# Multifractal Analysis of Infinite Products of Stationary Jump Processes

Authors: Petteri Mannersalo; Ilkka Norros; Rudolf H. Riedi
Journal: Journal of Probability and Statistics
 (2010)

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## Abstract

There has been a growing interest in constructing stationary measures with known multifractal properties. In an earlier paper, the authors introduced themultifractal 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.


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## Body

## 1. 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 ∫Qn(t)dσ(t), where Qn(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 assumption of the multipliers or randomizing the number of offsprings have also been studied. To give a short list without intention of being complete, we refer to Molchan [19] and Waymire and Williams [24, 25] regarding dependent multipliers, and to Peyrière [26], Arbeiter [27], and Burd and Waymire [28] regarding random numbers of offsprings.Classes of Kahane's martingales that move away from a tree-based structure include Gaussian chaos by Kahane [1], Lévy chaos by Fan [29], random Gibbs measures by Fan and Shieh [30], random coverings by Kahane and Fan [2, 3, 31, 32], multifractal product of cylindrical pulses by Barral and Mandelbrot [13, 14, 33], an infinitely divisible cascades by Bacry and Muzy [34, 35], Chainais et al. [36, 37] and Riedi and Gershman [38], as well as products of stochastic processes by the present authors [39] and subsequently by Anh et al. [40, 41] and Matsui and Shieh [42].Processes defined through iterative multiplication such as the above exhibit rich scaling properties and ubiquitous, fine details. Multifractal analysis, in a nutshell, strives at capturing the essence of such complex scaling behaviour through geometric and probabilistic descriptors. The multifractal spectrum of a given realization, on one hand, is given as the Hausdorff dimension of all points where the local Hölder regularity has a given degree; it is local and random in nature. The multifractal envelope, on the other hand, is defined in terms of the scaling of moments and constitutes, thus, a global and deterministic descriptor. It is well known that the envelope provides an almost sure upper bound to the spectrum, a fact that can easily be established through arguments akin to large deviation principles. The goal of multifractal analysis lies in establishing almost sure equality of spectrum and envelope.Motivated by natural, desirable properties of a process such as stationarity, this paper studies the multifractal spectrum of an infinite product of stationary jump processes, defined as the limit of a particular choice of martingales of Kahane:(1.1)A(t)=limn→∞∫0t∏i=1nΛ(i)(s)ds,
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 ℒ2 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)(·)=distΛ(bi·), where Λ is a stationary jump process with i.i.d. exponential interjump times.Overview and contributions: Section2 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].


## 2. Multifractal Spectra and Formalism

The multifractal spectrum of a process is defined pathwise. Therefore, we start in sections2.1–2.3 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.


### 2.1. Hausdorff Dimension

Let us start by recalling notions related to Hausdorff dimension. Consider a setF⊂ℝd and a class 𝒞 of subsets of ℝd. Then, the s-dimensional Hausdorff measure with respect to (w.r.t.)𝒞 is given by(2.1)hs𝒞(F)≐limδ→0inf{∑i|Ui|s:0<|Ui|≤δ,Ui∈𝒞∀i;F⊆⋃iUi},
where the diameter of set U is denoted by |U|=sup{|x-y|:x,y∈U}. Then (2.2)dim𝒞(F)≐sups{s≥0:hs(F)=∞}=infs{s≥0:hs(F)=0}
is called the Hausdorff dimension of setFw.r.t.𝒞. By definition, when choosing the powerset 2ℝd, that is, the class of all subsets of ℝd as the class 𝒞 in (2.1), then these notions reduce to the usual s-dimensional Hausdorff measure hs and dimension dim:(2.3)hs(F)=hs2ℝd(F),dim(F)=dim2ℝd(F).
Clearly, dim(F)≤dim𝒞(F). The Hausdorff dimension w.r.t. 𝒞 is σ-stable, that is, dim𝒞(∪jFj)=supjdim𝒞(Fj) for any countable family of sets Fj. This can be established as in the classical case. See Rogers [43] for more properties of Hausdorff measures.Numerous classes of sets𝒞 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)=dim𝒞(F). The simplest example of such a family is the b-ary cubes of the form
(2.4)[k1b-n,(k1+1)b-n)×⋯×[kdb-n,(kd+1)b-n)
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.
LetF⊂ℝd and 𝒞 be a nested collection of arbitrary cubes. Assume that for all x∈F there exists a sequence {Cn(x)}ℕ∈𝒞 such that x∈Cn(x)≠{x} for all n,
(2.5)limn→∞|Cn(x)|=0,limsupn→∞log|Cn(x)|log|Cn-1(x)|≤1.
Then dim(F)=dim𝒞(F).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)=dim𝒞(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 |Cn(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 asmass 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 2.1.Lemma 2.2 (see [45, Lemma 4.3.2.b]).
Letν be a Borel measure on ℝd, F a Borel subset, and 𝒞={Ckn:k=1,…,Nn,n∈ℕ} a nested collection of cubes. Assume  ν(F)>0, (i)
{Ckn}k=1Nn cover F for all n,(ii)
there existsb>1 such that (1/n)log|Ckn|→-logbas Ckn→{x} for all x∈F,(iii)
logν(Ckn)/log|Ckn|→s as Ckn→{x} for all x∈F. Then, dim(F)=dim𝒞(F)≥s.


