Multifractal analysis in a mixed asymptotic framework

Submitted to the Annals of Appl ie d P r ob abil ity MUL TIFRA CT AL ANAL YSIS IN A MIXED ASYMPTOTI C FRAMEW ORK By Emmanuel Ba cr y , Ar n aud Gloter , Marc Hoffmann and Jean Franc ¸ ois Muzy Ec ole P olyte chnique, U niversit ´ e Paris-Est and Universit ´ e de Corse Multifractal analysis of m ultiplicativ e random cascades is revis- ited within the framew ork of mixe d asymptotics . In this new frame- w ork, statistics are estimated ov er a sample which size increases as the resolution scale ( or the sampling perio d) b ecomes fi ner. This al- lo ws one to conti nuously interpolate b etw een the situation where one studies a single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (num ber of cascades realizations) b e- comes infinite. W e show t hat scaling exp onents of ”mixed” partitions functions i.e., the estimator of the cumulan t generating function of the cascade generator distribution, dep ends on some “mixed asymp- totic” exp onent χ resp ectively ab ov e and b eyond tw o critical va lue p − χ and p + χ . W e stu dy the con verg ence prop erties of partition functions in mixed asym totics regime and establish a cen tral limit theorem. These results are shown to remain v alid within a general wa v elet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreov er, within the mixed asymptotic framew ork, we establish a “box-coun ting” multifractal formalism that can b e seen as a rigorous formulatio n of Mandelbrot’s negative d imension theory . Numerical illustrations of our purp ose on sp ecific examples are also provided. 1. Intro duction. Multifractal pro cesses hav e b een used successfully in man y applications whic h inv olve series with inv ariance scaling prop erties. W ell k n o wn exa mples are fully devel op ed turbu lence wh ere suc h processes are used to mod el the v elocit y or the dissipation energy fields [6 ] or finance, where they ha v e b een sh own to repro d uce v ery accurately the ma jor “st yl- ized facts” of return time-series [4, 5, 21]. Since pioneering works of Mandel- brot [14, 15], Kahane and Peyri ` er e [11], a lot of mathematical studies h av e b een d evote d to m ultiplicativ e cascades, d en oted in s equ el as M -cascades (see e.g. refs [8 , 13, 18, 22]). One of the cen tral issues of these studies was to understand ho w the p artition function scaling exp onen ts (h er eafter denoted as τ 0 ( q )), are r elated, on one hand, to the cum u lan t generating fu nction of cascade w eigh t d istribution and, on the other h and, to the r egularit y prop- erties of cascade s amp les. Actually , the goal of the m ultifractal formalism is to directly relate the function τ 0 ( q ) to the so-called singularit y sp ectrum , 1 imsart-aap ver. 2007/12 /10 file: paper_ver5- 3_IMS.tex date: November 20, 2018 2 i.e., the Hausdorff dimension of th e set of all the p oin ts corr esp onding to giv en H¨ older exp onen t. Let us mentio n th at recen tly conti n uous v ersions of m ultiplicativ e cascades ha v e b een in trod uced [1 , 2] : they sh are m ost of prop erties w ith d iscrete cascades bu t do not inv olv e any preferen tial scale ratio and r emain inv arian t under time translation. In these constructions, the analog of the in tegral scale T , i.e., the coarsest scale where the cascade iteration b egins, is a correlation time. In all the ab o v e cited r eferences, the main results concern one single cas- cade o ve r one in tegral scale T in the limit of arbitrary small sampling scale. Ho wev er, in many app licatio ns (e.g., the ab o v e tur bulence exp erimen ts) there is no r eason a priori that the length of the exp erimental series cor- resp onds to one (or few) integ ral scal e(s). F rom a general p oin t of view, as long as mo deling a d iscrete (time or space) series with a cascade pro cess is concerned, thr ee scales are inv olve d : (i) the r esolution scale l whic h corre- sp onds to the samp ling p erio d of the series, (ii) the inte gr al (or c orr elation) sc ale T and (iii) the size L of the wh ole series. Using these n otations, the total num b er of samples of the series is N = L l . Therefore, when mo deling a discrete series with a m u ltifractal pr o cess, v ari- ous t yp es of asymptotics for N → + ∞ can b e defin ed. The “high resolution asymptotics” considered in the literature, corresp onds to l → 0 whereas L is fixed. On the other sid e, one could also consider the “infinite historic asymptotics” that corresp ond s to L → + ∞ whereas l is fixed. If w e define N T to b e the num b er of inte gral scales in v olv ed in the series N T = L T , and N l the num b er of samp les p er inte gral s cale (1) N l = T l , then we ha v e N = N T N l . Th us, the high resolution asymp totics corresp onds to N T fixed and N l → + ∞ wh er eas the in fi nite historic asymp totics corresp onds to N l fixed and N T → + ∞ . But in man y applications, it is clear that since th e relativ e v alues of N T and N l can b e arbitrary , it is not ob vious that one of th e tw o men tionned asymp totics can accoun t suitably for situation. Th is leads us imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 3 to consider an asymptotics according to w hic h N T and N l go to infin it y (and therefore N go es to in finit y) and at the s ame time preserve their rela- tiv e “v elocities”, i.e., the ratio of their logarithm. S ome of us, ha v e already suggested the follo w ing “mixed asymptotics” [12, 19, 20] : N T = N χ l , where χ ∈ R + is a fixed num b er that quan tifies the relativ e ve lo cities of N T and N l . Thus, • χ = 0 corresp onds to the high r esolution asymptotics, • χ → + ∞ corresp onds to the infin ite historic asymp totics, and all other v alues are truly “mixed” asymptotics. Successful applications of the mixed asymptotics ha v e already b een p erformed [19, 20]. In this pap er w e revisit the standard p roblems of (i) the estimation of cascade ge nerator cum ulan t generating function in the mixed asymptotic framew ork and of (ii) the multifractal formalism or of ho w to relate th is function to a dimension- lik e quantit y . The pap er is organized as follo ws: in section 2 w e recall basic definitions and prop erties of M -cascades. Section 3 cont ains the main results of this pap er. If we define a m ultifractal measure ˜ µ as the concatenation of ind e- p endent M -cascades of length T , with co mmon generator la w W , th en we sho w in Theorem 2: 1 log (1 /l ) log N − 1 N − 1 X k =0 ˜ µ ([ k l , ( k + 1) l ]) p ! → p − log 2 E [ W p ] := τ ( p ) + 1 for p in some range ( p − χ , p + χ ). These critical exp onen ts p − χ , p + χ are related to the tw o solutions, h − χ , h + χ of the equ ation D ( h ) = − χ w here D ( h ) = inf p { ph − τ ( p ) } is the Legendre transform of τ . The con vergence rate is studied in S ection 3.5. Let us stress th at the range of v alidit y on p of this con v ergence is wider in the m ixed asymptotic framew ork ( χ > 0) than in the high resolution asymptotic ( χ = 0). As a consequence we can relate D ( h ) to a ”b o x -counting dimension” (sometimes referred to as a b o x dimension [10] or a coarse- grain sp ectrum [24]), and derive, as stated in Theorem 4, a ”b ox-c oun ting m ultifractal formalism” for ˜ µ 1 N T # { k ∈ { 0 , . . . , N } | ˜ µ ([ k l, ( k + 1) l ]) ∈ [ l h − ε , l h + ε ] } ≃ l − D ( h ) imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 4 in the range of [ h + χ , h − χ ]. S ince for χ > 0, D ( h ) can tak e n egativ e v alues in previous equation, this can b e seen as a rigorous form ulation of Mandelbrot’s negativ e dimension theory [16, 17]. In section 4, w e extend previous r esults to partition functions relying on some arb itrary w a v elet decomp osition of the pro cess. In Section 5 w e giv e an interpretation of the r esults connected with the Beso v fr on tier asso ciated with ou r multifractal m easure. Finally , in section 6 we d iscuss some sp ecific examples where the la w of the cascade generator is resp ectiv ely log-normal, log-P oisson and log -Gamma. F or illus- tration pur p ose, we also rep ort, in eac h case, estimations p erformed from n umerical simulatio ns. Auxiliary useful Lemmas are mo v ed to App endices. 