Multifractal Analysis of Infinite Products of Stationary Jump Processes

There has been a growing interest in constructing stationary measures with known multifractal properties. In an earlier paper, the authors introduced the multifractal products of stochastic processes MPSP and provided basic properties concerning convergence, nondegeneracy, and scaling of moments. This paper considers a subclass of MPSP which is determined by jump processes with i.i.d. exponentially distributed interjump times. Particularly, the information dimension and a multifractal spectrum of the MPSP are computed. As a side result it is shown that the random partitions imprinted naturally by a family of Poisson point processes are sufficient to determine the spectrum in this case.


Introduction
The measures resulting from the limits of random multiplicative martingales have attracted much attention in the mathematical community since the the work by Kahane on positive martingales 1-3 ; these martingales are of the form Q n t dσ t , where Q n t forms a positive martingale for each t.Related early ideas go back to de Wijs 4, 5 , Kolmogorov 6 , Novikov and Stewart 7 , Yaglom 8 , and Mandelbrot 9-12 , and emerged mostly in the context of turbulence.Recently, Barral and Mandelbrot have published a series of papers 13, 14 completing Kahane's general theory of T -martingales.
Research on multiplicative cascades has been very active.Especially, Mandelbrot's martingale 9, 10 , a simple tree-based construction with independent random multipliers, has been considered in a large number of publications; first by Kahane and Peyrière 15 , and the story still continues see e.g.16-23 .Extensions such as relaxing the independence where the Λ i are independent rescaled versions of a mean one, stochastic mother process Λ.We refer to Anh et al. 40,41 for a collection of such products of stochastic processes for which the multifractal envelope is explicitly computed.Recently, the L 2 convergence of A t with a long range-dependent Λ is shown by Matsui and Shieh 42 .In this paper, we study the case Λ i • dist Λ b i • , where Λ is a stationary jump process with i.i.d.exponential interjump times.Overview and contributions: Section 2 reviews the multifractal analysis in the context of restricted coverings, recalls the less known semispectra needed in our approach, and establishes the corresponding multifractal formalism.The main purpose of this section is to provide solid grounds for the reader less familiar with the field of fractals and to clearly state results that are not found in common books.Section 3 develops the multifractal properties of the infinite product of jump processes and constitutes the heart of the paper.To compute the multifractal spectrum we establish the multifractal formalism, for the infinite product of jump processes.To our best of knowledge, these are the first processes that may exhibit temporal dependence over arbitrarily large lags and for which the formalism has been established.Notably, the absence of an a priori bound on the length of the temporal dependence precludes the use of common methods as found, for example, in 14 .To overcome this hurdle, we take advantage of a more general version of the multifractal formalism than is usually applied.More precisely, we base the multifractal computations on partitions created naturally by the Poisson processes underlying the processes at hand.While Poisson partitions can in general not be used to compute the ordinary Hausdorff dimension of sets, they will suffice here.Moreover, the Poisson partition yields the same multifractal envelope as the one obtained with the usual dyadic partitions in earlier work 39 .

Multifractal Spectra and Formalism
The multifractal spectrum of a process is defined pathwise.Therefore, we start in sections 2.1-2.3 by providing the relevant notions for a given deterministic increasing function, or its related measure.We also develop tools which allow for computing the multifractal spectrum of a measure in ways which are adapted to the inherent structure of the processes we will study in this paper.In 2.4 we proceed to random measures, providing almost sure upper bounds on the pathwise multifractal spectrum.It will be the task of the remainder of the paper to establish conditions under which these upper bounds are actually tight.
In this paper, we consider scaling exponents and multifractal spectra which are defined via nonhomogenous partitions.The results have analogous counter-parts in the standard setting but it should be noted that these are not fully equivalent notions.

