Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes

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1 Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes arxiv:4599v [mathpr] Dec 4 R Le Guével Université de Rennes - Haute Bretagne, Equipe de Statistique Irmar, UMR CNRS 665 Place du Recteur Henri Le Moal, CS 437, 3543 RENNES Cedex, France ronanleguevel@univ-rennesfr and J Lévy Véhel Regularity team, INRIA Saclay and MAS laboratory Ecole Centrale Paris, Grande Voie des Vignes, 99, Chatenay Malabry, France jacqueslevy-vehel@inriafr Abstract We compute the Hausdorff multifractal spectrum of two versions of multistable Lévy motions These processes extend classical Lévy motion by letting the stability exponent α evolve in time The spectra provide a decomposition of [, ] into an uncountable disjoint union of sets with Hausdorff dimension one We also compute the increments-based large deviations multifractal spectrum of the independent increments multistable Lévy motion This spectrum turns out to be concave and thus coincides with the Legendre multifractal spectrum, but it is different from the Hausdorff multifractal spectrum The independent increments multistable Lévy motion thus provides an example where the strong multifractal formalism does not hold Introduction and background Multifractal analysis gives a fairly complete description of the singularity structure of measures, functions or stochastic processes Various versions of multifractal analysis exist, which include the determinations of the so-called Hausdorff, large deviation, and Legendre multifractal spectra [] Multifractal analysis has been performed for various measures [, 7], functions [5], and stochastic processes [4, 5, 8, 9, 6] In the case of Lévy processes, substantially finer results have been obtained in [3] using -microlocal analysis This article deals with the multifractal analysis of extensions of Lévy stable motions known as multistable Lévy motions Generally speaking, multistable processes extend the well-known stable processes see, eg [3] by letting the stability index α evolve in time These processes have been introduced in [3] and have been studied for instance in [, 6, 4, 8, 9, ] They provide useful models in various applications where the data display jumps with varying intensity, such as financial records, EEG or natural terrains: indeed, multistability is one practical way to deal with increments- non-stationarities

2 observed in various real-world phenomena, since a multistable process X is tangent, at each time u, to a stable process Z u in the following sense [, ]: Xu+rt Xu lim = Z r r h u t for a suitable h the limit is taken either in finite dimensional distributions or, when X has a version with càdlàg paths, in distribution - one then speaks of strong localisability Without loss of generality, we shall consider our processes on [,] We will need the following ingredients: α : [,], is a C function Γ i i is a sequence of arrival times of a Poisson process with unit arrival time V i i is a sequence of iid random variables with uniform distribution on [,] γ i i is a sequence of iid random variables with distribution Pγ i = = Pγ i = = / The three sequencesγ i i,v i i, andγ i i are independent We denotec = inf αu, u [,] d = sup αu, u [,] We shall consider two versions of Lévy multistable processes: the first one has independent but non stationary increments It admits the following representation: Bt = + i= γ i C /αv i αv i Γ /αv i i Vi t, while the second one has correlated non stationary increments and reads: Dt = C /αt αt γ i Γ /αt i [,t] V i, 3 i= where C u = x u sinx dx Both processes are semi-martingales and are tangent, at each time t, to αt stable Lévy motion See [9] and the references therein for more details on these processes We shall also denote: Yt = i= γ i Γ /αt i [,t] V i Hausdorff multifractal spectra Let h Y t denote the pointwise Hölder exponent of Y at t The Hausdorff multifractal analysis of Y consists in measuring the Hausdorff dimension denoted dim H of the sets F h = {t [,] : h Y t = h} The Hausdorff multifractal spectrum is the function h f H h := dim H F h We will use the following notations: S = i {V i }, S = S N and R t = {r n n N S : r n t} If r n n R t, we put V φn = r n Finally, define the positive function δ for t / S, δt = inf liminf logφi V φn n R t i log V φi t

3 Main result The Hausdorff multifractal spectra of both B and D are described by the following theorem: Theorem With probability one, the common Hausdorff multifractal spectrum f H of B and D satisfies: for h < ; hd for h [, ]; d f H h = for h, ; 4 d c dim H {t [,] : αt = c} for h = c ; for h > c Theorem follows from a series of lemmas that are proven in the next section: Lemma Almost surely, t Yt is càdlàg Lemma 3 Almost surely, t [, ]\S, δt Lemma 4 Almost surely, t [,]\S, h Y t δt αt Lemma 5 Let g : [,] R be a càdlàg function, and f be the function defined on [,] by ft = t gxdx The pointwise Hölder exponent h f of f verifies: t,, h f t Lemma 6 Almost surely, t [,]\S, h Y t δt αt Lemma 7 Almost surely, h <,f H h = Lemma 8 Almost surely, f H = Lemma 9 Almost surely, h, d ],f Hh = hd Lemma Almost surely, h d, c,f Hh = Lemma Almost surely, f H c = dim H {t [,] : αt = c} Lemma Almost surely, h > c,f Hh = Proofs of the lemmas Proof of Lemma : Set Y N t = N γ i Γ /αt i [,t] V i Lemma 8 in [9] states that, amost surely, Y N i= converges to Yt uniformly on [, ] Fix ε > and choose N N such that, N N, sup Y N t Yt ε t [,] 3 5 3