2. M -cascades : Definitions and prop erties. 2.1. Definition of the M -c asc ades. Let us fir st int ro duce some n otations. Giv en a j -uplet r = ( r 1 , . . . , r j ), for all str ictly p ositiv e in teger i ≤ j , w e note r | i the restriction of the j -uplet to its fi rst i comp onents, i.e., r | i = ( r 1 , . . . , r i ) , ∀ i ∈ { 1 , . . . , j } . By conv enti on, if j = 0, w e consider th at r = ∅ and in the sequel, w e denote b y r r ′ the j + j ′ -uplet obtained by concatenati on of r ∈ { 0 , 1 } j and r ′ ∈ { 0 , 1 } j ′ . Moreo v er, w e note r = ( 2 j P j i =1 r i 2 − i , if r 6 = ∅ 0 if r = ∅ . Let fix T ∈ R + ∗ and k ∈ N . W e defin e I j,k as the interv al (2) I j,k = [ k 2 − j T , ( k + 1)2 − j T ] . Th us, for an y j ∈ N ∗ , the inte rv al [0 , T ] can b e decomp osed as 2 j dy adic in terv als : [0 , T ] = [ r ∈{ 0 , 1 } j I j, r . Let us n ow build the s o called M -cascade measur es in tro duced by Mandel- brot in 1974 [15]. Let { W r } r ∈{ 0 , 1 } j , j ∈ N ∗ b e a set of i.i.d random v ariables of mean E [ W r ] = 1. Give n j ∈ N ∗ , we define the random measure µ j on [0 , T ] suc h that, for all r ∈ { 0 , 1 } j , th e Radon-Nikodym deriv ativ e with r esp ect to the Leb esgue measure d µ j d x is constan t on I j, r with: (3) d µ j d x = j Y i =1 W r | i , on I j, r , for r ∈ { 0 , 1 } j . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 5 As it is w ell kno wn [11], the measures µ j ha v e a non-trivial limit measure µ ∞ , wh en j go es to ∞ , as so on as E [ W log 2 W ] < 1. Moreo v er, the total mass µ ∞ ([0 , T ]) = lim j → ∞ T 2 − j X r ∈{ 0 , 1 } j j Y i =1 W r | i , v erifies E [ µ ∞ ([0 , T ])] = T . Let u s remark that if r ∈ { 0 , 1 } j then b y con- struction we ha v e: µ ∞ ( I j, r ) = lim n →∞ T 2 − j j Y i =1 W r | i   X r ′ ∈{ 0 , 1 } n 2 − n n Y i =1 W r r ′ | ( j + i )   = 2 − j   j Y i =1 W r | i   µ ( r ) ∞ ([0 , T ]) , (4) where µ ( r ) ∞ is a M -cascade measure on [0 , T ] based on the random v ariables W r r ′ for r ′ ∈ ∪ j ≥ 1 { 0 , 1 } j . This equalit y is usually referr ed to as ”Mandelbrot star equation”. In the s equ el we need the follo wing set of assumptions: E [ W log 2 W ] < 1 , P ( W = 1) < 1 , (5) P ( W > 0) = 1 , E [ W p ] < ∞ for all p ∈ R . (6) Let τ ( p ) b e the smo oth and conca v e function d efi ned on R by (7) τ ( p ) = p − log 2 E [ W p ] − 1 . Let us notice that log 2 E [ W p ] is nothing but the cum u lan t generating func- tion (log- Laplace transf orm) of the logarithm of cascade generator distribu- tion. It is shown in [11] that for p > 1, the condition τ ( p ) > 0 implies the finiteness of E [ µ ∞ ([0 , T ]) p ]. By Theorem 4 in [18], the conditions (6) imply the existence of fin ite negativ e moments E [ µ ∞ ([0 , T ]) p ], for all p < 0. 2.2. Multifr actal pr op erties of M -c asc ades. A M-cascade is a multifrac- tal measure and the stud y of its m ultifractal prop erties red uces to the study of the p artitio n function (8) S µ ( j, p ) = 2 j − 1 X k =0 µ ∞ ( I j,k ) p . Basically , one can show [18, 22] that, for fixed p , this partition f unction b eha v es, when j go es to ∞ , as a p o wer la w function of the scale | I j,k | = T 2 − j . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 6 More precisely , let u s int ro duce the t w o f ollo wing cr itical exp onen ts: p + 0 = in f { p ≥ 1 | pτ ′ ( p ) − τ ( p ) ≤ 0 } ∈ (1 , ∞ ] p − 0 = s u p { p ≤ 0 | pτ ′ ( p ) − τ ( p ) ≤ 0 } ∈ [ −∞ , 0) . If p + 0 (resp. p − 0 ) is fi nite we set h + 0 = τ ′ ( p + 0 ) (resp. h − 0 = τ ′ ( p − 0 )). Theorem 1 . Scaling of the partition function [22] L et p ∈ R , the p ower law sc aling exp onent of S µ ( j, p ) is given by (9) lim j → ∞ log 2 S µ ( j, p ) − j − → a.s. τ 0 ( p ) , wher e τ 0 ( p ) is define d by (10) τ 0 ( p ) =      τ ( p ) , ∀ p ∈ ( p − 0 , p + 0 ) h + 0 p, ∀ p ≥ p + 0 h − 0 p, ∀ p ≤ p − 0 . The pro of can b e found in [22]. The conv ergence in probabilit y of (9) w as obtained in the ea rlier w ork [18]. This theorem basically states that S µ ( j, p ) b eha v es lik e S µ ( j, p ) ≃ 2 − j τ 0 ( p ) . Let us note that the partition fun ction (8) can b e rewritten in the follo wing w a y (11) S µ ( j, p ) = X r ∈{ 0 , 1 } j µ ∞ ( I j, r ) p and usin g (4), one gets (12) S µ ( j, p ) = 2 − j p X r ∈{ 0 , 1 } j j Y i =1 W p r | i  ¯ µ ( r ) ∞ ([0 , T ])  p , where the { ¯ µ ( r ) ∞ ([0 , T ]) } r ∈{ 0 , 1 } j are i.i.d. random v ariables with the same la w as µ ∞ ([0 , T ]). Thus, a simple computation shows that E [ S µ ( j, p )] = 2 − j p 2 j E [ W p ] j E [ µ ∞ ([0 , T ]) p ] = 2 − j τ ( p ) E [ µ ∞ ([0 , T ]) p ] . One sees that the last theorem states that, in the case p ∈ [ p − 0 , p + 0 ], S µ ( j, p ) scales as its mean v alue. On the other hand, the fact that for p / ∈ [ p − 0 , p + 0 ] the partition function scales as giv en in (10) instead of scaling as its mean v alue, is r eferred to imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 7 as the ’linearization effect’. A p ossible explanatio n of this effect is th at for p larger than the critical exp onen ts p + 0 (resp. smaller than p − 0 ), the sum in v olv ed in the partition function (11) is dominated b y its sup r em um (resp. infimum) term. Th us one should not expect a law of large num b er to hold for th e b ehavior of this su m. Another p ossible in terpretation of this theorem in the case p > p + 0 is give n in Section 5. 3. Mixed asymptotics for M -cascades. 3.1. Mixe d asymptotics : definitions and nota tions. A con v enient wa y to construct a multifrac tal measur e on R + , with an in tegral scale equal to T , is to patc h ind ep endent realizati ons of M -cascades measures. More precisely , consider { µ ( m ) ∞ } m ∈ N a sequence of i.i.d M -cascades on [0 , T ] as defined in Section 2.1 and d efine the sto c hastic measure on [0 , ∞ ) by: (13) ˜ µ ([ t 1 , t 2 ]) = + ∞ X m =0 µ ( m ) ∞ ([ t 1 − mT , t 2 − mT ]) , for all 0 ≤ t 1 ≤ t 2 . This mo del is en tirely defined as so on as b oth T and the la w of W are fix ed . The discretized time mod el for the N samples of the series is { ˜ µ [ k l , ( k + 1) l ] } 0 ≤ k < N − 1 . 