### 2.2. Multifractal Spectrum

Letν be a Borel measure in ℝd. Its Hölder exponent at x, or equivalently local dimension at x, is given by (2.6)dimlocν(x)=limr→0logν(B(x,r))logr
if this limit exists. In multifractal analysis, we are usually interested in the properties like fractal dimension of the set {x:dimlocν(x)=α}.A more approachable framework to study scaling properties is to consider a sequence of finer and finer partitionsCn≐{Ckn:k=1,…,Nn} of a set containing the support of measure. The standard example in ℝ is to partition the unit interval by the dyadic intervals [(k-1)2-n,k2-n), k=1,…,2n.Definition 2.3.
LetF be a Borel subset of ℝd and Cn={Ckn:k=1,…,Nn} a partition of F for each n=1,2,…. We call the Cnnested partitions, if any Ckn∈Cn (n≥2), k=1,…,Nn, is a subset of some Ck′n-1∈Cn-1, k′∈{1,…,Nn-1}.Let(2.7)αn𝒞(x)≐logν(Cn(x))log|Cn(x)|,
where 𝒞={Cn:n=1,…} is a collection of nested partitions Cn of supp,ν and Cn(x) is the unique Ckn∈Cn containing x. Then the local scaling exponent (w.r.t. 𝒞) of ν at x is given by (2.8)α𝒞(x)≐limn→∞αn𝒞(x)
if this limit exists. One should think of α𝒞(x) as giving approximately the degree of Hölder regularity of ν at x, that is, an approximation to dimlocν(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(2.9)Eα𝒞≐{x:limnαn𝒞(x)=α},Vα𝒞≐{x:liminfnαn𝒞(x)≤α}
characterizing the local scaling behaviour. The classical literature on multifractals studies typically the sets Eα𝒞, calling dimEα𝒞 the multifractal spectrum of the measure ν. The sets Vα𝒞 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:liminfnαn𝒞(x)=α} would be of interest since they form a partition of space. However, their dimension would lie between dim(Eα𝒞) and dim(Vα𝒞). Trivially, Eα𝒞⊆Vα𝒞 and dim(Eα𝒞)≤dim(Vα𝒞).