Hausdorff Dimension
Let us start by recalling notions related to Hausdorff dimension.Consider a set F ⊂ R d and a class C of subsets of R d .Then, the s-dimensional Hausdorff measure with respect to (w.r.t.) C is given by for any countable family of sets F j .This can be established as in the classical case.See Rogers 43 for more properties of Hausdorff measures.Numerous classes of sets C are known to result in the usual Hausdorff dimension see e.g., 44-46 .For our purposes, it will be sufficient to consider certain nested classes C and to establish a simple sufficient condition to warrant that dim F dim C F .The simplest example of such a family is the b-ary cubes of the form where b > 1.To generalize this fact, recall that a collection is called nested if for all pairs of its sets A, B, we have A ∩ B ∅A ⊆ B, or B ⊆ A. Also, we use the term cube for any product of intervals of equal lengths.
Lemma 2.1.Let F ⊂ R d and C be a nested collection of arbitrary cubes.Assume that for all x ∈ F there exists a sequence The use of nested covers with a condition akin to 2.5 is quite standard.In 45 , for example, a Frostman-type result based on nested cubes is derived.In 46 , the equality dim F dim C F is established under assumptions which are somewhat more restrictive than 2.5 and prevent the use of Poisson covers compare lemma 3.6 .As we do not need this result directly we leave the proof to the interested reader who will have no problems rewriting the arguments of 45 for a proof of the result given here.In doing so, exploit that the covers are nested to avoid the need of a lower bound on |C n x | as used in 45 .A proof is contained in the technical report 47 .
A standard method to get a lower bound for the Hausdorff dimension of a set is based on a scaling law, which appears as mass distribution principle in 44 .For our purposes, we need the following stronger lemma which follows from 45 .Note that condition ii can be replaced by the less restrictive 2.5 , following the classical argument of the Frostman lemma applied to the covers C, combined with lemma 2.

Multifractal Spectrum
Let ν be a Borel measure in R d .Its Hölder exponent at x, or equivalently local dimension at x, is given by if this limit exists.One should think of α C x as giving approximately the degree of H ölder regularity of ν at x, that is, an approximation to dim loc ν x .It is tempting to conjecture that these two notions are equal, except in a set of dimension zero, provided that the partitions satisfy the conditions of Lemma 2.2.While this conjecture is known to hold for certain random measures ν such as the binomial cascade, a general proof seems hard to come by.We consider the sets characterizing the local scaling behaviour.The classical literature on multifractals studies typically the sets E C α , calling dim E C α the multifractal spectrum of the measure ν.The sets V C α are somewhat easier to study, yet provide spectral information in the sense of multifractal analysis, as we will point out in this section.Also the sets {x : lim inf n α C n x α} would be of interest since they form a partition of space.However, their dimension would lie between dim E C α and dim

Pathwise Partition Function and Coarse Grain Spectra
In order to make the presentation simpler, we consider only the 1-dimensional case assuming that supp ν ⊆ 0, 1 , but all the definitions are easily extended to compactly supported measures on R d .
Consider nested partitions J n {J n k : k 1, . . ., N n } of the unit interval.Analogous to the previous section, denote C {J n : n 1, . ..} the nested collection of sets which is determined by the J n .The partition sum of the measure ν is defined by where we adapt the convention 0 q .0 for all q ∈ R. The only purpose of this convention is to provide a convenient way to ensure that the sets J n k with ν J n k 0 do not contribute to any of the S n q .Define then the partition function as We drop the index C whenever the choice of the partitions is clear from the context.The above functionals can be used to characterize multifractal properties through the Legendre transform Then τ * α denotes a pathwise large deviations spectrum.In the special case where the partitions of C consist of the b-ary intervals, the above definition of τ C reduces to the standard one for an overview see 48 .
To provide upper bounds on the multifractal spectra, it is useful to introduce the socalled "coarse grain" spectra.To this end, the following collections of intervals are of interest: The reference to large deviations relies on the fact that N n may be represented as a rare event or a "large deviation from the mean" if one considers log ν J n k / log |J n k | to be the random variable, where k is random but ν and J n are fixed and n → ∞.In the proper setting, the following spectrum f C becomes a large deviation rate function: In the following, we collect some properties needed to order the different spectra.Note that these results specifically hold for spectra and partition functions defined via nonhomogeneous partitions.The proofs are quite straightforward but they are shown for the sake of completeness.
Let m be an arbitrary positive integer.Then, for any t ∈ V C α there exists n > m such that α n t < α ε and, thus, there is k ∈ N n α ε such that J n t J n k .Since t ∈ J n t , we get This means that for every m we have constructed a cover of Lemma 2.5.Let ν be a Borel measure and let the J n be nested partitions.Then, for all real α, f α ≤ qα − τ q for q ≥ 0, and τ q ≤ f * q for q > 0.
Proof.Let α, q, and γ be arbitrary real numbers for the moment.By definition, ν J n k > |J n k | α for all k in N n α .For nonnegative q we may, thus, estimate 2.18 Fix a real α and q ≥ 0. Consider γ < τ q .By definition of τ, we find Consequently, qα − γ ≥ f α .Since γ can be chosen arbitrarily close to τ q , this actually implies that qα − τ q ≥ f α for all non-negative q, as claimed.
The convexity of τ q can be established for q > 0 under mild conditions.
Let ν be a bounded Borel measure.Then τ q f * q and τ is convex for q > 0.
Proof.Replacing ν by cν for a sufficiently small c, we may assume that α n t ≥ 0. Fix q > 0. Let γ < inf α qα − f α , let ε > 0 be such that γ < qα − f α − εq for every α, and let U j denote the interval j − 1 ε, jε .Let M be sufficiently large such that qM − γ > 2. Let p be large enough so that pε ≥ M. Then

