qa et (Yr sont les eXtr6mith

du support du spectre du premier ordre fi” (voir la definition (5)). Les resultats obtenus sont les suivants (les preuves completes sont disponibles dans [lo]) :

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Two-point statistics of statistically self-similar cascades

THBOR~ME 1

0 Si a0 < obP1, on a presque stirement, F~“((Y, a’) = si cf’ = a’, si a # a’.

l si a0 > fib-l, on a presque shement :

FL2)(o,a’) = si CY = a’, si a = [r&l et a’ < a(1 - (3) + ,hO.

sinon.

Par consequent, le spectre de Hausdorff du deuxieme ordre n’apporte aucune information sur les relations entre dimensions locales voisines.

Notons l’usage d’une limsup dans la definition de F12’. L’usage d’une lim simple donne lieu a

un spectre valant fjl)(n) si 0 = u’ et -cc sinon.

TH~OREME 2. - On n :

f/2)(“: a’) “” ininla(nq + dq’ - d2)(q,q’)),

avec

b-l b-l

Tc2)($ 4’) = -i 3 10 !?ab c ,+fiq’) -( 1 - B) log, c E($jE (p;‘) - P InaX cL(q> q’h ‘),

i=o i=o

+)(q + 9’) + lo& E(&

En revanche, le spectre de Legendre du deuxieme ordre presente des termes non triviaux. On gagne de I’information en mesurant les correlations entre dimensions locales par l’intermediaire des exposants de RCnyi d2)(q, 4’).

L’obtention de ces deux resultats est basee sur la decomposition b-adique de la somme d’un reel de [0, 1) et de b-‘“. Si t = 0.tlt2 . . ., alors t + 6” = O.trt2.. . (1 + t,,-l-i)O.. .Ot,, . . ., ou 1 = l(t, u) est le nombre de chiffres successifs Cgaux a b - 1 avant le rang r~ (non compris). On a lim inf Z(t, ~J)/T& = 0 et lim sup I(t, TX)/TL peut prendre n’importe quelle valeur dans [0, 11. C’est cette distinction entre lim inf et lim sup qui est a l’origine des deux cas du theoreme I.

3. Quelques rikltats plus gCnCraux

Les definitions donnees ici concernent des statistiques d’ordre deux, se generalisent a des ordres superieurs et dans un cadre plus general (voir [lo]). Les intervalles consider& ne sont plus b-adiques, et la suite (T,),, est arbitraire.

Dans [9], page 80, il a CtC montre que le spectre de Hausdot-ff ne permettait pas de caracteriser de facon unique la mesure sous-jacente. Le resultat est en fait bien plus precis (voir [lo]). II y est prouve qu’en general, une mesure ne peut etre caracteride de man&e unique B partir de tous ses spectres de correlations (d’ordre fini). On y montre comment construire une classe de mesures dont tous les spectres d’ordre tini coincident.

Enfin, les spectres de second ordre de cascades multiplicatives deterministes sont calcules, cascades dont on sait qu’elles verifient le formalisme multifractal classique (du premier ordre). En revanche, le formalisme multifractal du second ordre ne tient plus.

367

R. Vojak

We aim to refine the classical multifractal analysis of measures by generalizing the notion of multifractal spectrum. More precisely, for a given measure, if a(t) denotes its local dimension at t, we study the Hausdorff dimension of points having the same vector (a(t), a(t + 6)) (6 > 0) of local dimensions. Possible applications of such second order spectra for local dimensions prediction or the characterization of measures have been pointed out in [IO].

Let p be a positive (deterministic or random) measure with support in [0, 1). Let b and n be two integers, b, TL >_ 2, and C = (0, , . , . b - l}” be the code space over the indices 0,. . , b - 1,

L = (0,. . . , b - l}“, and C, = U,, C,, be the space of all finite sequences with entries from 0 . . ; b-l. Fora = (ai ,... :gTL) EC,, let 1~71 = 7~ denote the length of g, and a/X: = (gi, . . . ,ak), k’ i n, denote the truncation of n to the length k:. For 0 5 i < b, we note c * ,1 = (ITS! . . . , (I~, i) the juxtaposition of the word D and the digit %.