3.2. Sc aling pr op erties. In this section, we study the partition fun ction for the measure ˜ µ as defined in Eq. (13) in the m ixed asym p totic limit. T is fixed, we c hoose the sampling step l = T 2 − j , N l = 2 j , and the num b er of in tegral scales is related with th e sampling s tep as N T = ⌊ N χ l ⌋ ∼ 2 j χ , with χ > 0 fixed. According to (1), one gets for the total n umber of data: N = N T 2 j ∼ 2 j (1+ χ ) . The mixed asymptotics corresp onds to the limit j → + ∞ . Th e partition function of ˜ µ can b e written as (recall (2)): S ˜ µ ( j, p ) = N − 1 X k =0 ˜ µ ( I j,k ) p (14) = N T − 1 X m =0 S ( m ) µ ( j, p ) , (15) imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 8 where S ( m ) µ ( j, p ) is th e partition fu nction of µ ( m ) ∞ , i.e., (16) S ( m ) µ ( j, p ) = 2 j − 1 X k =0 µ ( m ) ∞ ( I j,k ) p . Let us state the results of this section. W e in tro duce the t wo critical expo- nen ts in the mixed asymptotic fr amew ork : p + χ = in f { p ≥ 1 | pτ ′ ( p ) − τ ( p ) ≤ − χ } ∈ (1 , ∞ ] (17) p − χ = s u p { p ≤ 0 | pτ ′ ( p ) − τ ( p ) ≤ − χ } ∈ [ −∞ , 0) , (18) and w e s et when these critical exp onents are fin ite h + χ = τ ′ ( p + χ ), h − χ = τ ′ ( p − χ ). Theorem 2 . Scaling of t he partition function in a mixed asymp- totics L et p ∈ R and ˜ µ b e the r andom me asur e define d by (13) wher e the law of W satisfies (5) – (6) . We assume that, either p + χ = ∞ , or p + χ < ∞ with τ ( p + χ ) > 0 . Then, the p ower law sc aling of S ˜ µ ( j, p ) is given by (19) lim j → ∞ log 2 S ˜ µ ( j, p ) − j − → a.s. τ χ ( p ) , wher e τ χ ( p ) is define d by τ χ ( p ) =      τ ( p ) − χ, ∀ p ∈ ( p − χ , p + χ ) h + χ p, ∀ p ≥ p + χ h − χ p, ∀ p ≤ p − χ . Remark 1 . If p + χ = ∞ then simple c onsider ations on the c onc ave func - tion τ shows that τ ( p ) > 0 for al l p > 1 , and henc e the c asc ade me asur e has finite moments of any p ositive or ders. Otherwise the assump tion τ ( p + χ ) > 0 is state d in The or em 2 to insur e E [ µ ∞ ([0 , T ]) p ] < ∞ for p ∈ [0 , p + χ ) . Such assumption was not ne e de d in The or em 1 , sinc e on c an che ck that ne c essarily τ ( p + 0 ) > 0 . Remark 2 . L et us str ess that the b ehavior of the p artition function is lar gely affe c te d by the c hoic e of a mixe d asymptotic: the ’line arization effe ct’ now o c curs f or p in the set ( −∞ , p − χ ) ∪ ( p + χ , ∞ ) , which is smal ler when χ incr e ases. imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 9 Theorem 3 . Scaling of the supremum and t he infimum of the mass in a mixed asymptotics Assume (5) – (6) . Then, if p + χ < ∞ , one has, (20) lim j → + ∞ log 2 sup k ∈ [0 ,N − 1] ˜ µ ( I j,k ) − j = h + χ , almo st su r ely, and if p − χ > − ∞ , one has, (21) lim j → + ∞ log 2 inf k ∈ [0 ,N − 1] ˜ µ ( I j,k ) − j = h − χ , almo st su r ely. The theorem 3 sh o ws that when the ’linearizati on effect’ occurs, the scal- ing of the p artition function (14 ) is go v erned by its supremum and infimum terms f or resp ectiv ely large p ositiv e and negativ e p v alues. These theorems will b e p ro v ed in thr ee parts. In Section 3.3.2, w e will pro v e Eq. (19) of Theorem 2 only for p ∈ ( p − χ , p + χ ). In Section 3.3.3, we will pro v e the case p / ∈ ( p − χ , p + χ ) and Theorem 3 is sho wn in S ection 3.3.4 to b e a simple corrolary of this last case. 3.3. Pr o of of The or em 2. First we need an auxiliary result w hic h is help- ful in the sequel. 3.3.1. Limit the or em for a r esc ale d c asc ade. F o r eac h m we denote as ( W ( m ) r ) r ∈∪ j { 0 , 1 } j the set of i.i.d. random v ariables used for the constru ction of the measure µ ( m ) ∞ . Moreo ver we assume that for eac h m ≥ 0, j ≥ 0, r ∈ { 0 , 1 } j w e are giv en a r an d om v ariable Z ( m,r ) , measur ab le with resp ect to the sigma-field σ  W ( m ) r r ′ | r ′ ∈ ∪ j { 0 , 1 } j  . W e mak e the assu mption th at the la w of Z ( m,r ) do es not dep end on ( m, r ), and denote b y Z a v ariable with this la w. Let us consider the qu an tities, for p ∈ R : (22) M ( m ) j ( p ) = 2 − j p X r ∈{ 0 , 1 } j j Y i =1  W ( m ) r | i  p Z ( m,r ) , and (23) N j ( p ) = N T − 1 X m =0 M ( m ) j ( p ) . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 10 Pr oposition 1 . Assu me that for some ǫ > 0 , E h | Z | 1+ ǫ i < ∞ and − pτ ′ ( p ) + τ ( p ) < χ , then: 2 j ( τ ( p ) − χ ) N j ( p ) j → ∞ − − − → E [ Z ] , almost su r ely. Pr oof. F rom (22)–(23) and the defin ition (7) we get, E [ N j ( p )] = N T 2 j 2 − j p E [ W p ] j E [ Z ] ∼ j → ∞ 2 j χ 2 − j τ ( p ) E [ Z ] . Hence th e prop osition will b e p ro v ed if w e sho w: (24) 2 j ( τ ( p ) − χ ) ( N j ( p ) − E [ N j ( p )]) j → ∞ − − − → 0 , almost surely . F or an arbitrary small ǫ > 0, w e study the L 1+ ǫ ( P ) norm of the difference. Set, (25) L 1+ ǫ N = E h |N j ( p ) − E [ N j ( p )] | 1+ ǫ i . Applying su ccessive ly lemmas 1 and 2 of App endix A, we get: L 1+ ǫ N ≤ C 2 j χ E     M (0) j ( p )    1+ ǫ  ≤ C 2 − j [(1+ ǫ ) τ ( p ) − χ ] j X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) . W e dedu ce that 2 j ( τ ( p ) − χ )(1+ ǫ ) L 1+ ǫ N is b ounded b y the quan tit y: C 2 − j χǫ j X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) . Clearly , as so on as 2 − χǫ 2 − τ ( p (1+ ǫ )) 2 (1+ ǫ ) τ ( p ) < 1, this qu antit y is, in turn, b ound ed by C 2 − j χǫ ′ for some ǫ ′ > 0. T aking the log, a sufficien t condition is τ ( p )(1 + ǫ ) − τ ( p (1 + ǫ )) ǫ < χ whic h is implied for ǫ small enough by − pτ ′ ( p ) + τ ( p ) < χ. Th us we ha v e shown that 2 j ( τ ( p ) − χ )(1+ ǫ ) L 1+ ǫ N is asymptotically smaller than 2 − j ǫ ′ with some ǫ ′ > 0. Using the Biena ym ´ e-Cheb yshev inequalit y leads to P { 2 j ( τ ( p ) − χ ) |N j ( p ) − E [ N j ( p )] | ≥ η } ≤ 2 j ( τ ( p ) − χ )(1+ ǫ ) L 1+ ǫ N η 1+ ǫ ≤ C 2 − j ǫ ′ η 1+ ǫ for any η > 0. A simple use of the Borel Cantelli lemma sh o ws (24). imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 11 3.3.2. Pr o of of The or em 2 for p ∈ ( p − χ , p + χ ) . F rom (15)–(16) and th e represent ation (12) for the p artition function of a sin gle cascade, we see that that S ˜ µ ( j, p ) exactly has the same structure as the quan tit y N j ( p ) of section 3.3.1 wh ere Z ( m,r ) = µ ( m,r ) ∞ ([0 , T ]) p are random v ariables distributed as Z = µ ∞ ([0 , T ]) p . By definition (reca ll (17)–(18)), the condition − pτ ′ ( p ) + τ ( p ) < χ h olds for any p ∈ ( p − χ , p + χ ), and b y Remark 1, E  | Z | 1+ ǫ  < ∞ for ǫ small enough. Th us, an application of Prop osition 1 yields the almost sure conv ergence: (26) 2 j τ χ ( p ) S ˜ µ ( j, p ) = 2 j ( τ ( p ) − χ ) S ˜ µ ( j, p ) j → ∞ − − − → E [ µ ∞ ([0 , T ]) p ] . This p r o ves the theorem 2 for the case p ∈ ( p − χ , p + χ ). 3.3.3. Pr o of of The or em 2 for p / ∈ ( p − χ , p + χ ) . Th e follo wing pro of is an adaptation of the corresp onding pro of in [23]. W e need th e follo wing n ota- tions: S ˜ µ ∗ ( j ) = sup k ∈ [0 ,N − 1] ˜ µ ∞ ([ k 2 − j T , ( k + 1)2 − j T ]) , m sup ( p ) = lim sup j → ∞ log 2 S ˜ µ ( j, p ) − j , m inf ( p ) = lim inf j → ∞ log 2 S ˜ µ ( j, p ) − j , m ∗ sup = lim sup j → ∞ log 2 S ˜ µ ( j ) ∗ − j , m ∗ inf = lim inf j → ∞ log 2 S ˜ µ ( j ) ∗ − j . In Section 3.3.2 w e pro v ed that for all p ∈ ( p − χ , p + χ ) the follo wing holds almost su rely: m sup ( p ) = m inf ( p ) = τ χ ( p ) . W e may assu m e th at on a ev ent of pr obabilit y one, this equalit y holds for all p in a countable and dense sub set of ( p − χ , p + χ ). F rom the sub -additivit y of x 7→ x ρ , ∀ ρ ∈ ]0 , 1[ , ∀ p ∈ R S ˜ µ ( p, j ) ρ ≤ S ˜ µ ( ρp, j ) , and thus m inf ( p ) ≥ m inf ( ρp ) ρ . But we hav e seen that m inf ( ρp ) = τ χ ( ρp ), for a dense subset of ρp ∈ ( p − χ , p + χ ). Assume now for simp licit y th at p ≥ p + χ and let ρ → ( p + χ /p ), we get (27) ∀ p ≥ p + χ , m inf ( p ) p ≥ τ χ ( p + χ ) p + χ = τ ( p + χ ) − χ p + χ = h + χ , imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 12 where we ha v e u sed (17) On the other hand , let p > 0, q ∈ [0 , p + χ ), and q ′ ∈ [0 , q ), we ha v e S ˜ µ ( j, q ) = N − 1 X k =0 ˜ µ ∞ ([ k 2 − j T , ( k + 1)2 − j T ]) q ≤ S ˜ µ ∗ ( j ) q − q ′ S ˜ µ ( j, q ′ ) ≤ S ˜ µ ( j, p ) q − q ′ p S ˜ µ ( j, q ′ ) . Th us m sup ( q ) ≥ ( q − q ′ ) m sup ( p ) p + m inf ( q ′ ) , then m sup ( p ) p ≤ m sup ( q ) − m inf ( q ′ ) q − q ′ = τ χ ( q ) − τ χ ( q ′ ) q − q ′ . T aking the limit q ′ → q − m sup ( p ) p ≤ inf q ∈ [0 ,p + χ ) τ ′ χ ( q ) ≤ τ ′ χ ( p χ ) = h + χ . Merging th is last relation with (27) leads to (28) ∀ p ≥ p + χ , h + χ ≤ m inf ( p ) p ≤ m sup ( p ) p ≤ h + χ , whic h prov es Theorem 2 for p ∈ [ p + χ , + ∞ [. The pro of for p ≤ p − χ is similar. 