2.20
By choice of M, and since the J n k are disjoint subsets of −1, 2 for n large enough, By choice of γ, it follows that τ q ≥ γ, thus τ q ≥ f * q .The equality follows from Lemma 2.5 and a Legendre transformation is always convex.

Deterministic Envelope for Random Measures
Let us now consider a random measure μ defined on 0, 1 and a random nested collection C. The asymptotics of ensemble moments are given by Then T * α denotes the deterministic large deviation spectrum.The following lemma shows that if T is convex and τ is convex a.s., then T * is an upper bound to τ * .Recall that τ * provides a pathwise upper bound to f see Lemma 2.5 which in turn bounds the multifractal spectrum from above.
To check if T q is convex, one can usually calculate the function explicitly, whereas establishing the convexity of τ q is more subtle.In the previous section, we stated Lemma 2.6 which provides a way to guarantee a.s.convexity.Lemma 2.7.If T is convex and τ is convex a.s. in an interval I, then a.s.τ q ≥ T q for all q ∈ I.
Proof.Fix q ∈ I such that T q < ∞ and take any γ < T q .Since S n q, γ is positive, and thus n≥m S n q, γ < ∞ a.s., that is, τ q > γ a.s.Next, choose γ m T q .In conclusion, τ q ≥ T q a.s., for any countable set of q ∈ I. Since τ and T are a.s.convex, the statement holds a.s.for all q ∈ I.
Combining lemmas 2.4, 2.5, and 2.7, we can now order the multifractal spectra.Theorem 2.8.If T q is convex and τ q is convex a.s.for q > 0, and lim n → ∞ max k |J n k | 0 a.s., then the multifractal spectra are ordered as follows: with probability one, for all α, The basic arguments that lead to Theorem 2.8 are sometimes addressed as "steepest ascent" methods or as large deviation arguments.They are quite standard in multifractal analysis 12, 14-17, 19-22, 24, 30, 34, 47-50 to obtain an upper bound on the multifractal spectrum dim E α in a setting of interest.Notably, 2.24 is quite generally applicable since a mild condition on the covers asserts that τ is convex without the need of strong assumptions on the measure.Note also that some of the inequalities of 2.24 may be strict 51, 52 .
Arbitrary partitions in a deterministic setting have been studied by Brown et al. 53 .Using their results, the pathwise inequalities of Theorem 2.8, that is, the parts not including T C , could have been established under slightly different technical assumptions.However, our setting leads to the assumptions that are more easily verified for processes studied in this paper.
In certain settings, different partition functions might become useful, yet still provide upper bounds akin to the above 20, 53-56 .A very typical setting is to form the partitions C via dyadic cubes.The formalism 2.24 is most useful only for the increasing part of the spectrum as the right-hand side is convex and increasing.Using similar arguments one can easily obtain an upper bound which uses the negative range of q values and is decreasing and convex, as is done, for example, in 47 .