### 2.3. Pathwise Partition Function and Coarse Grain Spectra

In order to make the presentation simpler, we consider only the 1-dimensional case assuming thatsuppν⊆[0,1], but all the definitions are easily extended to compactly supported measures on ℝd.Consider nested partitionsJn={Jkn:k=1,…,Nn} of the unit interval. Analogous to the previous section, denote 𝒞={Jn:n=1,…} the nested collection of sets which is determined by the Jn. The partition sum of the measure ν is defined by (2.10)Sn𝒞(q,γ)≐Sn(q,γ)=∑k=1Nnν(Jkn)q|Jkn|-γ,
where we adapt the convention 0q≐0 for all q∈ℝ. The only purpose of this convention is to provide a convenient way to ensure that the sets Jkn with ν(Jkn)=0 do not contribute to any of the Sn(q). Define then the partition function as(2.11)τ𝒞(q)≐τ(q)=sup{γ:∑nSn𝒞(q,γ)<∞}.
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(2.12)τ*(α)≐infq(αq-τ(q)).
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 τ𝒞 reduces to the standard one (for an overview see [48]).To provide upper bounds on the multifractal spectra, it is useful to introduce the so-called “coarse grain” spectra. To this end, the following collections of intervals are of interest:(2.13)Nn+(α)≐{k:Jkn∩[0,1]≠∅,logν(Jkn)log|Jkn|<α}.
The reference to large deviations relies on the fact that Nn+ may be represented as a rare event or a “large deviation from the mean” if one considers logν(Jkn)/log|Jkn| to be the random variable, where k is random but ν and Jn are fixed and n→∞. In the proper setting, the following spectrum f𝒞+ becomes a large deviation rate function: (2.14)f𝒞+(α)≐f+(α)≐inf{δ>0:∑n≥m∑k∈Nn+(α)|Jkn|δ→0asm→∞}.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 non-homogeneous partitions. The proofs are quite straightforward but they are shown for the sake of completeness.Lemma 2.4.
For Borel measureν, the following holds. If maxk|Jkn|→0 as n→∞ then
(2.15)dim𝒞(Vα𝒞)≤f𝒞+(α+)≐limε↓0f𝒞+(α+ε).Proof.
Fixα and take any γ such that γ>f+(α+). Then there exists ε>0 such that f+(α+ε)<γ. Let m be an arbitrary positive integer. Then, for any t∈Vα𝒞 there exists n>m such that αn(t)<α+ε and, thus, there is k∈Nn+(α+ε) such that Jn(t)=Jkn. Since t∈Jn(t), we get
(2.16)Vα𝒞⊆⋃n≥m⋃k∈Nn+(α+ε)Jkn.
This means that for every m we have constructed a cover of Vα𝒞 with Jkn such that
(2.17)maxn≥m,k∈Nn+(α+ε)|Jkn|→0,∑n≥m∑k∈Nn+(α+ε)|Jkn|γ→0asm→∞.
We conclude that hγ𝒞(Vα𝒞)=0 and dim𝒞(Vα𝒞)≤γ.Lemma 2.5.
Letν be a Borel measure and let the Jn 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.
Letα, q, and γ be arbitrary real numbers for the moment. By definition, ν(Jkn)>|Jkn|α for all k in Nn+(α). For nonnegativeq we may, thus, estimate
(2.18)Sn(q,γ)≥∑Nn+(α)ν(Jkn)q|Jkn|-γ≥∑Nn+(α)|Jkn|qα-γ.
Fix a realα and q≥0. Consider γ<τ(q). By definition of τ, we find
(2.19)∑n≥m∑Nn+(α)|Jkn|qα-γ→0asm→∞.
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 ofτ(q) can be established for q>0 under mild conditions.Lemma 2.6.
LetJn form nested partitions such that ∑nmaxk|Jkn|<∞. Let ν be a bounded Borel measure. Then τ(q)=(f+)*(q) and τ is convex for q>0.Proof.
Replacingν by cν for a sufficiently small c, we may assume that αn(t)≥0. Fix q>0. Let γ<infα(qα-f+(α)), let ε>0 be such that γ<qα-f+(α)-εq for every α, and let Uj 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)Sn(q,γ)≤(∑j=1p∑k:logν(Jkn)/log|Jkn|∈Uj+∑k:logν(Jkn)/log|Jkn|>M)ν(Jkn)q|Jkn|-γ≤∑j=1p∑k∈Nn+(jε)|Jkn|q((j-1)ε)|Jkn|-γ+∑k:Jkn∩[0,1]≠∅|Jkn|qM|Jkn|-γ.
By choice of M, and since the Jkn are disjoint subsets of [-1,2] for n large enough,
(2.21)∑Jkn∩[0,1]≠∅|Jkn|qM|Jkn|-γ≤∑Jkn∩[0,1]≠∅|Jkn|2≤maxk|Jkn|∑Jkn∩[0,1]≠∅|Jkn|≤3maxk|Jkn|.
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.