3.3.4. Pr o of of The or em 3. Th e follo wing pro of is an adaptation of the corresp onding pro of in [23 ]. W e hav e for p > 0, S ˜ µ ∗ ( j ) p ≤ S ˜ µ ( j, p ) ≤ N S ˜ µ ∗ ( j ) p = ⌊ 2 j χ ⌋ 2 j S ˜ µ ∗ ( j ) p , th us pm ∗ inf ,sup ≥ m inf ,sup ( p ) ≥ 1 + χ + p m ∗ inf ,sup , whic h means that m inf ,sup ( p ) p − 1 + χ p ≥ m ∗ inf ,sup ≥ m inf ,sup ( p ) p , and taking the limit p → + ∞ and us ing (28) pro v es that m ∗ sup = m ∗ inf = h + χ , whic h pro ves (20). The pro of of (21) is obtai ned analogo usly by considering p < 0. imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 13 3.4. Multifr actal formalism and “ne gative dimensions”. Let D ( h ) b e th e Legendre tran s form of τ ( p ) : D ( h ) = min p ( ph − τ ( p )) , The multifr actal f ormalism [7 ] giv es an in teresting interpretation of D ( h ), as soon as D ( h ) > 0, in terms of dimension of set of p oints with the same regularit y . F or M -cascades, this formalism holds [18], i.e., D ( h ) corresp onds to th e Hausd orff d imension of th e p oin ts t ∈ [0 , T ] aroun d w hic h µ ∞ scales with the exp onent h : (29) D ( h ) = dim H  t, lim sup ǫ → 0 log 2 µ ∞ ([ t − ǫ, t + ǫ ]) log 2 ( ǫ ) = h  . The r.h.s. of (29) is usu ally referred to as the singularity sp e ctrum and there- fore th e multifracta l f orm alism simply states that D ( h ) can b e identified with the sin gu larity sp ectrum of the cascade. In a mixed asymptotic fr amew ork, our next result sh o ws that some kind of multifracta l formalism s till holds for D ( h ) < 0 in th e sense that D ( h ) go verns th e b ehavior of the p opulation histogram p e r sample of measur e v alues at scale 2 − j as esti mated o v er 2 j χ cascade samples. In other w ord s, D ( h ) coincides w ith a b ox-c oun ting d imension (sometimes referr ed to as a b o x dimension [10] or a coarse-grain sp ectrum [24]). Hence the Legendre transform of τ ( p ) can b e interpreted as a ”p opu lation” dimension ev en for singularit y v alues ab o v e and b elo w h + 0 and h − 0 . Since for these v alues, one h as D ( h ) < 0 they ha v e b een called ”negativ e d imensions” by Mandelbrot [16 ]. This simply means that they cannot b e observe d on a single cascade sample but one needs at least 2 j χ realizatio ns to observ e them with a ”cardinalit y” lik e 2 j ( χ + D ( h )) . In th at resp ect, they ha ve also b een r eferred to as ”laten t” singularities [17]. Theorem 4 . Assume p + χ < ∞ , p − χ > −∞ and τ ( p + χ ) > 0 . L et h ∈ ( h + χ , h − χ ) , then: (30) lim ε → 0 lim j 1 j log # n k ∈ { 0 , . . . , N − 1 } | 2 − j ( h + ε ) ≤ ˜ µ ( I j,k ) ≤ 2 − j ( h − ε ) o = χ + D ( h ) , (31) lim ε → 0 lim j 1 j log # n k ∈ { 0 , . . . , N − 1 } | 2 − j ( h + ε ) ≤ ˜ µ ( I j,k ) ≤ 2 − j ( h − ε ) o = χ + D ( h ) . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 14 Pr oof. 1 st step: W e fo cus on the cases that yield to negativ e dimen- sions, i.e. h ∈ ( h + χ , h + 0 ) ∪ ( h − 0 , h + χ ). F or simplicit y assume h ∈ ( h + χ , h + 0 ) = ( τ ′ ( p + χ ) , τ ′ ( p + 0 )). W e can wr ite h = τ ′ ( p ) for some p ∈ ( p + 0 , p + χ ) and if we define χ ′ = τ ( p ) − p τ ′ ( p ) ∈ (0 , χ ) w e easily get that h = τ ′ ( p + χ ′ ) and D ( h ) = − χ ′ . Thus the theorem amoun ts to assess the magnitud e of #  k ∈ { 0 , . . . , N − 1 } | 2 − j ( τ ′ ( p + χ ′ )+ ε ) ≤ ˜ µ ( I j,k ) ≤ 2 − j ( τ ′ ( p + χ ′ ) − ε )  as ≃ 2 j ( χ − χ ′ ) . First w e d eriv e a lo w er b oun d for this cardinalit y . The idea is to split the data into blo c ks of size 2 j j 2 j χ ′ k and rely on the b eha vior of the supr em um of ˜ µ ( I j,k ) und er m ixed asymptotic with ind ex χ ′ . More p recisely let N ′ = 2 j j 2 j χ ′ k and define th e blo cks B a = { aN ′ , . . . , ( a + 1) N ′ − 1 } , f or a = 0 , . . . , M − 1 :=  N/ N ′  − 1 . Fix a ∈ { 0 , . . . , M − 1 } , then for an y p 1 < p 2 < p + χ ′ w e hav e (32) sup k ∈ B a ˜ µ ( I j,k ) p 2 − p 1 ≥ P k ∈ B a ˜ µ ( I j,k ) p 2 P k ∈ B a ˜ µ ( I j,k ) p 1 := 2 j ( τ χ ′ ( p 1 ) − τ χ ′ ( p 2 )) Q ( a ) j ( p 1 , p 2 ) , where Q ( a ) j ( p 1 , p 2 ) = P k ∈ B a ˜ µ ( I j,k ) p 2 2 j τ χ ′ ( p 2 ) P k ∈ B a ˜ µ ( I j,k ) p 1 2 j τ χ ′ ( p 1 ) . Clearly the la w of Q ( a ) j ( p 1 , p 2 ) do es not dep end on a , and Q (0) j ( p 1 , p 2 ) is the ratio of t wo rescaled p artition fun ctions in mixed asymptotic with index χ ′ . Hence by (26), Q (0) j ( p 1 , p 2 ) con v erges almost sur ely to th e non zero constant E [ µ ∞ ([0 , T ]) p 2 ] / E [ µ ∞ ([0 , T ]) p 1 ]. W e deduce that for all fi x ed a ≤ M − 1, and p 1 , p 2 ∈ (0 , p + χ ′ ), the sequence (1 /Q ( a ) j ( p 1 , p 2 )) j ≥ 0 is b ounded in probabilit y . W rite now, b y (32 ), and τ χ ′ ( p ) = τ ( p ) − χ ′ for p < p + χ ′ , sup k ∈ B a ˜ µ ( I j,k )2 j ( τ ′ ( p + χ ′ )+ ǫ ) ≥ Q ( a ) j ( p 1 , p 2 )2 − j [ τ ( p 2 ) − τ ( p 1 ) p 2 − p 2 − τ ′ ( p + χ ′ ) − ǫ ] and choose p 1 , p 2 fixed but close enough to p + χ ′ . Th is yields, for all j ≥ 0: sup k ∈ B a ˜ µ ( I j,k )2 j ( τ ′ ( p + χ ′ )+ ǫ ) ≥ Q ( a ) j ( p 1 , p 2 )2 j ǫ / 2 . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 15 Then using that 1 /Q ( a ) j ( p 1 , p 2 ) is b oun ded in pr ob ab ility , we get P (sup k ∈ B a ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ )+ ε ) ) ≥ P (1 /Q ( a ) j ( p 1 , p 2 ) ≤ 2 j ǫ / 2 ) ≥ 1 / 2 f or j large enough. Remark n o w that the cardinalit y of the set { k ∈ { 0 , . . . , N − 1 } | ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ )+ ε ) } is imm ediately lo w er b ound ed b y the su m M − 1 X a =0 1  sup k ∈ B a ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ )+ ε )  of i.i.d. Bernoulli v ariables with parameter greater than 1 / 2. Then it is easily deduced, using the Borel Cantell i lemma and M ∼ j → ∞ 2 ( χ − χ ′ ) j that with probabilit y one: (33) lim j 2 − ( χ − χ ′ ) j #  k ∈ { 0 , . . . , N − 1 } | ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ )+ ε )  ≥ 1 / 4 . W e now focu s on up p er b ounds f or the cardinalit y of the set { k ∈ { 0 , . . . , N − 1 } | ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ )+ η ) } where η is some real n u m b er in a neighborho od of zero. It is s imply deriv ed from th e connection with the partition function that for an y p > 0 this cardinalit y is low er than S ˜ µ ( j, p )2 j ( τ ′ ( p + χ ′ )+ η ) p . App lying this with p = p + χ ′ and since, by (17), τ χ ( p + χ ′ ) = τ ( p + χ ′ ) − χ = pτ ′ ( p + χ ′ ) + χ ′ − χ we get the follo win g u pp er b ound, (34) # { k ∈ { 0 , . . . , N − 1 } | ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ )+ η ) } ≤ S ˜ µ ( j, p + χ ′ )2 j τ χ ( p + χ ′ ) 2 j ( χ − χ ′ + ηp + χ ′ ) . By p + χ ′ < p + χ , the con v ergence resu lt (26) with p = p + χ ′ applies and we deduce for η = − ǫ < 0 that: (35) lim j 2 − ( χ − χ ′ ) j # { k ∈ { 0 , . . . , N − 1 } | ˜ µ ( I j,k ) ≥ 2 − j ( τ ′ ( p + χ ′ ) − ε ) } = 0 . Then (30) is a consequence of (33) and (35). Finally , th e up p er b ound (31) is directly obtained b y app lying (34) with η = ε . 2 nd step: W e no w deal with the more classical case h ∈ [ h + 0 , h − 0 ]. It is kno wn, fr om the m ultifractal formalism for a single M-cascade on [0 , T ] (see imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 16 [3, 24]), that w ith probabilit y 1: (36) lim ε → 0 lim j 1 j log # n k ∈ { 0 , . . . , 2 j − 1 } | ˜ µ ( I j,k ) ∈ [2 − j ( h + ε ) , 2 − j ( h − ε ) ] o = D ( h ) . F or m ∈ { 0 , . . . , N T − 1 } , an d ε > 0, η > 0, denote by A ( m ) j ( η , ε ) the ev en t: # n k ∈ { 0 , . . . , 2 j − 1 } | ˜ µ ( I j, 2 j m + k ) ∈ [2 − j ( h + ε ) , 2 − j ( h − ε ) ] o ≥ 2 j ( D ( h ) − η ) . Using the indep endence of the M -ca scades ( µ ( m ) ) m , these ev ent s are inde- p endent and P ( A ( m ) j ( η , ε )) do es not dep end on m . Moreo v er, by (36), for an y η > 0, th ere exists ǫ > 0 suc h that P ( A ( m ) j ( η , ε )) j → ∞ − − − → 1. W e easily deduce that for any η > 0 and ǫ small enough: lim j 1 N T N T − 1 X m =0 1 { A ( m ) j ( η,ε ) } ≥ 1 / 2 , almost sur ely . Since the cardinalit y of n k ∈ { 0 , . . . , N − 1 } | ˜ µ ( I j,k ) ∈ [2 − j ( h + ε ) , 2 − j ( h − ε ) ] o is greater than 2 j ( D ( h ) − η ) P N T − 1 m =0 1 { A ( m ) j ( η,ε ) } w e dedu ce that the left hand side of (30) is great er than D ( h ) + χ − η , for an y η > 0. T o end the pro of, it suffices to sho w that the left h and side of (31) is lo wer than D ( h ) + χ . This is ea sily done, as in the end of the fir st step, b y relying on the asymptotic b eha vior of the partition function. 