Definition and Basic Properties
We start from T -martingales defined by independent multiplication as in 2 .Consider a family of independent positive processes Λ i which are independent rescaled copies of a stationary mother process Λ defined on R .In this paper, we restrict to the class of jump processes satisfying the following assumptions: Recall that a real-valued stationary process is weakly mixing if for all B 1 , B 2 ∈ B the σ-algebra generated by the process lim 0 where S t is the shift operator.Examples of interest include the cases where the sequence {M k } k forms an i.i.d.sequence of random variables Λ is a renewal reward process or a finite state irreducible Markov chain.The requirement that the multiplier processes are strictly independent of the partitioning is added for convenience here and more general cases can Journal of Probability and Statistics be studied.For example, if Λ is a finite state Markov process, then analogous results can be derived.
Next define the finite product processes For t ∈ 0, 1 , the cumulative processes can be associated with positive measures defined on the Borel sets B of 0, 1 :

3.3
We note that the restriction to the unit interval is purely for convenience, and extensions to compact intervals and to the real line are straightforward.
In the context of the martingales of Kahane 2 and in multifractal analysis, we are interested in the limit measure μ .lim n μ n and its associated cumulative process A. The existence of the limiting objects possibly degenerate is established in 2 .
Definition 3.1.Assume A1 and A2 .Then, the multifractal jump process MJP is the limit A n t a.s.

3.4
An MJP is called degenerate if A 1 0 almost surely, and nondegenerate otherwise.
It can be shown that A is non-degenerate if and only if E A 1 1 see 39 .The convergence of A n 1 in L p for some p > 1 naturally implies nondegeneracy since then To provide criteria for convergence in L p is thus of central importance.As pointed out in 2 , criteria for convergence in L 2 are particularly manageable and useful.In the following theorem, which is a straightforward adaption of 39, Corollary 3 , the L 2 conditions are stated for MJPs.
then A n t converges in L 2 if and only if b > 1 σ 2 where σ 2 Var M .
Condition 3.5 allows a very large family of time series M k .For example, a sufficient condition for convergence is , j 0, 1, 2, . . ., 3.6 for some constant C > 0.
The majority of the following results are stated assuming a converging MJP in some L p , p > 0. The conditions for p 2 are, of course, sufficient for p ∈ 0, 2 .In principle, setting explicit conditions for p > 2 would be possible.However, even for the simplest possible case when the M j are i.i.d. and the L 3 convergence is considered, long and tedious calculations are needed.Thus the general convergence considerations are out of the scope of this paper.
The assumption of weak mixing in A2 is needed to show that the limit measure μ is ergodic with respect to the time shift operator.Ergodicity of μ, together with the positivity of the multipliers M k , guarantees that the random variable μ 0, t does not have an atom at zero.This result, in turn, will be used in the multifractal decomposition.
Proposition 3.3.The associated measure μ on R of an MJP is ergodic.
Proof.Consider the spaces Ω i X i , B i , P i , i 0, 1, . . ., and Ω Π ∞ i 0 Ω i X, B, P , where X i is the set of piece-wise continuous functions, B i its Borel-algebra, and P i is the law of Λ i .The shift operator is weakly mixing in Ω i and thus it is also weakly mixing in i n Ω i form a semialgebra generating B, it follows that the shift operator is also weakly mixing in the infinite product space Ω see e.g., 57, Theorem 1.7 .Weak mixing implies ergodicity which is then trivially inherited to the subsystem determining the random measure μ.
The following measures are instrumental in our analysis: that is, the measure where n first terms of the product are neglected.Thus Proposition 3.4.Assume that A is a non-degenerate MJP.Then, with probability one, μ 0, t > 0 for all t ∈ 0, 1 .
Proof.First consider an arbitrary t > 0 and make a change of variable to get where μ and μ are identically distributed and μ independent of Λ n .Then by the positivity of Λ n and ergodicity of μ, we have

Random Partition and Multifractal Envelope
In order to determine the prevalence of scaling exponent in an entire interval rather than one single point, we use the formalism developed in Section 2. In analogy with related multiplicative processes 15 , we should expect moments of the multipliers to affect the multifractal properties through the structure function Indeed, as a direct consequence of 39, Propositions 7 and 5 , the structure function gives the deterministic envelope of the multifractal formalism based on the dyadic partitions D.
Corollary 3.5 see 39 .If E Λ p E M p < ∞ and A n t converges in L p for some p > 1, then, for 0 < q ≤ p, Ct β q 1 ≤ E A t q ≤ Ct β q 1 ∀t ∈ 0, 1 , 3.13 3.14 Note that under the assumption 3.5 of Theorem 3.2, convergence of the MJP A t in L 2 is equivalent with β 2 > 0.
In order to analyze the MJP, it is natural to consider the nested sequence of partitions J n induced by Λ n .By assumptions A1 and A2 , Proof.Without loss of generality, assume λ n λb n and t 0 ∈ 0, 1 . 1 • : Clearly P No points in t 0 − x, t 0 x ≤ P |J n t 0 | > x ≤ 2 P No points in t 0 , t 0 x/2 .