4 st case : t S Let i N be such that t = V i, and N = maxi,n Then, for h R, Yt Yt+h = Yt Y N t+y N t Y N t+h+y N t+h Yt+h Since lim h +Y N t Y N t+h =, there exists h > such that h,h, Y N t Y N t+h ε 3 As a consequence, h,h, Yt Yt+h ε and thus lim +Yt Yt+h = lim h Y N t Y N t+h = lim h N i= h γ i Γ /αt i t+h,t] V i = γ i Γ /αv i, thus lim h Y N t Y N t+h γ i Γ /αv i i = Choose h < such that h h,, Thus, Yt Yt+h γ i Γ /αv i i ε lim h Yt Yt+h = γ i Γ /αv i i nd case : t / S Since lim h Y N t+h Y N t =, there exists h > such that h < h, Y N t Y N t+h ε 3 Using 5, one thus has, for h < h, Yt Yt+h ε, and thus lim h Yt+h Yt = Note We have shown precisely that Y is càdlàg with set of jump points exactly equal to S The jump at point V i is of size γ i Γ /αv i i i Proof of Lemma 3: For j, k =,, j, let E k,j denote the interval [ k E k,j = j, k + +j+ j j Let us show that liminf j j j+ j k= j+ +j+ j {V i E k,j } { t / S,δt } first, and then that i= j P {V i E k,j } = We denote a j = j k= i= j +j+ j Assume that there exists J N such that for all j J, and all k =,, j, we can fix ik,j [ j, j+ ] with V ik,j E k,j Let t / S For all j J, there exists kj [, j ] such that t E kj,j, because a j j 4

5 As a consequence, t V ikj,j j+ j ikj,j j Hence logikj,j log t V ikj,j This entails liminf logikj,j j j log t V ikj,j, and, since V ikj,j tends to t, δt Finally, distinguishing the cases k =, k =,, j and k = j, one estimates j P k= j+ {V i / E k,j } i= j j k= P {V i / E k,j } i= j j+ a j + j j a j j + P{V / E k,j } j k= a j + j +j a j j Since + j a j j < +, Borel-Cantelli lemma allows us to conclude j= Proof of Lemma 4: Recall Note Lemma of [5] entails that, for all sequences V φi R t, and all t / S, h Y t liminf i + log Γ αv φi φi log V φi t = liminf i + αv φi log Γ φi log V φi t Since α is continuous, φi tends to infinity, the sequences V φi i converges to t, and almost surely Γ i i i tends to when i tends to infinity, one obtains h Y t αt liminf i + log φi log V φi t This inequality holds for all sequences V φi R t, and thus, t / S, h Y t δt αt Proof of Lemma 5: Since g is càdlàg, h g is non negative for all t Integration increases pointwise regularity by at least one, and thus h f t for all t An alternative direct proof goes as follows: let t,, and h > One computes ft+h ft h gt + = h = 5 t t+h gx gt + dx gt+sh gt + ds

6 s,, lim h +gt+sh gt+ = Since g is càdlàg, it is bounded and thus: Likewise, for h <, ft+h ft lim = gt + h + h and This entails h f t ft+h ft h Proof of Lemma 6: Theorem 7 of [9] states that: where At = gt = gt+sh gt ds, ft+h ft lim = gt h h t + i= Dt = At+Bt, d C /α α Γ /α i γ i dt In addition, Lemma 8 of [9] entails that converges to + i= d γ i C /α α Γ /α i dt of Lemma shows that s + N d γ i i= s [,s V i ds s dt [,s V i C /α Γ /α α i N s [,s V i uniformly on [,] The same proof as the one d C /α Γ /α α i γ i dt i= s [,s V i is càdlàg Lemma 5 then entails that h A t Since c >, it is thus sufficient to show that h B t δt αt Write Bt = Wt+Zt where Wt = + i= γ i C /αv i αv i Γ /αv i i i /αv i Vi t 6 and Zt = + i= γ i C /αv i αv i i /αv i Vi t 7 SetI k,m = [ k m, k+ m,n m,j,k = Card{V i,i = j,, j+,v i I k,m },d k,m = max u I k,m αu and C = max t [,] C/αt αt Define It is easily seen that j+ M m,j,k t = γ i C /αvi αv i i /αvi [,t] Ik,m V i i= j 6