3.5. Centr al Limit The or ems. In this sectio n, we br iefly study the rate of the con v ergence of S ˜ µ ( j, p ) as j → ∞ of (19) in Theorem 2. Using the same notations as in the p r o of of Theorem 2 , we write: (37) S ˜ µ ( j, p ) = j 2 j χ k 2 − j τ ( p ) E [ µ ∞ ([0 , T ]) p ] + A j + B j where A j = N T − 1 X m =0 2 − j p X r ∈{ 0 , 1 } j   j Y i =1 W ( m ) p r | i    ¯ µ ( m,r ) ∞ ([0 , T ]) p − E [ µ ∞ ([0 , T ]) p ]  and B j = N T − 1 X m =0  2 − j p X r ∈{ 0 , 1 } j j Y i =1 W ( m ) p r | i − 2 − j τ ( p )  E [ µ ∞ ([0 , T ]) p ] . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 17 Pr oposition 2 . Assume (5 ) – (6) and that, either p + χ = ∞ , or p + χ < ∞ with τ ( p + χ ) > 0 . If p − χ / 2 < p < p + χ / 2 then, 2 j ( τ (2 p ) − χ ) / 2 A j j → ∞ − − − → N  0 , V ar ( µ ∞ ([0 , T ]) p )  . Pr oof. Consistently with the n otatio ns of Section 3.3.1, we define, for ev ery r ∈ { 0 , 1 } j and m = 0 , . . . , N T − 1, the rand om v ariables ˜ Z ( m,r ) = ¯ µ ( m,r ) ∞ ([0 , T ]) p − E [ µ ∞ ([0 , T ]) p ] and denote b y ˜ Z = µ ∞ ([0 , T ]) p − E [ µ ∞ ([0 , T ]) p ] their common la w . F ur th ermore, w e will need the quantit y (38) η m,r,j ( p ) = 2 j ( τ (2 p ) − χ ) / 2 2 − j p   j Y i =1 W ( m ) p r | i   ˜ Z ( m,r ) , and the follo wing family of σ -fields: for j ≥ 0 F − 1 ,j := σ  W ( m ) r , | r | ≤ j, m = 0 , . . . , N T − 1  and for ev ery k = 0 , . . . , n ( j ) = 2 j ( N T − 1) F k ,j = F − 1 ,j ∨ σ  ˜ Z ( m,r ) , ¯ r + 2 j m ≤ k  . F or fixed j , w e ha v e a one-to-one corresp onden ce b et ween ( m, r ) and k = ¯ r + 2 j m , so abusing notation s ligh tly , w e write η k ,j ( p ) in stead of η m,r,j ( p ) in (38) wh en no confusion is p ossib le. With these notations, 2 ( τ (2 p ) − χ ) / 2 A j = n ( j ) X k =0 η k ,j ( p ) where η k ,j ( p ) is F k ,j -measurable and E [ η k ,j ( p ) | F k − 1 ,j ] = 0 , ∀ k = 0 , . . . , n ( j ) . Th us, w e are dealing with a triangular arr a y of m artin gale in cremen ts. Let us consider the su m of the cond itional v ariances: (39) V j = n ( j ) X k =0 E h η k ,j ( p ) 2 | F k − 1 ,j i . W e hav e V j = V ar( ˜ Z )2 j ( τ (2 p ) − χ ) 2 − 2 j p N T − 1 X m =0 X r ∈{ 0 , 1 } j j Y i =1  W ( m ) r | i  2 p , imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 18 th us b y app lication of Prop osition 1 (with the c hoice of Z ( m,r ) equal to 1) w e get, V j j → ∞ − − − → V ar ( ˜ Z ) . Hence the prop osition will b e pro v ed , if w e can sho w that th e tr iangular arra y satisfies a Lind eb erg condition: for some ǫ > 0, V ( ǫ ) j = n ( j ) X k =0 E h | η k ,j ( p ) | 2+ ǫ | F k − 1 ,j i j → ∞ − − − → 0 . But, we ha v e V ( ǫ ) j = E h | W | 2+ ǫ i 2 j ( τ (2 p ) − χ )(1+ ǫ/ 2) 2 − j (2+ ǫ ) p X r ∈{ 0 , 1 } j j Y i =1  W ( m ) r | i  2+ ǫ and b y application of the Prop osition 1, the order of magnitude of V ( ǫ ) j is 2 j  ( τ (2 p ) − χ )(1+ ǫ/ 2) − ( τ ((2+ p ) ǫ ) − χ )  . Thus, it can b e seen that V ( ǫ ) j con v erges to zero, for ǫ small enough, by the condition 2 pτ ′ (2 p ) − τ (2 p ) > − χ . This end s the p r o of of the prop osition. Pr oposition 3 . Assume (5 ) – (6) and that, either p + χ = ∞ , or p + χ < ∞ with τ ( p + χ ) > 0 . Then: 1. If τ (2 p ) − 2 τ ( p ) > 0 we have, (40) 2 j ( τ ( p ) − χ/ 2) B j → N  0 , c ( p )  , wher e c ( p ) > 0 dep ends on the la w of W and p . 2. If τ (2 p ) − 2 τ ( p ) = 0 , we have V ar ( B j ) = O ( j 2 − j ( τ (2 p ) − χ ) ) . 3. If τ (2 p ) − 2 τ ( p ) < 0 , we have V ar ( B j ) = O (2 − j ( τ (2 p ) − χ ) ) . Pr oof. Denote ν ( m ) j the measures d efined at the step j of th e construc- tion of the M-cascade on [0 , 1] based on W p / E [ W p ]: ν ( m ) j ([0 , 1]) = 2 − j X r ∈{ 0 , 1 } j j Y i =1 ( W ( m,r ) r | i ) p E [ W p ] − j , for m ∈ { 0 , . . . , N T − 1 } . With this notation we ha v e, (41) B j = E [ µ ∞ ([0 , T ]) p ] 2 − j τ ( p ) N T − 1 X m =0  ν ( m ) j ([0 , 1]) − 1  imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 19 and us in g Kahane and Peyri ` ere results [11], we kno w that for eac h m the sequence ( ν ( m ) j ([0 , 1])) j is b ounded in L q as so on as τ ( pq ) − q τ ( p ) > 0. W e first fo cus on the case τ (2 p ) − 2 τ ( p ) > 0. Hence the sequence ( ν ( m ) j ([0 , 1])) j is b ounded in L 2+ ǫ -norm for some ǫ > 0. Usin g that ( ν ( m ) j ([0 , 1]) − 1) m is a cen tered i.i.d. sequence and classical considerations for triangular array of martingale increments, one can show that a central limit theorem holds: N − 1 / 2 T N T − 1 X m =0 ( ν ( m ) j ([0 , T ]) − 1) j → ∞ − − − → N (0 , V ar ( ν (0) ∞ ([0 , 1]))) , where ν (0) ∞ ([0 , 1]) = lim j → ∞ ν (0) j ([0 , 1]). F rom this and (41) w e d educe (40) with c ( p ) = E [ µ ∞ ([0 , T ]) p ] 2 V ar ( ν (0) ∞ ([0 , 1])). In the cases τ (2 p ) − 2 τ ( p ) ≤ 0, by (41) again we ha v e V ar ( B j ) = j 2 j χ k 2 − j 2 τ ( p ) E [ µ ∞ ([0 , T ]) p ] 2 V ar ( ν (0) j ([0 , 1])) . No w V ar( ν (0) j ([0 , 1])) = E h ν (0) j ([0 , 1]) 2 i − 1 is un b ounded as j → ∞ , but a careful lo ok at the computations in Lemma 2 with ǫ = 1 yields to E h ν (0) j ([0 , 1]) 2 i ∼ j → ∞ j X l =0 2 − l 2 l (2 τ ( p ) − τ (2 p )) . W e d educe that V ar( B j ) = O  2 − j (2 τ ( p ) − χ ) P j l =0 2 − l 2 l (2 τ ( p ) − τ (2 p ))  . Th en, the theorem follo ws in the cases τ (2 p ) − 2 τ ( p ) = 0 and τ (2 p ) − 2 τ ( p ) < 0. Remark 3 . By (37) the differ enc e b etwe en 2 ( τ ( p ) − χ ) j S ˜ µ ( j, p ) and its limit is de c omp ose d into two dissimilar err or terms: p articularly th e fact that the c ontribution of B j c onver ges to zer o is due to the observa tion of a lar ge numb er of inte gr al sc ales, wher e as the c ontribution of A j vanishes as the sampling step tends to zer o. In the c ase τ (2 p ) − 2 τ ( p ) > 0 , the c ontribution of B j strictly dominates and 2 − ( τ ( p ) − χ ) S ˜ µ ( j, p ) − E [ µ ∞ ([0 , T ]) p ] is of magnitude 2 − j χ ∼ N − 1 / 2 T . If τ (2 p ) − 2 τ ( p ) < 0 , the magnitude of A j and B j ar e the same and 2 − ( τ ( p ) − χ ) S ˜ µ ( j, p ) − E [ µ ∞ ([0 , T ]) p ] is asympto tic al ly b ounde d by terms of magnitude 2 j / 2( − χ +2 τ (2 p ) − τ (2 p )) . This r ate of c onver genc e i s slower than N − 1 / 2 T . 4. Ext ension t o w a v ele t based part ition functions. Th e b eha vior of the partition fun ction provi des an ev aluatio n for the regularit y of the sample path of the pr o cess t 7→ ˜ µ ([0 , t ]) . Ho w ev er, it is more natur al to assess this regularit y via the b eha vior of wa ve let co efficient s. imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 20 4.1. Notations. In this section we assum e, for notational con v enience, that T = 1. Consider n ow g a “generali zed b ox” function. It is a real v alued function that satisfies the follo wing assumptions (H1) g has compact su pp ort included in [0 , 2 J ], for some J ≥ 0. (H2) g is piecewise contin uous. (H3) g is at least non zero on an in terv al. F ollo wing the common wa v elet notation, we define g j,k ( t ) = g (2 j t − k ) . The sup p ort of g j,k ( t ) is (42) Supp g j,k = [2 − j k , 2 − j k + 2 J − j ] . In th e sequel, if µ is a random measure, f or an y Borel function f w e will use the n otation h µ, f i = Z f ( t ) dµ ( t ) . 