3.18
Thus for n large enough,

2
• : Since the maximal interval is always longer than a fixed interval, for n large enough, On the other hand, if the maximal interval is longer than a given x, then at least one of the subintervals i − 1 x/2 , i x/2 , i 1, . . ., 2/x , has to be empty.Thus k 1 be a partition generated by a uniform random sample of size m on interval 0, 1 .It is straightforward to show, for example, by induction, that

3.22
Let N n denote the number of points of a Poisson process with density λb n located in 0, 1 .Applying 3.22 gives

3.23
where the last inequality holds when n is large enough.On the other hand, if the minimal interval is larger than some given x, then there is at most one point in each subinterval i − 1 x, ix , i 1, . . ., 1/x .Thus

4
• : Apply Borel-Cantelli to the above three cases to complete the proof.
This result sheds some light on the use of such Poisson partitions in the context of computing the Hausdorff dimensions and multifractal spectra.They do not satisfy the homogeneous scaling assumption of 46 ; yet, being nested they allow still to treat sets for which all points satisfy 2.5 , as for such sets dim • dim P • .In addition, by lemma 2.6 and theorem 2.8, the multifractal envelope T * P provides an upper bound on the multifractal spectrum dim P E α , almost surely.
The connection to the measures μ n defined in 3.7 is especially practical when considering sets J n t because then where μ is distributed as μ and is independent of J n and Λ i , i 0, . . ., n − 1.On the other hand, by corollary 3.5 and 3.8 for a fixed interval I.We find that T P T D under mild conditions.
Proposition 3.7.If E M p < ∞ and A n t converges in L p for some p > 1, then . ., j 1} the partition resulting from j uniformly distributed points on 0, 1 .Then Using the above result and conditioning with respect to the number of intervals gives

3.29
Now assume γ β q − with > 0. Conditioning with respect to the partition J n helps us to split the addends into three factors, that is,

3.30
Use the scaling law 3.26 together with the independence of μ n with respect to J n and that in order to estimate

Journal of Probability and Statistics
Next, assume that γ β q ε.Condition again with respect to the partition J n and apply the lower bound of 3.13 to get 3.33 Thus, we have found that n≥m E S n q, γ → 0 whenever γ < β q and n≥m E S n q, γ 0 whenever γ > β q .

Information Dimension
This section is devoted to analyzing E P α 1 , the set of points with scaling exponent which is the most relevant of all E P α .The upper bound of the multifractal spectrum dim P E P α ≤ T * P a.s.follows from lemma 2.6, lemma 3.6, and theorem 2.8.While T P T D under mild conditions see above , we need to introduce the auxiliary set a subset of E P α for which dim K P α dim P K P α due to Lemma 2.1 or due to Lemma 2.2 as we will see .
To perform a local analysis of path properties and establish facts which hold for almost all paths at almost all locations t, it is convenient to introduce a measure Q which is referred to as the "Peyrière measure" by Kahane.The approach is to apply the LLN to Q which provides a pathwise measure ν which allows us to bound dim K P α , and thus dim E P α , from below almost surely using Lemma 2.2.The measure Q lives on the space Ω × 0, 1 , defined as the unique probability measure which satisfies for all positive measurable functions ϕ t, ω .
Lemma 3.8.Assume that A t is a non-degenerate MJP.If ω, T is picked according to the Peyrière measure Q, then

3.37
Moreover, if A n t converges in L q for some q > 1 and b q−1 > E Λ q , that is, β q > 0, then 3.39 1 • Condition with respect to the partition J n in order to get

3.40
where C 1 is independent of n.Here, the last inequality follows by estimating the probability that a truncated exponential random variable is less than x, with x < 1/2.The other direction is easy as well.First we condition as above, and represent the sum as an integral: where C 2 is independent of n.Here, the last inequality follows from estimating the probability that a sum of two exponential random variables exceeds x.Since λ n O b n , we have