In addition, let I( (~1, . , aT1) (or simply I(a)) be the n-th generation b-adic interval of sidelength b-” (see Definition I). For t E [0, 1) and 7). E N, let I”(t) be the n-th generation b-adic interval containing t.

Let r = (r.,l)rlE~ be a sequence of (0,l) whose general term is ‘r,& = b-“sl, where (u,~,),, is a non-decreasing sequence of integers such that ‘u,, < 7~ and such that /j = lim,,, ,u,,/u exists (the

case u,, > 7~ is trivial).

1. Second order spectra

For all t E [O! l), let I be defined by (2) when the considered limit exists. Let (n, n’) E R+ x R+. Set K(2)(tu: (Y’) = {t E [0,1)/a: (‘j(t) = (0, cr’)} (the number 2 represents the order of the

correlation). The second order Huusdoflspectrum of /L is defined as FL2)(n, 11’) = dim Kc2)(rr, ~2’) if K(*)(c~! a’) # 0 and --o(j otherwise. Let r(‘)( 9, y’) be the R&y’ e p 1 x onent defined by (4). The word 6 E C, is given by the relation t E I(u) m t + T,, E I(6). We define the second order Legendre

(2) spectrum f, of IL as the Legendre transform of rc2) (see (3)).

2. Random cascades: definition and properties

In this section, we recall the definition of homogeneous statistically self-similar measures in the sense of Arbeiter and Patzschke [I], and give some of their first order multifractal properties.

2.1. Definition and notations

We assume that we are given b positive random variables pa, , . , pb-1 defined on a probability b-l

space (62, F, Pr) such that E c p.~ ( >

= 1. i=O

Let j~ be a positive random measure defined on E = [0, l), and let l>(g) = ,u(~((T)) for all cr E C,. We assume that 0 < p(E) < CC with probability 1.

The measure p is a statistically self-similar cascade if, for every D E C,, the random vector

i

p(fl* ;I

P(O) : 0 < % < b

1 has the same joint distribution as the vector 1~; ; 0 5 i < b}.

We also assume that the vector random variables

distinct g E C,. 1

~(ff * i) ~ ; 0 5 i < b P(a) I

are independent for

This class of random cascades is a particular case of the statistically self-similar measures studied by Arbeiter and Patzschke [l], and was considered by F. Ben Nasr [2], generalizing the construction of Mandelbrot’s martingale [6], [7], [8]. [5], [4].

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Two-point statistics of statistically self-similar cascades

2.2. First order properties of the random cascades

Many first order properties of random cascades can be found in [3], pp. 684-689. We only give results necessary to the computation of the multifractal correlation spectra.

Let t E [0, l), and g(t) E C such that t E l(a(t)]n) f or all n, i.e. g(t) is the b-adic expansion of t (only terminating ones are considered here). For each % E (0,. . . : b - l}, let

be the relative frequency of the digit i in the b-adic expansion of t up to order n. It is well-known that, for almost all t E [0, I), the limit cp.i(t) = ,,‘i, im cpi(t, n) exists. The random cascades possess

the following first order properties:

PROPOSITION 1 b-l

1. For almost all t E [0, l), lim 71 -’ logb p,(r’(t)) = &go qi(t) E(logb p.l) with probability 1.

2. Let q E W be such that E[p(E)g) < cc. We have

r(l)(y) ‘Z - limsuprl-l logbE n

3. .fjl’(cy) “g inf, (~19 - (

b-l

T(I)(q)) = sup - c vi log, pi , i=o >

the supremum being taken over the

b-l b-l

set C

vi 2 0: Lgo (Pi = 1, %F:, viE@gb PL) = -Cl I

4. On the domain of f,(,“, we have, with probability 1,

f:‘)(n) %’ dim {t E [O? 1) ; - li;u n-l log, p(P(t)) = LY} = f/‘)(o). (5)

3. Results

Set o. = -E(lo&,pa) and dZ&1 = -E(log,p&l). When b = 2 for instance, o. and cyl represent the boundary of the support of the first order spectrum fi”. We have the following results (the proofs are given in [lo]):

THEOREM 1 (Fh(‘) spectrum) l If (~0 < ab-1, we have, with probability 1:

p(Q) iftv = fi’, -cm if0 # a’.

l If a0 > a&l, we have, with probability 1:

Fh(2)(cl,a’) = f(l)(a) ifa! = a’ oh if 0 = nb-1 and (Y’ < (~(1 -- 0) + @ao, --x otherwise.