7 sup s,t I k,m For t,, let α n t = n j+ γ i C /αvi αv i i /αvi [s,t V i i= sup j ] n Vi t t [ n i= M m,j,k t t [,] 8 denote the empirical process, and w n a = sup α n t α n s denote the oscillation modulus of α n We apply Lemma 4 t s a of [5] with m 3,a = m,s = m /4 j,n = j,δ = This yields that there exists M N such that m M, jm with m jm m such that j jm : m /4 j j m and P sup α jt α js > m/4 j 56 m m e j 64 9 t s m m j+ We need to estimaten m,j,k and sup γ i C /αvi s,t I αv i i /αvi [s,t V i k,m i= fork =,,m j and j m Study of N m,j,k for m j jm : Set X i = k m V i and n = k+ j X i i is an iid sequence of Bernoulli random m variables with parameter p = For m j jm, one has mj j m = np, and m thus, for a >, using a classical bound on the sum of iid Bernoulli random variables, P N m,j,k > a mj j+ P X i a j m i= j j+ = P X i anp i= j As a consequence, P jm j=m m k= a n a j { Nm,j,k > a mj } 7 m j=m m k= m + j=m a a a j a j a m

8 Choosing a = 3, Borel Cantelli lemma entails that jm m { P lim sup Nm,j,k > 3 mj } = m + Study of N m,j,k for j jm: j=m P N m,j,k > j m = P j using 9 As a consequence, P + j=jm k= α j j P w j k + α m j m j m = P w j m j m m P w j m m/4 j m m k= Borel Cantelli lemma yields Study of sup s,t I k,m P 56 m m e j 64 lim sup m + { Nm,j,k > j m} + j=jm m k= + j=m 564 m { Nm,j,k j m} k + j m j m m m 56 m m e j 64 m e m 64 e m 64 = j+ γ i C /αvi αv i i /αvi [s,t V i for m j jm : i= j Almost surely, there exists m such that, for m m and for m j jm, k =,, m, N m,j,k 3 mj, and s,t I k,m, i = j,, j+, thus sup s,t I k,m γ i C /αv i αv i i /αvi [s,t V i C j d k,m j+ γ i C /αvi αv i i /αvi [s,t V i i= 3 mjc j j d k,m 8

9 j+ Study of sup γ i C /αvi s,t I αv i i /αvi [s,t V i for j jm: k,m i= j We consider the events and F m,j,k = sup s,t I k,m G m,j,k = E m,j,k = { N m,j,k j m}, j+ γ i C /αvi αv i i /αvi [s,t V i i= jc j m j j d k,m { sup M m,j,k t t [,] jc } j m j d k,m Relation 8 entails that PF m,j,k E m,j,k PG m,j,k E m,j,k In the following computation, l corresponds to the number of terms V i belonging to I k,m, andncorresponds to the number of thosev i among them that contribute to the supremum of G m,j,k G m,j,k E m,j,k j m l= l,,l l [ j, j+ ] Using independence of the V i, PG m,j,k E m,j,k j m l= l,,l l [ j, j+ ] G m,j,k i {l,l l } P G m,j,k V i I k,m i {l,l l } i/ {l,l l } V i I k,m P V i / I k,m i/ {l,l l } Now, P V i / I k,m = j l Let us fix an order on the V m i belonging i/ {l,l l } to I k,m : there are l! possibilities, all equiprobable, and thus P G m,j,k V i I k,m = l!p G m,j,k k < V m l < < V ll < k + m i {l,l l } V i / I k,m Let A l k,m denote the event { k m < V l < < V ll < k+ m } Then P l n G m,j,k A l k,m P A l k,m γ li C /αv l i αv li l /αv l i i jc j m n= i= j d k,m l n γ li C /αx i αx i l /αx i i jc n dx n= i= j dx l d k,m k m <x <<x l < k+ m P The probability inside the integral above may be estimated with the help of Lemma 5 in [7]: 9

10 One then computes PG m,j,k E m,j,k n P i= j m l= j m l= e j 8 C /αx i αx γ i li C l,,l l [ j, j+ ] j d k,m l /αx i i j n e j 8 l l! mj l n= j! l! j l! l l!le j 8 mj j m l= 4 j m e j 8 4 j m e j 8 j! l l! j l! l mj j m l= ml e j 8 ml l! j! l! j l! l mj ml dx dx l k m <x <<x l < k+ m As a consequence, P + j=jm m k= F m,j,k E m,j,k 4 + j=m + e j j=m j e j 8 for m Borel Cantelli lemma entails that P lim sup m + + j=jm m k= F m,j,k = Computation of the Hölder exponent: Let t,, t / S and let U be an open interval of, containing t Denote = max αt If δt =, then h Yt = and the formula holds Suppose now δt > t U Let ε > be such that δt > ε and > ε There exists i N such that i i, t V i Choose m large enough so that < min{ t V iδt ε m i,i =,,i } Let j = [mδt ε] Increasing m if necessary, we may and will assume that i j, and j j, j jε Let s, be such that t s < There exists m+ m+ k {,, m } such that t,s Ik,m Increasing again m if necessary, we may assume that I k,m U Then d k,m, and, for i j, [s,t V i = One computes:

11 + Zt Zs = γ i C /αv i αv i i /αvi [s,t V i i= + j+ = γ i C /αvi αv i i /αvi [s,t V i j=j i= j jm j+ γ i C /αvi αv i i /αvi [s,t V i j=j i= + j jm 3 mj + jc j m + j=j j d k,m j=jm j d k,m 3+ C + j m m 6+C m m 6+C m m K U,ε,C m m Since δt ε >, [ j j=j + jε+ j=j j d k,m j ε+ ε+ +δt εε+ ] + j=jm K U,ε,C log t s t s δt ε ε+ δt ε Zt Zs K U,ε,C log t s t s δt ε ε Let us now study W recall 6: there exists i such that, for all i i, One then computes: Γ /αv i i i /αv i i K c,d + d k,m K c,d i + j+ γ i C /αvi αv i i /αvi [s,t V i i= j

12 Wt Ws = + j=j jm j+ γ i C /αvi αv i Γ /αv i i i /αvi [s,t V i i= j mj + K c,d C j m j + + jε j=j j=j 3K c,dc K m + j=jm + K m + j + j=j j K log t s mδt εε ε +K t s mδt ε K log t s t s δt ε + ε +K t s +δt ε Gathering our results, we have shown that: Bt Bs K log t s t s δt ε ε In other words, h B t δt for any open interval U containing t Letting the diameter of U go to, one gets h B t δt αt Proof of Lemma 7: Lemma entails that Y is almost surely a càdlàg process Thus, for all t,, h Y t Proof of Lemma 8: We seek to compute the Hausdorff dimension of F = {t [,]\S : δt = } S j+ ] Let E γ = limsup [V i i γαv i,v i +i γαv i Since E γ {t : h Y t γ}, j + Now, i i= j ] [V i i γd,vi +i γd {t [,]\S : δt = } γ>e γ i is a covering of E γ, and thus dim H E γ γd As a consequence, dim H {t [,]\S : δt = } = Since dim H S =, we find that f H = Proof of Lemma 9: Following [4], set λ i = +Γ i For the system of points P = {V i,λ i } i and t [,], define the approximation rate of t by P as δ t P = sup{δ : t belongs to an infinite number of balls BV i,λ δ i } Let us show that δ t P = δt Γ In that view, note first that, since lim i i + i = almost surely,

13 δt = inf liminf log +Γ φi V φn n R t i log V φi t Let δ be such that t belongs to an infinite number of balls BV i,λ δ i There exists φ : N N such that V φn n R t and t V φi λ δ φi, ie log +Γ φi log V φi As a t δ consequence δt Taking the supremum over all admissible δ, one gets δ δ tp δt For the reverse inequality, consider two cases: δt = : since δ t P, one gets δ t P δt ; δt < : choose ε > such that δt +ε There exists φ : N N such that for all large enough i N, log +Γ φi log V φi δt+ε, ie t V t φi By definition, this entails δ t P δt+ε, and finally δ tp δt +Γ φi δt+ε by letting ε go to We now apply Theorem of [4]: since hd, one has hαt for all t, The function t is continuous and thus hαt ie f H h = hd dim H {t, : δ t P = } = sup{hαt,t,} = hd hαt Proof of Lemma : Since hc < < hd and α is C, there exist t,t, such that hαt = and hαt for all t I := t,t or t,t Define g : t min, Theorem in [4] yields: hαt dim H {t I : δ t P = gt} = sup{,t I} gt = sup{hαt,t I} = hαt = One gets that f H h dim H {t I : δt = hαt} =, ie f H h = Proof of Lemma : By definition and Lemma 3, F /c = {t [,] : δt = αt } = {t [,] : αt = c} {t [,] : δt = } c Set E = {t [,] : αt = c}, E = {t [,] : δt < } and E = {t [,] : δt = } Then [,] = E E, F /c = E E and thus f H c dim HE Now, if dim H E =, the lemma holds true since F /c is not empty and thus f H /c Suppose then that dim H E > Choose s < dim H E This implies that H s E = +, and Theorem 4 in [] entails that there exist a compact set E c E such that < H s E c < + 3