3.43
Scaling the μ n J n k up by b −n e −nε and applying 3.39 lead to

3.44
This implies P Q μ n J n T ≤ Ce −nε , because E N n λ n O b n .On the other hand, by 3.26 and 3.29 ,

3.45
Then apply the Markov inequality to get To establish the Q-independence required for the LLN, we need the following result.
Lemma 3.9.Assume that A t is a non-degenerate MJP.Then for all n ∈ N and positive Borel functions g i defined on R .
Proof.Note: this is a replication of the proof appearing in 2 .Condition with respect to the partition J n and use E μ n J n k | J n |J n k |.In addition to this, utilize independence w.r.t.Ω and stationarity:
Proof.Fix j and set g i x ≡ 1 when i / j, leaving g j arbitrary.Lemma 3.9 gives E Q g j Λ j T E g j Λ j t Λ j t .

3.49
Thus for all n ∈ N and positive Borel functions g i defined on R .
Corollary 3.11.Assume that A t is a non-degenerate MJP.Then E Q log Λ j T E Λ log Λ for all j ∈ N.

Proof. Set g i x
1 if i / j and g j x log x and apply Lemma 3.9.
We now show that the pathwise ν to be used in Lemma 2.2 is the realization of μ itself.
Lemma 3.12.Assume that A n t converges in L q for some q > 1.If β q > 0, then the set K P α 1 has full μ-measure Ω a.s., that is, Ω-a.s.

3.51
Proof.Notice first that log μ J n t 1/n log|J n t | .

3.52
Then by Corollary 3.10, 3.11, and LLN, On the other hand, by lemma 3.8,

3.54
Lemma 3.13.Assume that A n t converges in L q for some q > 1.If β q > 0, then, with probability one, we have either μ 0, 1 0 or dim K Proof.Condition i of lemma 2.2 is obvious; conditions ii and iii are satisfied by definition of K α 1 .Finally, either condition o or μ 0, 1 0 holds due to lemma 3.12.
The main result concerning the information dimension is stated in the following theorem.
Theorem 3.14.Assume that A n t converges in L q for some q > 1.If β q > 0, then, with probability one, For example, if A t is an MJP satisfying 3.5 and β 2 > 2, then dim Proof.First, note that β * α 1 α 1 since α 1 β 1 ; also β q > 0 for 1 < q < q since β is convex; also, μ ∈ L q for 1 < q < q.Second, by Lemmas 3.6 and 2.6, τ P q is convex for q > 0 a.s. and theorem 2.8 applies.By Lemma 3.13 we have 3.56

Multifractal Decomposition
Following the traditional approach in multifractal analysis, we generalize the above analysis of the information dimension of MJP to a larger set of scaling exponents through a change of the measure.Doing so, we are able to compute dim V P α for a range of exponents α.To this end, we fix q > 0 for the remainder of the section.Introduce the auxiliary mother process and the scaling exponent α q .β q .

3.59
We denote the associated MJP by A, its measure by μ, and the corresponding Peyrière measure by Q.Let us quickly summarize how our earlier results translate from A to A.
Clearly, E Λ 1, and by definition of β 3.60 From this we find It is quite typical in multifractal analysis that the Legendre transform should appear at this point.
By analogy with the previous section, we should expect that under appropriate assumptions with probability one the information dimension of μ is qα q −β q .We should also expect to find that with probability one, the measure μ assumes the scaling exponent α t α q for μ-almost all t, which is not a simple translation from earlier sections.If established, all this would imply that the dimension of the set E α q of points with H ölder exponent α q almost surely has at least Hausdorff dimension qα q − β q , since μ assigns full mass to it.
We are able to establish such a lower bound for the Hausdorff dimension of a natural subset of V P α , that is, the set for α α q .Combining this with an upper bound provided by inequality 2.24 , we will arrive at a formula for the dimension of V P α .As a first step towards this goal, we mention a sufficient condition for convergence in L p , 1 ≤ p ≤ 2, which follows directly from theorem 3.2, that is, Var Λ .Being equivalent to β 2 > 0 or β 2q > 2β q , this condition holds by convexity for all 0 < q < q if it holds for q.Next, replicating the results of the previous section gives the following corollary.Corollary 3.15.Assume that A n t converges in L p for some p > 1.If β p > 0 for some p > 0, then Q-a.s.,