### 2.4. Deterministic Envelope for Random Measures

Let us now consider a random measureμ defined on [0,1] and a random nested collection 𝒞. The asymptotics of ensemble moments are given by (2.22)T𝒞(q)≐T(q)=sup{γ:∑n=1∞E[Sn𝒞(q,γ)]<∞}.
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 ifT(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.
IfT is convex and τ is convex a.s. in an interval I, then a.s. τ(q)≥T(q) for all q∈I.Proof.
Fixq∈I such that T(q)<∞ and take any γ<T(q). Since Sn(q,γ) is positive,
(2.23)∑n≥mE[Sn(q,γ)]=E[∑n≥mSn(q,γ)]<∞,
and thus ∑n≥mSn(q,γ)<∞ a.s., that is, τ(q)>γ a.s. Next, choose γm↗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 lemmas2.4, 2.5, and 2.7, we can now order the multifractal spectra.Theorem 2.8.
IfT(q) is convex and τ(q) is convex a.s. for q>0, and limn→∞maxk|Jkn|=0 a.s., then the multifractal spectra are ordered as follows: with probability one, for all α,
(2.24)dim𝒞(Eα𝒞)≤dim𝒞(Vα𝒞)≤f𝒞+(α+)≤infq>0(qα-τ𝒞(q))≤infq>0(qα-T𝒞(q)).The basic arguments that lead to Theorem2.8 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(Eα) 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 T𝒞, 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].


## 2.1. Hausdorff Dimension

Let us start by recalling notions related to Hausdorff dimension. Consider a setF⊂ℝd and a class 𝒞 of subsets of ℝd. Then, the s-dimensional Hausdorff measure with respect to (w.r.t.)𝒞 is given by(2.1)hs𝒞(F)≐limδ→0inf{∑i|Ui|s:0<|Ui|≤δ,Ui∈𝒞∀i;F⊆⋃iUi},
where the diameter of set U is denoted by |U|=sup{|x-y|:x,y∈U}. Then (2.2)dim𝒞(F)≐sups{s≥0:hs(F)=∞}=infs{s≥0:hs(F)=0}
is called the Hausdorff dimension of setFw.r.t.𝒞. By definition, when choosing the powerset 2ℝd, that is, the class of all subsets of ℝd as the class 𝒞 in (2.1), then these notions reduce to the usual s-dimensional Hausdorff measure hs and dimension dim:(2.3)hs(F)=hs2ℝd(F),dim(F)=dim2ℝd(F).
Clearly, dim(F)≤dim𝒞(F). The Hausdorff dimension w.r.t. 𝒞 is σ-stable, that is, dim𝒞(∪jFj)=supjdim𝒞(Fj) for any countable family of sets Fj. This can be established as in the classical case. See Rogers [43] for more properties of Hausdorff measures.Numerous classes of sets𝒞 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)=dim𝒞(F). The simplest example of such a family is the b-ary cubes of the form
(2.4)[k1b-n,(k1+1)b-n)×⋯×[kdb-n,(kd+1)b-n)
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.
LetF⊂ℝd and 𝒞 be a nested collection of arbitrary cubes. Assume that for all x∈F there exists a sequence {Cn(x)}ℕ∈𝒞 such that x∈Cn(x)≠{x} for all n,
(2.5)limn→∞|Cn(x)|=0,limsupn→∞log|Cn(x)|log|Cn-1(x)|≤1.
Then dim(F)=dim𝒞(F).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)=dim𝒞(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 |Cn(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 asmass 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 2.1.Lemma 2.2 (see [45, Lemma 4.3.2.b]).
Letν be a Borel measure on ℝd, F a Borel subset, and 𝒞={Ckn:k=1,…,Nn,n∈ℕ} a nested collection of cubes. Assume  ν(F)>0, (i)
{Ckn}k=1Nn cover F for all n,(ii)
there existsb>1 such that (1/n)log|Ckn|→-logbas Ckn→{x} for all x∈F,(iii)
logν(Ckn)/log|Ckn|→s as Ckn→{x} for all x∈F. Then, dim(F)=dim𝒞(F)≥s.