14 Set µ s = H s E c This is a finite and positive Borel measure on [,] Theorem 37 in [8], along with Lemmas 4 and 6, entail that for all t,, Pt E = and thus Pt E c E = t Ec One computes: [ E[µ s E ] = E ] t E µ s dt = E[ t E ]µ s dt = Pt E µ s dt E c = Thus, µ s E is a positive random variable with vanishing expectation: almost surely, µ s E = Since µ s E c = µ s E c E +µ s E c E, one obtains that, almost surely, µ s E c = µ s E c E Now, H s E E H s E c E = H s E c > Thus dim H E E s, s dim H E and dim H E E dim H E Proof of Lemma : For all t [,], αt c and almost surely, for all t [,], δt, thus, almost surely, for allt [,], h Y t c As a consequence, forh > c, F h = andf H h = 3 Large deviation and Legendre multifractal spectra We compute in this section the large deviation and Legendre multifractal spectra of the process B on an interval Recall that we consider the process on [,], and that the large deviation multifractal spectrum of a process X on [,] is the random function f g defined on R by logn f g β = lim ε +liminf n εβ, where, for a positive integer n and ε >, N ε n log Xj+ n β = #{j {,,n } : β ε Xj n β +ε} Other large deviation multifractal spectra can be defined by replacing the increments X j+ n Xj by other measures of the variation of X, such as its oscillations, but we n will not consider these in this work We do not recall the definition of the Legendre multifractal spectrum, and refer the reader to [, ] instead We shall denote Pn j = P β ε log Yj+ n Yj n β +ε + = P Yj nβ+ε n Yj n n β ε 4

15 Set also X j = { n β+ε Bj+ n B j n n β ε} which follows a Bernoulli law with parameter P j n Clearly, N ε nβ = For U an open interval of,, we write n X j j= J n U = {j : j n U} There exists a constant K U > such that, for n large enough, #J n U K U n Finally, we will make use of the characteristic function of B, which reads [9]: m E exp i θ j B = exp m αs θ j [,tj ]s ds j= where m N,θ,,θ m R m,t,,t m R m 3 Main result The large deviation and Legendre multifractal spectra of B are described by the following theorem: Theorem 3 With probability one, the large deviation and Legendre multifractal spectra of B satisfy: for β < ; βd for β [, ]; d f g β = f l β = for β, ]; d c j= + c β for β,+ ]; c c for β > c The fact that f l = f g stems from the general result that f l is always the concave hull of f g when the set {β : f g β } is bounded The part concerning f g in Theorem 3 follows from a series of lemmas that are proven in the next sections We note in passing that, comparing with Theorem, we see that the weak multifractal formalism holds for B, but the strong one does not, that is, f H f g and f H f g The decreasing part with slope - for large exponents present inf g but not inf H is a common phenomenon when variations are measured with increments In order to prove Theorem 3, we will first show in each case of that the equality holds true for any given β with probability one Permuting for all β and almost surely will then often be achieved thanks to the two following general simple but useful lemmas on the large deviation spectrum, which are of independent interest 5

16 Lemma 4 The large deviation spectrum of any real function is an upper semicontinuous function Proof Let f g be the large deviation spectrum of a real function Consider β R, x j j a sequence such that lim x j = β, and set ε j = sup β x k j + k j Note first that lim liminf lognnβ ε = lim ε liminf j + logn ε j n β For all j and all l j, N ε j n β N ε j n x j N ε l n x j As a consequence, Letting l tend to infinity, one gets liminf one finally obtains logn ε j n β liminf logn ε j n β f g β limsupf g x j j + logn ε l n x j f g x j and letting j tend to infinity Lemma 5 Assume that there exist four functions h,h,g,g with lim gu = lim gu = u u such that, for all β in some interval I and all sufficiently small ε >, almost surely hβ+gε liminf logn ε nβ hβ+gε Then, almost surely, for all β in I, hβ f g β hβ Proof Define M n β,β = #{j {,,n } : β log Xj+ n Xj n Then, for β < β < β, and The set N β β β β n β M n β,β N β β β β n β f g β = inf liminf β,β Q,β <β<β A = β,β Q Mβ,β logm n β,β β } is countable, and, thus f g is obtained by a countable infimum for all β I This yields the result 6

17 3 Preliminary lemmas 3 Statements Define Ht = H λ,p t = λt log p+pe t Lemma 6 If < p < λ <, then Lemma 7 If < λ < p <, then supht = λlog λ t> p + λlog λ p supht = λlog λ t< p + λlog λ p Lemma 8 If p = pn = Kn b and λ = λn = n a, where K > and > a > b, then there exists n N such that, for all n n, sup t> Ht a b n a Lemma 9 If p = pn = K n b and λ = λn = K n a, where K >, K > and > b > a, then there exists n N such that, for all n n, supht K t< 4 nb Lemma If p = pn = Kn b and λ = λn = n a, where K > and > a > b, then there exists n N such that, for all n n, inf t> e λtn p+pe t n e a b n +a Lemma If p = pn = K n b and λ = λn = K n a, where K >, K > and > b > a, then there exists n N such that, for all n n, inf t< e λtn p+pe t KUn e K K U n +b 4 Lemma Assume there exist b, and K > such that, for all n n and for all j,n, P j n Knb Then, almost surely, logn ε nβ +b Lemma 3 Assume there exist an open interval U, a real b, and K > such that, for all n n and for all j J n U, P j n Knb Then, almost surely, logn ε nβ +b 7