4.2. The gener alize d p artition function : sc aling pr op erties. W e define the generalized partition function of an M-cascade µ ∞ on [0 , 1] at scale 2 − j as (43) S µ,g ( j, p ) = 2 j − 2 J − 1 X k =0 |h µ ∞ , g j,k i| p . Remark that for simp licit y w e r emo ved a finite num b er of b order terms, and that, in th e case g ( t ) is the “box” fun ction g ( t ) = 1 [0 , 1] ( t ) w e reco ver the partition fu nction of S ection 2.1. Let us stud y the scaling of E [ S µ,g ( j, p )] . Pr oposition 4 . Assume (5) – (6) . Then, we have K 1 2 − j τ ( p ) ≤ E [ S µ,g ( j, p )] ≤ K 2 2 − j τ ( p ) for K 1 , K 2 , two p ositive c onstants dep ending on p , W and g . Pr oof. Since | g ( t ) | is clearly a b oun ded function, we ha ve E [ |h µ ∞ , g j,k i| p ] ≤ C E h µ ∞ ([2 − j k , 2 − j k + 2 J − j ]) p i , where C is a constan t. W e write µ ∞ ([2 − j k , 2 − j k +2 J − j ]) = P 2 J − 1 l =0 µ ∞ ( I j,k + l ) , and dedu ce (44) E [ |h µ ∞ , g j,k i| p ] ≤ C E h | µ ∞ [0 , 2 − j ] | p i = K 2 − j ( τ ( p )+1) , imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 21 where K only d ep ends on g and the la w of W . By (43) w e get the up p er b ound f or E [ S µ,g ( j, p )] . F or the low er b oun d, let us write that S µ,g ( j, p ) is greater than 2 j − J − 1 X k ′ =0 | D µ ∞ , g j, 2 J k ′ E | p . But g j, 2 J k ′ is supp orted on [ k ′ 2 J − j , ( k ′ + 1)2 J − j ], thus app lying L emm a 3 in the App endix B with a = j − J , we deduce: D µ ∞ , g j, 2 J k ′ E = 2 J − j   j − J Y i =1 W r | i   Z where, in la w , Z is equal to h µ ∞ , g J, 0 i . Thus E [ |h µ ∞ , g j,k i| p ] is greater than 2 p ( J − j ) E [ W p ] j − J E [ |h µ ∞ , g J, 0 i| p ] = K 2 − j ( τ ( p )+1) E [ |h µ ∞ , g J, 0 i| p ] . Applying Lemma 4 with f = g J, 0 sho ws th at E [ |h µ ∞ , g J, 0 i| p ] is some p ositiv e constan t. Th en the lo w er b ound for E [ S µ,g ( j, p )] easily follo ws. 4.3. The p artition fu nction in the mixe d asymp totic f r amewo rk. F ollo w- ing (15), we define the p artition function in the mixed asymp totic framework as S ˜ µ,g ( j, p ) = N T − 1 X m =0 S ( m ) µ,g ( j, p ) , where S ( m ) µ,g ( j, p ) is th e partition fu nction of µ ( m ) ∞ , i.e., S ( m ) µ,g ( j, p ) = 2 j − 2 J − 1 X k =0    D µ ( m ) ∞ , g j,k E    p . W e hav e th e follo wing result. Theorem 5 . Scaling of the genera lized partition function in a mixed asymptotic L et p > 0 , then under the same assumptions as The or em 2 the p ower law sc aling of S ˜ µ,g ( j, p ) is given by lim j → ∞ log 2 S ˜ µ,g ( j, p ) − j − → a.s. τ χ ( p ) . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 22 Pr oof. Using P rop osition 4 w e h a ve, lim j → ∞ 1 − j log 2 E [ S ˜ µ,g ( j, p )] = τ χ ( p ) , and we just need to prov e that almost surely , S ˜ µ,g ( j, p ) − E [ S ˜ µ,g ( j, p )] = o (2 − j τ χ ( p ) ) . Using lemma 5 and 6 of app endix B , this is done in th e exact same wa y as the p r o of of (24) in P rop osition 1. 5. Link with Beso v spaces. F ollo wing [9], one ma y defi ne f or a mea- sure µ on [0 , T ], the b oundary of its Beso v d omain as the function s µ : (0 , ∞ ) → R ∪ {∞} giv en by s µ (1 /p ) = sup      σ ∈ R | sup j ≥ 0 2 j σ   2 − j 2 j − 1 X k =0 | µ ( I j,k ) | p   1 /p < ∞      . The follo wing prop osition can b e shown (see [9]). Pr oposition 5 . The function s µ is an incr e asing, c onc ave fu nction, with a derivative b ounde d by 1 . Let us stress th at th e condition s ′ µ (1 /p ) ≤ 1 is a simp le consequence of the Sob olev embed d ing for Beso v spaces. Th e T heorem 1 c haracterizes the Beso v d omain for µ ∞ a M-cascade on [0 , T ]: ∀ p > 0 , s µ ∞ (1 /p ) =    τ ( p )+1 p if 1 p > 1 p 0 h + 0 + 1 p if 1 p ≤ 1 p 0 . If we den ote s ( 1 p ) = τ ( p )+1 p then, it is simply c hec k ed th at the condition 1 /p > 1 /p + 0 is equiv alen t to s ′ (1 /p ) < 1. Hence Prop osition 5 explains wh y for 1 /p ≤ 1 /p + 0 the b oundary of the Besov d omain must b e linear with a slop e equ al to one. In mixed asymptotic the supp ort of the measure gro ws with j but we can still d efine, using the notations of Section 3 , the ind ex: s χ ˜ µ (1 /p ) = su p      σ ∈ R | sup j ≥ 0 2 j σ   N − 1 T 2 − j N T 2 j − 1 X k =0 | ˜ µ ( I j,k ) | p   1 /p < ∞      . Then, it is simply c hec k ed that Theorem 2 implies s χ ˜ µ (1 /p ) = s (1 /p ) when s ′ (1 /p ) < 1 + χ , and s χ ˜ µ (1 /p ) = h + χ + 1+ χ p otherwise. This s h o ws how the linear part in s χ ˜ µ is shifted to larger v alues of p under the mixed asymptotic framew ork. imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 23 Fig 1 . Thr e e synthetic samples of M -c asc ades with T = 2 13 and λ 2 = 0 . 2 : (a) lo g-Normal sample, (b) lo g-Poisson sample wi th δ = − 0 . 1 and (c) lo g-Ga mma sample wi th β = 10 . In fact we use d µ j max +5 [ n, n + 1] n =0 ...L as a pr oxy of µ ∞ [ n, n + 1] with j max = log 2 ( T ) = 13 (se e Eq. (3 ) for the definition of µ j ). imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 24 6. Numerical examples a nd applications. Our goal in this section is not to focus on statistical issues and notably on precise estimate s of mul- tifractal exp onent s from empirical data. W e rather aim at illustrating the results of theorem 2 on p recise examples, n amely random cascades with re- sp ectiv ely log-normal, log-P oisson and log-Gamma statistics. F or the sak e of simplicit y we will consider exclusively scaling of p artition function for p ≥ 0 1 . In order to facilitate the comparison of the thr ee m o dels, λ 2 will represent the so-called in termittency co efficien t, i.e., λ 2 = − τ ′′ (0) where τ ( p ) is defin ed in Eq. (7). This v alue will b e fixed for the three con- sidered mo dels. Let { W r } b e the cascade r andom generators as defined in Eq. (3) and let ω r = ln W r . In the simplest, log-Normal case the { ω r } r are normally distribu ted ran- dom v ariables of v ariance λ 2 ln(2). Thanks to the condition E [ W r ] = E [ e ω r ] = 1, their mean is necessarily − λ 2 ln(2) / 2. In that case, the cum u lant gener- ating f u nction τ ( p ) defin ed in Eq. (7) is simply a parab ola: τ n ( p ) = p (1 + λ 2 2 ) − λ 2 2 p 2 − 1 In the log-P oisson case, the v ariables ω r are written as ω r = m 0 ln(2) + δ n r where the n r are in tegers distribu ted according to a Poi sson la w of mean γ ln(2). It results that τ ( p ) = p (1 − m 0 ) + γ (1 − e pδ ) − 1. If one s ets τ (1) = 0 and τ ′′ (0) = − λ 2 , one fi nally gets th e expression of τ ( p ) of a log-P oisson cascade with in termittency co efficient λ 2 : (45) τ p ( p ) = p 1 + λ 2 δ 2 ( e δ − 1) ! + λ 2 δ 2 (1 − e pδ ) In third case the v ariables ω r are dr a w n from a Gamma distribution. If x is a random v ariable of p d f β α ln(2) x α ln(2) − 1 e − β x / Γ( α ln(2)), then one chooses ω r = x + m 0 ln(2) and it is easy to sh o w that τ ( p ) is d efined only for p < β and in this case τ ( p ) = p (1 − m 0 ) + α (1 − p/β ). By fixing τ (1) = 1 and τ ′′ (0) = λ 2 , on obtains: (46) τ g ( p ) = p  1 − λ 2 β 2 ln β − 1 β  + λ 2 β 2 ln β − p β Notice that one reco vers the log-normal case f r om b oth log-P oisson and log-Ga mma statistics in the limits δ → 0 and β → + ∞ r esp ectiv ely . 1 Numerical method s for estimating τ ( p ) for p < 0 are tric kier to handle imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 25 F or the 3 cases, one can exp licitely compute all the mixed asymptotic exp onent s as functions of χ : In particular the v alues of p ± χ read: p ± χ,n = ± s 2(1 + χ ) λ 2 p ± χ,p = W  ± , δ 2 (1+ χ ) − λ 2 eλ 2  + 1 δ = p ± χ,n + 2(1 + χ ) 3 λ 2 δ + O ( δ 2 ) p ± χ,g = β " 1 + 1 + χ λ 2 β 2 − e 1+ W ( ± , − e − 1 − 1+ χ λ 2 β 2 ) # = p ± χ,n − 4(1 + χ ) 3 λ 2 β + O ( β − 2 ) where suffixes n, p, g stand for resp ectiv ely log-normal, log-P oisson and log- Gamma cascades and W ( ± , z ) represent th e t w o branches of the Lam b ert W ( z ) fu nction, solution of W ( z ) e W ( z ) = z that tak e (resp ectiv ely p ositiv e and negativ e) r eal v alues for the consid er ed arguments. F or log-P oisson and log-Ga mma cases, w e h a ve also indicated the asymp totic b eha vior in the limits δ → 0 and β → ∞ . The v alues h ± χ can be easily dedu ced form their definition: h ± χ = τ ′ ( p ± χ ). In Fig. 1 is plotte d a sample of eac h of the three examples of M -cascades. W e chose T = 2 13 and λ 2 = 0 . 