## 2.2. Multifractal Spectrum

Letν be a Borel measure in ℝd. Its Hölder exponent at x, or equivalently local dimension at x, is given by (2.6)dimlocν(x)=limr→0logν(B(x,r))logr
if this limit exists. In multifractal analysis, we are usually interested in the properties like fractal dimension of the set {x:dimlocν(x)=α}.A more approachable framework to study scaling properties is to consider a sequence of finer and finer partitionsCn≐{Ckn:k=1,…,Nn} of a set containing the support of measure. The standard example in ℝ is to partition the unit interval by the dyadic intervals [(k-1)2-n,k2-n), k=1,…,2n.Definition 2.3.
LetF be a Borel subset of ℝd and Cn={Ckn:k=1,…,Nn} a partition of F for each n=1,2,…. We call the Cnnested partitions, if any Ckn∈Cn (n≥2), k=1,…,Nn, is a subset of some Ck′n-1∈Cn-1, k′∈{1,…,Nn-1}.Let(2.7)αn𝒞(x)≐logν(Cn(x))log|Cn(x)|,
where 𝒞={Cn:n=1,…} is a collection of nested partitions Cn of supp,ν and Cn(x) is the unique Ckn∈Cn containing x. Then the local scaling exponent (w.r.t. 𝒞) of ν at x is given by (2.8)α𝒞(x)≐limn→∞αn𝒞(x)
if this limit exists. One should think of α𝒞(x) as giving approximately the degree of Hölder regularity of ν at x, that is, an approximation to dimlocν(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(2.9)Eα𝒞≐{x:limnαn𝒞(x)=α},Vα𝒞≐{x:liminfnαn𝒞(x)≤α}
characterizing the local scaling behaviour. The classical literature on multifractals studies typically the sets Eα𝒞, calling dimEα𝒞 the multifractal spectrum of the measure ν. The sets Vα𝒞 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:liminfnαn𝒞(x)=α} would be of interest since they form a partition of space. However, their dimension would lie between dim(Eα𝒞) and dim(Vα𝒞). Trivially, Eα𝒞⊆Vα𝒞 and dim(Eα𝒞)≤dim(Vα𝒞).


## 2.3. Pathwise Partition Function and Coarse Grain Spectra

In order to make the presentation simpler, we consider only the 1-dimensional case assuming thatsuppν⊆[0,1], but all the definitions are easily extended to compactly supported measures on ℝd.Consider nested partitionsJn={Jkn:k=1,…,Nn} of the unit interval. Analogous to the previous section, denote 𝒞={Jn:n=1,…} the nested collection of sets which is determined by the Jn. The partition sum of the measure ν is defined by (2.10)Sn𝒞(q,γ)≐Sn(q,γ)=∑k=1Nnν(Jkn)q|Jkn|-γ,
where we adapt the convention 0q≐0 for all q∈ℝ. The only purpose of this convention is to provide a convenient way to ensure that the sets Jkn with ν(Jkn)=0 do not contribute to any of the Sn(q). Define then the partition function as(2.11)τ𝒞(q)≐τ(q)=sup{γ:∑nSn𝒞(q,γ)<∞}.
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(2.12)τ*(α)≐infq(αq-τ(q)).
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 τ𝒞 reduces to the standard one (for an overview see [48]).To provide upper bounds on the multifractal spectra, it is useful to introduce the so-called “coarse grain” spectra. To this end, the following collections of intervals are of interest:(2.13)Nn+(α)≐{k:Jkn∩[0,1]≠∅,logν(Jkn)log|Jkn|<α}.
The reference to large deviations relies on the fact that Nn+ may be represented as a rare event or a “large deviation from the mean” if one considers logν(Jkn)/log|Jkn| to be the random variable, where k is random but ν and Jn are fixed and n→∞. In the proper setting, the following spectrum f𝒞+ becomes a large deviation rate function: (2.14)f𝒞+(α)≐f+(α)≐inf{δ>0:∑n≥m∑k∈Nn+(α)|Jkn|δ→0asm→∞}.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 non-homogeneous partitions. The proofs are quite straightforward but they are shown for the sake of completeness.Lemma 2.4.
For Borel measureν, the following holds. If maxk|Jkn|→0 as n→∞ then
(2.15)dim𝒞(Vα𝒞)≤f𝒞+(α+)≐limε↓0f𝒞+(α+ε).Proof.
Fixα and take any γ such that γ>f+(α+). Then there exists ε>0 such that f+(α+ε)<γ. Let m be an arbitrary positive integer. Then, for any t∈Vα𝒞 there exists n>m such that αn(t)<α+ε and, thus, there is k∈Nn+(α+ε) such that Jn(t)=Jkn. Since t∈Jn(t), we get
(2.16)Vα𝒞⊆⋃n≥m⋃k∈Nn+(α+ε)Jkn.
This means that for every m we have constructed a cover of Vα𝒞 with Jkn such that
(2.17)maxn≥m,k∈Nn+(α+ε)|Jkn|→0,∑n≥m∑k∈Nn+(α+ε)|Jkn|γ→0asm→∞.
We conclude that hγ𝒞(Vα𝒞)=0 and dim𝒞(Vα𝒞)≤γ.Lemma 2.5.
Letν be a Borel measure and let the Jn 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.
Letα, q, and γ be arbitrary real numbers for the moment. By definition, ν(Jkn)>|Jkn|α for all k in Nn+(α). For nonnegativeq we may, thus, estimate
(2.18)Sn(q,γ)≥∑Nn+(α)ν(Jkn)q|Jkn|-γ≥∑Nn+(α)|Jkn|qα-γ.
Fix a realα and q≥0. Consider γ<τ(q). By definition of τ, we find
(2.19)∑n≥m∑Nn+(α)|Jkn|qα-γ→0asm→∞.
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 ofτ(q) can be established for q>0 under mild conditions.Lemma 2.6.
LetJn form nested partitions such that ∑nmaxk|Jkn|<∞. Let ν be a bounded Borel measure. Then τ(q)=(f+)*(q) and τ is convex for q>0.Proof.
Replacingν by cν for a sufficiently small c, we may assume that αn(t)≥0. Fix q>0. Let γ<infα(qα-f+(α)), let ε>0 be such that γ<qα-f+(α)-εq for every α, and let Uj 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)Sn(q,γ)≤(∑j=1p∑k:logν(Jkn)/log|Jkn|∈Uj+∑k:logν(Jkn)/log|Jkn|>M)ν(Jkn)q|Jkn|-γ≤∑j=1p∑k∈Nn+(jε)|Jkn|q((j-1)ε)|Jkn|-γ+∑k:Jkn∩[0,1]≠∅|Jkn|qM|Jkn|-γ.
By choice of M, and since the Jkn are disjoint subsets of [-1,2] for n large enough,
(2.21)∑Jkn∩[0,1]≠∅|Jkn|qM|Jkn|-γ≤∑Jkn∩[0,1]≠∅|Jkn|2≤maxk|Jkn|∑Jkn∩[0,1]≠∅|Jkn|≤3maxk|Jkn|.
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.