18 3 Proofs Proof of Lemma 6 supht = Ht where t = log t> Proof of Lemma 7 supht = Ht where t = log t< λ p p λ λ p p λ Proof of Lemma 8 With t = log λ p, one has, for n large enough, p λ n Ht = n a a b n log + n a a log K Kn b n = n a a b log + n a log + Knb n a K Kn b n n a a b Kn log + n a b n a K Kn [ b ] = a bn a +n a n a Knb a logk Kn b Since lim na Knb a Kn b logk = logk, there exists n N such that, for all n n, n a Ht a b+ [ ] n a Knb a logk Kn b a b Proof of Lemma 9 With t = log λ p, one has, for n large enough, p λ Ht = K n a K n a b log + K n a log + K n b K n a K K n b K n a K n a b log + K K n a K n b K n a K n b = n [K b n a b K n a b log + K n a K ] K n a b K n b K which yields the result since lim K n a b log K n a b K + K n a K K n a b K = K n b Proof of Lemma One has Lemma 8 then implies that, for n n, inf t> e λtn p+pe t n = inf t> e nht = e nsup t>ht inf t> e λtn p+pe t n e a b n +a 8

19 Proof of Lemma Write inf t< e λtn p+pe t K Un = e K UnsupHt t< where Ht = λ K U t log p+pe t Lemma 9 ensures that, for n n, inf t< e λtn p+pe t n e K K U n 4 +b Proof of Lemma Fix a b, Then, for all t >, P N ε n β n+a = P e t n j= e ntλ E X j e tn +a [ n where λ = n a The X j are independent and E [ e tx j] = P j n +P j n et, thus j= e tx j P n Nn ε β n+a e ntλ Pn j +Pj n et, t > For t >, the function p p+pe t is increasing and so, by assumption on P j n, j= P N ε nβ n +a e ntλ p+pe t n, where p = Kn b Minimizing over t > and using Lemma, one gets and thus n n,p N ε n β n+a e a b n +a P Nn ε β n+a < + n N The Borel-Cantelli lemma then ensures that, almost surely, or n N, n n,nn ε β n+a logn ε n β +a Since this inequality holds true for any a b, one has indeed that, almost surely, ] logn ε n β +b 9

20 Proof of Lemma 3 Fix a < b For all t <, P Nn ε β n+a P = P e tn+a j J nu e t j JnU X j n +a j J nu X j e tn +a P j n +P j ne t When t <, the function p p + pe t is decreasing As a consequence, by assumption on P j n and with p = Knb, λ = n a, one has, for n large enough, P N ε n β n+a e ntλ p+pe t #JnU Minimizing over t < and using Lemma, one gets and thus e ntλ p+pe t K Un n n,p N ε nβ n +a e K K U 4 n +b P Nn ε β n+a < + n N As in the proof of Lemma, this leads to and finally, almost surely, a < b, almost surely, logn ε nβ logn ε n β +b +a 33 Estimates of P j n For U an open interval, denote c U = inf t U αt and = sup t U αt Set also = j n 33 Lemmas Lemma 4 Assume β < d Then, K >, n N such that n n, j,n, P j n Kn αβ+α ε Lemma 5 Assume β < Then, K >, n N such that n n, j J n U, P j n Kn αβ α ε Lemma 6 Assume β > c and ε,β c Then, K >, n N such that n n, j,n, P j n Kn α +ε β Lemma 7 Assume β > c U and ε,β c U Then, K >, n N such that n n, j J n U, Pn j Kn α +ε β

21 33 Proofs Proof of Lemma 4 Set µ j = α β + α ε Using the truncation inequality [, Section 3, p 9], one computes Pn j = P n Y+ Yt β+ε j P Y+ Y n β+ε 7 n β+ε = n β+ε e + e + + ξ αx dx dξ n β ε n β+εαx v αx dx dv n β+εαx v αx dxdv n β+εαx dx = 7n µ j + n +β+εαx αtj dx Since α is C, there exists a constant K such that, for all x,+, As a consequence, for a constant K β +ε αx α K x K n P j n 7n µ j + = 7n µ j n K n K n µ j n + K n dx n β ε Proof of Lemma 5 Set µ = + n β+ε n and σ = β ε Choose a function ϕ that satisfies the following properties: suppϕ [,] n β+ε