3.63
To provide a counterpart to Lemma 3.12, we need the following result concerning the local scaling of μ under the measure Q. Adding extra points at the locations x 1, 2, . . .makes the intervals even smaller.Thus the partition ∪ m i 1 I ∞, i−1,i is finer than I n, 0.m for all n 0, 1, . . .and m.In other words, for any m ∈ Z , I ∞, i−1,i t ⊆ I n, 0,m t ∀n ∈ N, ∀i ∈ 1, m , ∀t ∈ i, i − 1 .
3.65 2 • : By a change of variables, and using E Λ n T n k | J n 1 together with the independence of Λ n of J n , μ n , and μ n , we get where J n is distributed as J n and independent of μ and μ.
For each fixed n, construct a nested sequence of partitions I i, 0,b n , i 0, 1 . . ., which are independent of μ and μ and satisfy I n, 0,b n b n J n .Then write 3.66 as an integral split into parts of length 1 and apply 3.65 to get Combining the previous results, we find the dimension of the set V P α q .
Theorem 3.17.Assume that A t is a non-degenerate MJP and that A n t converges in L p for some p > 1.If β p > 0, then, with probability one, dim P V P α q dim V P α q qα q − β q . 3.72 Proof.The proof follows the lines of lemmas 3.12 and 3.13 and theorem 3.14.To get the lower bound, the set K α 1 is replaced by W α q and Lemma 2.2 is applied to μ.The upper bound follows directly from inequality 2.24 .In summary, dim P V P α q dim V P α q f P α q T * P α q T * D α q β * α q qα q − β q .

Lemma 3 . 16 . 64 Proof. 1 •
Assume that A t is a non-degenerate MJP and that A n t converges in L p for some p > 1. Thenlim inf n → ∞ log b μ n J n T −n ≤ 1 Q-a.s.3.: Let us start with a simple observation on nested Poisson point processes.Given a sequence of independent Poisson point processes of densities b −k λ, their superposition forms a sequence of Poisson point processes of densities 1 • • • b −n λ.Denote I n, x,y {I n, x,y k } the partition of the interval x, y determined by a Poisson point process with density 1 • • • b −n λ.

b n 0 1 μ≤ E b −n b n i 1 i i− 1 1
I n, 0,b n t < b −nε dμ t μ I ∞, i−1,i t < b −nε dμ t E b −n μ b n , b n ≤ P Q μ I ∞, 0,1 T < b −nε b −n .3.67 3 • : By proposition 3.4, μ I > 0 a.s.for any fixed interval I. Thus,P Q μ I ∞, 0,1 T a.s.μ I ∞, 0,1 T > 0.Applying this to 3to see by contradiction that the previous equation implies lim inf n log b μ n J n T −n ≤ 1 Q-a.s.3.70 4 • : By the LLN, by the analogs of Corollaries 3.10 and 3.11 for Q, and by 3.60 , we have Q-a.s.b Λ i T −→ E Q Λ T 1 − α q .3.71 By Corollary 3.15 we have 1/n log b |J n T | → −1 Q-a.s.Applying these results and 3.70 to 3.52 completes the proof.

1. Lemma 2.2 see
45, Lemma 4.3.2.b .Let ν be a Borel measure on R d , F a Borel subset, and : n 1, . ..} is a collection of nested partitions C n of supp, ν and C n x is the unique C n k ∈ C n containing x. Then the local scaling exponent w.r.t.C of ν at x is given by n . processes, distributed as Λ; A2 Λ is a weakly mixing Markov process which is defined by Λ t .M k for T k ≤ t < T k 1 , where T 0 0 and {T k 1 − T k } k are i.i.d.exponential random variables of mean 1/λ, and where the M k form a stationary positive time series independent of {T k } k satisfying E M k 1 for all k.
. From the construction of {T n k } k , it follows that the partitions J n {J n k : k 1, . ..} are nested.We denote the resulting collection of nested Poisson partitions J n by P. Consider a nested Poisson partition P. Let t o be arbitrary.With probability one, as n → ∞,