Thus, the second order Hausdorff spectrum does not give much information about “correlations” between local dimensions. Note that changing limsup for lim in the definition of FL” worsens the

shape of the spectrum. In that case, the spectrum is equal to f:“(n) if cu’ = a, and -cc otherwise.

369

R. Vojak

THEOREM 2 (Legendre spectrum). - We have:

fi(2)(cl.,a’) = ,JnnR(aq+a’q’-T(2)(q>q’)),

with

b-l b- 1

P)(q, q’) = -Iljllog, c E (PYt”) -( 1 - /-I) log, c E(p:)E(p;‘) - pmax (L(q, q’), O), id) i=O

qq. q’) +J $1) (q + 4) + ~~Ei, E (d-1) E (1);‘)

The second order Legendre spectrum is not as trivial as the Hausdorff spectrum. Clearly, information is gained through the computation of the RCnyi exponents Tc2)(q, q’).

The proofs of both theorems are based upon the b-adic expansion of the sum of a real number in [O,l) and b-“. If 1’ = O.trta.. .t,, . . ., then t + b-” = O.trt*. (1 + t,-l-t)O.. .Ot,. . . with 1 = I(t, 71) being the number of successive digits equal to b - 1, starting backwards from position 71 - 1. We have liru inf Z(t, 7),)/n = 0, and limsup I(t, n)/n can be any real number in [O? I], which explains why we have to consider two cases in the statement of Theorem 1.

4. Further results

Two-point statistics have been extended to finite order statistics in [IO]. General definitions of spectra are considered. The intervals are no longer b-adic, and the sequence (,r,),, is arbitrary.

In [9], p. 80, it was shown that the sole Hausdorff spectrum does not fully characterize the underlying measure. In [lo], it is proven that, in general, a measure cannot be uniquely recovered from the knowledge of finite order spectra. It is shown how to construct a class of measures having the same finite order Hausdorff spectra.

Finally, second order spectra of deterministic cascades are computed, and it is shown that the second order multifractal formalism does not hold in general.

References

[I] Arbeiter M., Patzschke N.. Self-Similar Multifractals, Math. Nachr. 181 (1886) 542. [2] Ben Nasr F., Mesures aleatoires de Mandelbrot associees a des substitutions, C. R. Acad. SC. Paris 304 (10) Serie 1

(1987) 255-258. [3] Falconer K.J., The Multifractal Spectrum of Statistically Self-Similar Measures, J. Theor. Probability 7 (3) (1994) 681-702. [4] Holley R., Waymire E., Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl.

Prob. 2 (4) (1992) 819-845.

[5] Kahane J.-P., Peyriere J., Sur Certaines Martingales de B. Mandelbrot, Adv. Math.. 22 (1979) 131-145. [6] Mandelbrot B.B., Intermittent turbulence in self similar cascades : divergence of high moments and dimension of the

carrier. J. Fluid Mech. 62 (1974) 331-358. 171 Mandelbrot B.B., Multiplications aleatoires it&es et distributions invariantes par moyenne pond&&e aleatoire, C. R. Acad.

Sci. Paris A 278 (1974) 2X9-292.

181 Mandelbrot B.B., Multiplications aleatoires it&&es et distributions invariantes par moyenne ponderer aleatoire : quelques extensions, C. R. Acad. Sci. Paris A 278 (1974) 355-358.

191 Vojak R., Analyse et modelisation multifractales de signaux complexes : applications au tratic routier, PhD thesis, University of Paris IX Dauphine, January 9 1996.

[IO] Vojak R., Higher order multifractal analysis of deterministic and statistically self-similar cascades. preprint, 1996.

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