22 ϕ is even 3 ϕ is C 4 4 x R, ϕx 5 ϕ is not identically These properties imply in particular that ˆϕ is real and even In addition, for all x R: ϕ x+µ x µ +ϕ σ σ [ n β+ε, n ]x+ β ε [n β ε,n ]x β+ε Since the Fourier transform of ϕ x µ σ is σexp iµξˆϕσξ, Parseval formula yields P j n R π σ ϕ x+µ σ +ϕx µ σ PY+ Y dx cosµξˆϕσξe + t ξ αx dx j dξ = π R + cos µ + σ ηˆϕηe t η j σ αx dx dη Now, cos µ σ ηˆϕηdη = ϕµ σ = since µ > σ One may thus write: R P j n π + cos µ + σ ηˆϕηe t η j σ αx dx dη =: π nαβ α ε I j n, where I j n = I j n, +I j n, +I j n,3 with I j n, = + ˆϕη cos µ σ η cosη n αβ+εα e + η σ αx dx dη, I j n, = + I j n,3 = ˆϕηcosηn αβ+εα + tj+ ˆϕηcosηn αβ+εα e + η σ αx dx + η σ αx dxdη tj+ η σ αx dx dη,

23 Let us show that there exists K U > such that, for all j J n U, tj+ n αβ+εα η σ αx dx K U η c +η d Since σ = nβ+ε n ε and, there exits K such that, for all x,+, αx α K n, one has σ α αx K and tj+ n αβ+εα As a consequence, I j n, η σ αx dx = n αtjβ+εαtj tj+ n η αx σ αtj αx dx σ α + α n εα n ε α ηc +η d n Kη c +η d ˆϕη µ σ n αβ+εα K µ + σ ˆϕη η c +η d dη tj+ tj+ η σ αx dxdη σ α αx dx One finally obtains that sup I j n, K and lim sup I j n ε n, = j J nu j J nu Let us now deal with In, j I j n, + K U ˆϕη n αβ+εα tj+ η σ αx dx dη + K U n αβ ε α K U n β ε and thus lim sup I j n, = j J nu Fubini s theorem implies that I j n,3 = = ˆϕη n αβ+εα η c +η d n αβ εα dη tj+ n αtjβ+εαtj σ αx tj+ n αtjβ+εαtj σ αx + Fαxdx 3 ˆϕηcosηη αx dη dx

24 + where Fδ = ˆϕηcosηη δ dη The appendix contains a proof that min Fδ > As a consequence, δ [c,d] I j n,3 min Fδn αtjβ+εαtj δ [c,d] σ α ] αtj = min Fδ δ [c,d] K U [ n ε n ε [ n ε n ε ] [ c n tj+ tj+ n σ αtj αx dx n tj+ e α αxlogσ dx e α αxlogσ dx+ ] Now lim consequence, n ε n ε = and lim sup j J nu n + e αtj αxlogσ dx = As a and thus K U >, n N : n n, inf j J nu Ij n,3 K U > inf j J n αβ+εα Pn j > nu Proofs of Lemma 6 and Lemma 7 Parseval formula yields Pn j = sin ξ sin ξ n β ε n β+ε e + t ξ αx dx j dξ π ξ Set µ j = one gets Define α R +ε β with ε,β c U Using the change of variable ξ = n /α v, P j n = π R sinn µ j v sinn µ j ε v v I j n = R + tj e αx v αx n + αx v αx αt n j dx t e j dv α dx dv Since α is Lipschitz and c αx d, one deduces that Now, KU >, K U >, n N, KU inf j J Ij n sup In j K U nu j J nu P j n = n µ j I j n + I j n = n µ j I j n +Lj n R sinnµ j v sinn µ j ε v n µ j v 4 + tj e αx v αx αt n j dx dv

25 One computes: L j n [ sinnµ j v KU R t j+ KU R 6 nµ j v + n εe tj 6K U n +ε β c U ] n µ j v n µ j v + sinnµj ε v R + v tj e αx v αx n α dx dv + tj e αx v αx αt n j dx dv + K U KU There exist K 3 U R and n N such that, for all n n, t j+ sup v tj e j J nu R As a consequence, lim sup L j n = j J nu αx v αx n α dx dv K 3 U < + αx v αx n n ε α dx dv 34 Estimates for the number of increments and determination of the spectrum 34 Lemmas Lemma 8 Almost surely, β <, f g β = Lemma 9 Almost surely, β, d, f gβ = βd Lemma 3 Almost surely, f g = Lemma 3 Almost surely, β [ d, c ], f gβ = Lemma 3 Almost surely, β c,+ c, f gβ = + c β Lemma 33 Almost surely, f g + c = Lemma 34 Almost surely, β > + c, f gβ = 34 Proofs Proof of Lemma 8 Fix β < Denote E β =, β If ε E β, then PNn ε β = PNε n β = n = Pn j j= 5