2 for all mo dels while, in the log-P oisson case we ha v e set δ = − 0 . 1 and β = 10 in th e log-Gamma m o d el. In eac h case, an appro ximation of the M -cascade samp le is generated. W e c hose to generate µ 18 (as defin ed b y Eq. (3)) so th at the smallest scale inv olve d is l min = 2 − 18 T = 2 − 5 (w e h a ve chec k ed that the results rep orted b elo w do not dep end on l min ). An app ro ximation of ˜ µ is generated b y concatenating i.i.d. realizatio ns of µ 18 . Then, for eac h mo del and for eac h c hosen v alue of χ , τ χ ( p ) ( p = 0 . . . 6) wa s obtained fr om a least square fit of the cur v e log 2 S ˜ µ ( j, p ) v ersus j ov er the range j = 0 . . . 6. Let us recall that, for ea c h v alue of j , the mixed asymptotic regime corresp onds to sampling ˜ µ at scale l = 2 − j T and o v er an inte rv al of size L = 2 j χ T . The exp onents r ep orted in figs. 2 and 3 represent the mean v alues of exp onents estimated in that wa y using N = 130 exp erimen ts. The log-Normal mixed asymptotic scaling exp onen ts for χ = 0 , 0 . 5 , 1 are represent ed in Fig. 2. F or illustration p u rp ose w e ha ve p lotted τ χ ( p ) + χ as a function of p : one clearly observes that, as the v alue of χ increases, the v alue of p + χ b elo w wh ich the function is linear, also increases while the v alue of the slop e h + χ decreases. As exp ected, when χ increases τ χ ( p ) + χ matc h es τ ( p ) ov er an increasing r ange of p v alues. Noti ce that the estimated exp onent s are very close the analytica l pr ed ictions as represen ted b y the dashed lines. Er ror b ars on the mean v alue estimates are s imply compu ted from th e estimated r.m.s. o v er the 130 trials and are rep orted only for the imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 26 Fig 2 . Estimates of the f unction τ χ ( p ) of lo g-normal c asc ades with λ 2 = 0 . 2 for χ = 0 ( • ) χ = 0 . 5 ( ◦ ) and χ = 1 ( N ). Dashe d lines r epr esent the c orr esp onding analytic al expr ession fr om the or em 2 and the solid line r epr esent s the f unction τ ( q ) as define d in Eq. (7 ) . τ χ is estimate d f r om the aver age over 130 trials of 2 j χ c asc ades samples. Err or b ars ar e r ep orte d on the χ = 0 curve as vertic al solid b ar. These err ors ar e of or der of symb ol size. Fig 3 . Estimates of the function τ χ ( p ) of l o g-Poisson for χ = 0 ( • ) and χ = 1 ( ◦ ). In b oth mo dels we chose T = 2 13 and λ 2 = 0 . 2 . (a) L o g-Poisson c ase with δ = − 0 . 1 . (b) L o g-Gamma c ase with β = 10 . Soli d lines r epr esent the curve s τ ( p ) . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 27 χ = 0 curve . W e can see that these errors are smaller or close to the symbol tic kn ess. In Fig. 3 are r ep orted estimates of τ χ ( p ), χ = 0 , 1 for log-P oisson (fi g. 3(a)) and log-Gamma (fig. 3(b)) samples. The solide lines represent the theoretical τ ( p ) functions for b oth mo dels as pr o vid ed by Eqs. (45) and (46). W e u sed the same estimatio n pro cedure as for the log- normal case. One sees that, in b oth cases, since the intermittency co efficien t is the same for the three mo dels, the classical τ 0 ( p ) curve s are v ery similar to the log-normal curve (Fig. 2). Ho wev er, these mod els b eha v e v ery differently in mixed regime: for χ = 1, log-P oisson and log-Gamma b oth estimated scaling exp onents b e- come closer to th e r esp ectiv e v alues of τ ( p ) an d are v ery easy to distinguish. Let us men tion that suc h a analysis has b een recen tly p erformed b y tw o of us in order to distin gu ish t w o p opular log-normal and log-P oisson mo d els for s patial fluctuations of en er gy dissip ation in f u lly develo p ed turb ulence [19]. APPENDIX A: LEMMA USED FOR THE PR OOF OF THEOREM ?? Lemma 1 . We have L 1+ ǫ N ≤ C 2 j χ E     M (0) j ( p )    1+ ǫ  wher e L 1+ ǫ N is define d by (25) and C is a c onsta nt that dep ends only on ǫ . Pr oof. According to [25], if ǫ ∈ [0 , 1] and if { X i } 1 ≤ i ≤ P are centered indep end en t random v ariables one has E        P X i =1 X i      1+ ǫ   ≤ C P X i =1 E h | X i | 1+ ǫ i , where C is a constan t that dep ends only on ǫ (and neither on the la w of X n or on P ). Applying it with P = N T = ⌊ 2 j χ ⌋ to the expression (23 ) of N j ( p ), and using the fact th at the ran d om v ariables {M ( m ) j ( p ) } m defined by (22) are i.i.d, one gets L 1+ ǫ N ≤ C 2 j χ E     M (0) j ( p ) − E h M (0) j ( p ) i    1+ ǫ  ≤ C 2 j χ  E     M (0) j ( p )    1+ ǫ  + E h    M (0) j ( p )    i 1+ ǫ  . Using the Jensen inequality we get the resu lt. imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 28 Lemma 2 . Assume that E  | Z | 1+ ǫ  < ∞ . Then we have f or al l m , E     M ( m ) j ( p )    1+ ǫ  ≤ C 2 − j (1+ ǫ ) τ ( p ) j X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) , wher e C is a c onstant that dep ends only on p and ǫ . Pr oof. The pr o of of this r esu lt is very muc h inspired from [23]. Since the la w of M ( m ) j ( p ) is indep enden t of m , we forget th e sup script m throughout the p r o of. Using the definition (22) , one gets (47) E h |M j ( p ) | 1+ ǫ i = 2 − j p (1+ ǫ ) E          X r ∈{ 0 , 1 } j j Y i =1 W p r | i Z ( r )       1+ ǫ    . Let (48) X = 2 − j p X r ∈{ 0 , 1 } j j Y i =1 W p r | i Z ( r ) , then X 2 = 2 − 2 j p X r 1 ∈{ 0 , 1 } j X r 2 ∈{ 0 , 1 } j j Y i =1 W p r 1 | i W p r 2 | i Z ( r 1 ) Z ( r 2 ) . It can b e rewritten as (49) X 2 = Y + D , where Y corresp onds to the non diagonal terms : (50) Y = 2 − 2 j p X r 1 ∈{ 0 , 1 } j X r 2 ∈{ 0 , 1 } j r 2 6 = r 1 j Y i =1 W p r 1 | i W p r 2 | i Z ( r 1 ) Z ( r 2 ) , and D to the diagonal terms (51) D = 2 − 2 j p X r ∈{ 0 , 1 } j j Y i =1 W 2 p r | i  Z ( r )  2 . The left hand side of (47) is nothing bu t E h | X | 1+ ǫ i . By writing th at E h | X | 1+ ǫ i = E   X 2  1+ ǫ 2  , using the sub -additivit y of x 7→ x (1+ ǫ ) / 2 , w e get (52) E h | X | 1+ ǫ i ≤ E h | Y | 1+ ǫ 2 i + E h D 1+ ǫ 2 i . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 29 Let us first w ork with the Y te rm. W e f actorize the common b eginnin g of the words r 1 and r 2 in the expression (50) of Y Y = 2 − 2 j p j − 1 X k =0 X r ∈{ 0 , 1 } k k Y i =1 W 2 p r | i X r 1 ,r 2 ∈{ 0 , 1 } j − k r 1 | 0 6 = r 2 | 0 j Y i = k +1 W p r r 1 | i W p r r 2 | i Z ( r 1 ) Z ( r 2 ) . Again by the sub -additivit y of x 7→ x (1+ ǫ ) / 2 and using the fact that th e W r | i are i.i.d., one gets E h | Y | 1+ ǫ 2 i ≤ 2 − j p (1+ ǫ ) j − 1 X k =0 E h W p (1+ ǫ ) i k X r ∈{ 0 , 1 } k E       X r 1 ,r 2 ∈{ 0 , 1 } j − k r 1 | 0 6 = r 2 | 0 j Y i = k +1 W p r r 1 | i W p r r 2 | i | Z ( r 1 ) Z ( r 2 ) |  (1+ ǫ ) / 2      , and by using Jensen inequalit y E h | Y | 1+ ǫ 2 i ≤ 2 − j p (1+ ǫ ) j − 1 X k =0 E h W p (1+ ǫ ) i k X r ∈{ 0 , 1 } k  X r 1 ,r 2 ∈{ 0 , 1 } j − k r 1 | 0 6 = r 2 | 0 j Y i = k +1 E h W p r r 1 | i W p r r 2 | i i E h | Z ( r 1 ) Z ( r 2 ) | i  (1+ ǫ ) / 2 . (53) The v ariables Z ( r 1 ) and Z ( r 2 ) are ind ep endant w ith finite exp ectation, thus the term E h | Z ( r 1 ) Z ( r 2 ) | i is b ounded b y a constan t C . Using Q j i = k +1 E h W p r r 1 | i W p r r 2 | i i = E [ W p ] 2( j − k ) , w e ded uce: E h | Y | 1+ ǫ 2 i ≤ C 2 − j p (1+ ǫ ) j − 1 X k =0 E h W p (1+ ǫ ) i k E [ W p ] ( j − k )(1+ ǫ ) X r ∈{ 0 , 1 } k | X r 1 ,r 2 ∈{ 0 , 1 } j − k r 1 | 0 6 = r 2 | 0 1 | (1+ ǫ ) / 2 . There are 2 k p ossible v alues for r and less than 2 2( j − k ) v alues for the couple ( r 1 , r 2 ), thus E h | Y | 1+ ǫ 2 i ≤ 2 − j p (1+ ǫ ) K j − 1 X k =0 E h W p (1+ ǫ ) i k E [ W p ] ( j − k )(1+ ǫ ) 2 k 2 ( j − k )(1+ ǫ ) . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 30 Since 2 − j τ ( p ) = 2 − j p 2 j E [ W p ] j (54) E h | Y | 1+ ǫ 2 i ≤ K 2 − j (1+ ǫ ) τ ( p ) j − 1 X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) . Let us no w tak e care of th e diagonal terms of X (51). First, w e write D 1+ ǫ 2 ≤ 2 − j p (1+ ǫ ) X r ∈{ 0 , 1 } j j Y i =1 W p (1+ ǫ ) r | i | Z ( r ) | 1+ ǫ , and usin g the E h | Z | 1+ ǫ i < ∞ w e ded u ce that (55) E h D 1+ ǫ 2 i ≤ C 2 − j p (1+ ǫ ) 2 j E h W p (1+ ǫ ) i j = C 2 − j τ ( p (1+ ǫ )) . Merging (54) and (55) in to (52) leads to E h | X | 1+ ǫ i ≤ K 2 − j (1+ ǫ ) τ ( p ) j X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) , and since E h |M j ( p ) | 1+ ǫ i = E h | X | 1+ ǫ i , it completes the p r o of. APPENDIX B: LE MMA US ED FOR THE PR OOF O F THEOREM ?? Lemma 3 . L et f : [0 , 1] → R b e some Bor el function whose supp ort is include d in I a, r = [ r 2 a , ( r +1) 2 a ] for r ∈ { 0 , 1 } a , a ≥ 0 . Then (56) h µ ∞ , f i = 2 − a a Y i =1 W r | i ! D µ ( r ) ∞ , ˜ f E wher e ˜ f ( x ) = f (2 − a ( x + r )) and µ ( r ) ∞ is a c asc ade me asur e on [0 , 1] me asur able with r esp e ct to the sigma field σ { W r r ′ , r ′ ∈ { 0 , 1 } a ′ , a ′ ≥ 1 } . Pr oof. The s caling relat ion (56 ) is easily obtained, by the defin ition of the measur e µ ∞ , if f is the c haracteristic fu nction of some inte rv al I a + a ′ , r r ′ where r ′ ∈ { 0 , 1 } a ′ , a ′ ≥ 0. This relation extends to any Borel function f by standard argum ents of measure theory . Lemma 4 . L et h : [0 , 1] → R b e a pie c ewise c ontinuous, non zer o, func- tion. Then E [ |h µ ∞ , h i | p ] > 0 for al l p > 0 . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 31 Pr oof. By con tradictio n, assum e that for some p > 0, E [ |h µ ∞ , h i| p ] = 0. Hence h µ ∞ , h i = 0, P -almost sur ely . But using Lemm a 3, 0 = h µ ∞ , h i = D µ ∞ , h 1 [0 , 1 / 2] E + D µ ∞ , h 1 (1 / 2 , 1] E = 1 2 W 0 D µ (0) ∞ , h (0) E + 1 2 W 1 D µ (1) ∞ , h (1) E , where h (0) ( · ) = h (2 − 1 · ), h (1) ( · ) = h (2 − 1 ( · + 1)) an d µ (0) ∞ , µ (1) ∞ are indep en den t cascade m easur es on [0 , 1]. Th us we deduce W 0 D µ (0) ∞ , h (0) E = − W 1 D µ (1) ∞ , h (1) E almost surely , and since W > 0 this shows that with probabilit y one the tw o indep end en t v ariables D µ (0) ∞ , h (0) E and D µ (1) ∞ , h (1) E v anish sim ultaneously . This is only p ossible either, if they b oth v anish on a set of full p robabil- it y , or if they b oth v anish on a negligible set. Assume the latter, then the follo win g id en tit y h olds almost sur ely D µ (0) ∞ , h (0) E D µ (1) ∞ , h (1) E = − W 1 W 0 where the v ariables on right and left hand sid e are indep end en t. Th ese v ari- ables must b e constan t, whic h is excluded b y the assum ption P ( W = 1) < 1 (recall (5) ). Th us w e deduce that the v ariables D µ ( i ) ∞ , h ( i ) E are almost surely equal to zero. Hence: E h    D µ ∞ , h ( i ) E    p i = 0 , for i = 0 , 1 . Iterating the argum en t w e dedu ce the f ollo wing p rop ert y: for any j ≥ 0 and k ≤ 2 j − 1, if we defin e a fun ction on [0 , 1] by h ( j,k ) ( x ) = h (2 − j ( x + k )) we ha v e E h    D µ ∞ , h ( j,k ) E    p i = 0 . This is clearly imp ossible if w e c ho ose j, k suc h th at h r emains p ositiv e (or negativ e) on [ k 2 − j , ( k + 1)2 − j ]. By the assumptions on h one can fin d suc h an interv al, yielding to a contradict ion. Lemma 5 . We have E h |S ˜ µ,g − E [ S ˜ µ,g ] | 1+ ǫ i ≤ C 2 j χ E h |S µ,g ( j, p ) | 1+ ǫ i wher e C is a c onstant that dep ends only on ǫ . Pr oof. The pro of is the exact same pro of as for Lemma 1. imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 32 Lemma 6 . F or any ǫ > 0 smal l enough we have, E h |S µ,g ( j, p ) | 1+ ǫ i ≤ K 2 − j (1+ ǫ ) τ ( p ) j X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) , wher e K is a c onstant that dep ends only on p and ǫ . Pr oof. The pro of basically follo ws the same lines as th e pro of of Lemma 2. The only d ifficult y , compared to this latter pro of, comes from the fact that the quanti t y h µ ∞ , g j,k i a p r iori inv olv es several no des of lev el j of the M- cascade. W e ha v e to reorganize the sum (43). Since we are interested in the limit j → + ∞ , we can s u pp ose, with no loss of generalit y that j > J . In the follo w ing, we note 1 ( n ) the n -u plet 1 ( n ) = 11 . . . 1 , where the 1 is rep eated n times . The p artition function (43) can b e written S µ,g ( j, p ) = P k | D µ ∞ , g j, k E | p , where the sum is o v er k ∈ { 0 , 1 } j suc h th at k l = 0 for some l ≤ j − J . The sum can b e regroup ed in th e follo wing w ay , where a + 1 denotes the p osition of the last 0 in the j − J first comp onents of k : S µ,g ( j, p ) = j − J − 1 X a =0 X r ∈{ 0 , 1 } a q = r 0 X s ∈{ 0 , 1 } J    D µ ∞ , g j, q 1 ( j − J − 1 − a ) s E    p . W e set (57) X a,s = X r ∈{ 0 , 1 } a q = r 0    D µ ∞ , g j, q 1 ( j − J − 1 − a ) s E    p and consequently S µ,g ( j, p ) = j − J − 1 X a =0 X s ∈{ 0 , 1 } J X a,s . Actually , a exactly corresp onds to the leve l of the “highest” no de that is common for dy adic interv als in the sup p ort of g j, q 1 ( j − J − 1 − a ) s . In deed, let us pro v e that (58) a ≥ 0 , ∀ s ∈ { 0 , 1 } J , Supp g j, q 1 ( j − J − 1 − a ) s ⊂ I a, r , imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 33 where q = r 0. Indeed, according to (42), the sup p ort of g j, q 1 ( j − J − 1 − a ) s is included in [2 − j q 1 ( j − J − 1 − a ) s, 2 − j q 1 ( j − J − 1 − a ) s + 2 − ( j − J ) ] . Th en, 2 − j q 1 ( j − J − 1 − a ) s = 2 − a r + j − J X i = a +2 2 − i + 2 − j s = 2 − a r + 2 − a − 1 − 2 − ( j − J ) + 2 − j s. Since s v aries in [0 , 2 J − 1], and a ≤ j − J − 1, it is easy to show that 0 ≤ 2 − a − 1 − 2 − ( j − J ) + 2 − j s, and 2 − a − 1 + 2 − j s ≤ 2 − a , whic h prov es (58). W e are n o w ready to compute the upp er b ound for kS µ,g ( j, p ) k L 1+ ǫ ( P ) ≤ P j − J − 1 a =0 P s ∈{ 0 , 1 } J k X s,a k L 1+ ǫ ( P ) . Using (57), (58) and L emma 3, we get X a,s = 2 − ap X r ∈{ 0 , 1 } a a Y i =1 W p r | i ! | D µ ( r ) ∞ , g j, 01 ( j − J − 1 − a ) s E | p , where the µ ( r ) ∞ are indep end en t cascade measures on [0 , T ]. Let us identify X a,s with X as defined in Lemma 2 by (48) in wh ich j pla ys th e role of a and Z ( r ) of D µ ( r ) ∞ , g j − a, 01 ( j − J − 1 − a ) s E p . As in (49), we can decomp ose X 2 a,s as the su m of the non d iagonal terms Y a,s and the diago nal terms D a,s (59) X 2 a,s = Y a,s + D a,s . Using the exact same dev elopmen t as the one w e used for E  X 1+ ǫ  starting at Eq. (47), we get the b ound of the non diagonal term s corresp onding to (53) in whic h the term E h Z ( rr 1 ) Z ( rr 2 ) i has to b e replaced by E h    D µ ( rr 1 ,s ) ∞ , g j − a, 01 ( j − J − 1 − a ) s E    p    D µ ( rr 2 ,s ) ∞ , g j − a, 01 ( j − J − 1 − a ) s E    p i whic h can b e b oun ded (u sing (44)) by K 2 − 2( j − a )( τ ( p )+1) . Going on with the same arguments as in Lemma 2, we fi nally ge t, the b ound for the non diagonal terms corresp ondin g to (54) E h | Y a,s | 1+ ǫ 2 i ≤ K 2 − j (1+ ǫ ) τ ( p ) 2 ( a − j )(1+ ǫ ) a − 1 X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) ≤ K 2 − j (1+ ǫ ) τ ( p ) 2 ( a − j )(1+ ǫ ) j − 1 X k =0 2 − k τ ( p (1+ ǫ )) 2 k (1+ ǫ ) τ ( p ) . imsart-aap ver. 2007/12/10 file: paper_ver5-3 _IMS.tex date: November 20, 2018 34 F ollo wing the argumen ts in Lemma 2 for the diagonal terms, w e get (60) E  D 1+ ǫ 2 a,s  ≤ 2 − j τ ( p (1+ ǫ )) 2 a − j . 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