26 Lemma 4 implies that, for n large enough, PN ε nβ Kn dβ+dε n e 3 Kndβ+dε 3 Kndβ+dε, and thus lim PNε nβ = Since Nnβ ε tends to in probability when n tends to infinity, there exists a subsequence σn such that Nσn ε β tends to almost surely logn This implies that, almost surely, liminf nβ ε = We have proved that: logn β <, ε E β, almost surely, liminf nβ ε Let Ω β,ε = {ω : holds}, Ω β = Ω β,ε and Ω = ε E β Q β, Q probability, and consider ω Ω Choose β < and j N Set β j = [βj] j enough, N /j n β N /j n β j In addition liminf n logn /j n β n logn /j n β j log n = and, almost surely, for all β <, f g β = = Ω β Note that Ω has For j large = As a consequence, Proof of Lemma 9 Fix β, Denote E d β = {ε,minβ, β such that d d εβ ε,} Choose ε E β By Lemma 4, there exists K > and n N such that, for all n n and all j,n, Lemma then implies that, almost surely, P j n Kn dβ+dε logn ε n β dβ +dε 3 There exists an open interval U such that, for all t U, αt d ε Using Lemma 5, there exist K > and n N such that, for all n n and all j J n U, P j n Kn d εβ ε, and Lemma 3 then implies that, almost surely, logn ε n β d εβ ε We thus have proved that, for all β, d and all ε E β, almost surely, d εβ ε liminf logn ε n β dβ +dε 4 Then Lemma 5 ensures that almost surely, for all β, d, f gβ = dβ 6

27 Proof of Lemma 3 We obtain Inequality 3 for β = by applying Lemma 4 and Lemma This yields that, almost surely, logn ε n dε As a consequence, f g Then apply Lemma 4 and Lemma 9 to obtain f g limsupf g j + j d = limsup j + j = Proof of Lemma 3 Let β, and ε, d c Choose an open interval U such that β > c U and β < +ε Lemma 7 then ensures that there exist K > and n N such that, for all n n and all j J n U, P j n Kn α +ε β Kn +ε β Kn ε By Lemma 3, for all β,, almost surely, d c logn ε n β We conclude by applying Lemma 5 For β =, we apply Lemma 4 to obtain c ε f g c limsup f g [βj] j + j = limsup j + For β =, Lemma 4 and Lemma 9 lead to d f g d limsup f g [βj] j + j = limsupd [βj] j + j = 7

28 Proof of Lemma 3 Let β, + Denote E c c β =,minβ, + β Fix c c ε E β By Lemma 6, there exist K > and n N such that, for all n n and all j,n, Pn j Kn c +ε β Lemma then implies that, almost surely, logn ε nβ + +ε β 5 c Choose an open interval U such that c ε β ε By Lemma 7, there exist K > and n N such that, for alln n and all j J n U, Lemma 3 then implies that, almost surely, logn ε n β P j n Kn +ε β + +ε β + c ε β We have proved that, for all β c,+ c and all ε E β, almost surely, + c ε β liminf logn ε nβ + c +ε β 6 The result then follows from Lemma 5 Proof of Lemma 33 We obtain Inequality 5 for β = + c Lemma This yields that, almost surely, by applying Lemma 6 and logn ε n + c dε As a consequence, f g + c Then, Lemma 4 and Lemma 3 lead to f g + c limsup j + f g [+ j] c j = limsup+ j + c [+ j] c j = Proof of Lemma 34 Let β > + and denote E c β =,β For ε E c β, PNnβ ε = PNnβ ε = n = Pn j j= 8

29 Lemma 6 ensures that, for n large enough, PN ε nβ Kn c +ε β n 3 Kn+ c +ε β Thus, lim PNε nβ =, which implies that Nnβ ε tends to in probability when n tends to infinity There exists a subsequence σn such that Nσn ε β tends to almost surely As a consequence, for all β > + and all ε E c β, almost surely, logn ε n β We conclude as in the last part of the proof of Lemma 8 Appendix The following result is due to R Schelling [4]: Lemma 35 For all β >, Fβ = Proof The Lévy Khintchine formula yields η β = cosyην β dy with By Fubini s theorem, y = 7 η β cosη ϕηdη > 8 ν β dy = c βdy y +β η β cosη ϕηdη = cosη cosyηcosη ϕηdην β dy y = cosη cos+yη cos yη ϕηdην β dy y = c ϕ ϕ+y ϕ y ν β dy, y where c is a positive constant By definition, ϕ is smooth and supported on[, ], thus ϕ = As a consequence, we find that η β cosη ϕηdη = c ϕ+y+ϕ yν β dy < This is inequality 8 y It is easy to see that the function β Fβ is continuous min δ [c,d] Fδ > As a consequence, 9

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