## 2.4. Deterministic Envelope for Random Measures

Let us now consider a random measureμ defined on [0,1] and a random nested collection 𝒞. The asymptotics of ensemble moments are given by (2.22)T𝒞(q)≐T(q)=sup{γ:∑n=1∞E[Sn𝒞(q,γ)]<∞}.
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 ifT(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.
IfT is convex and τ is convex a.s. in an interval I, then a.s. τ(q)≥T(q) for all q∈I.Proof.
Fixq∈I such that T(q)<∞ and take any γ<T(q). Since Sn(q,γ) is positive,
(2.23)∑n≥mE[Sn(q,γ)]=E[∑n≥mSn(q,γ)]<∞,
and thus ∑n≥mSn(q,γ)<∞ a.s., that is, τ(q)>γ a.s. Next, choose γm↗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 lemmas2.4, 2.5, and 2.7, we can now order the multifractal spectra.Theorem 2.8.
IfT(q) is convex and τ(q) is convex a.s. for q>0, and limn→∞maxk|Jkn|=0 a.s., then the multifractal spectra are ordered as follows: with probability one, for all α,
(2.24)dim𝒞(Eα𝒞)≤dim𝒞(Vα𝒞)≤f𝒞+(α+)≤infq>0(qα-τ𝒞(q))≤infq>0(qα-T𝒞(q)).The basic arguments that lead to Theorem2.8 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(Eα) 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 T𝒞, 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].


## 3. Multifractal Jump Processes

### 3.1. Definition and Basic Properties

We start fromT-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: (A1)
Λ is stationary with Λ>0 and E[Λ]=1; there exists b>1 such that Λ(i)(·/bi) are i.i.d. processes, distributed as Λ;(A2)
Λ is a weakly mixing Markov process which is defined by Λ(t)≐Mk for Tk≤t<Tk+1, where T0=0 and {Tk+1-Tk}k are i.i.d. exponential random variables of mean 1/λ, and where the Mk form a stationary positive time series independent of {Tk}k satisfying E[Mk]=1 for all k.Recall that a real-valued stationary process is weakly mixing if for allB1,B2∈ℬ (the σ-algebra generated by the process) limT→∞T-1∫0T|P(B1∩StB2)-P(B1)P(B2)|dt=0 where St is the shift operator. Examples of interest include the cases where the sequence {Mk}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 be studied. For example, if Λ is a finite state Markov process, then analogous results can be derived.Next define the finite product processes(3.1)Λn(t)≐∏i=0n-1Λ(i)(t).
For t∈[0,1], the cumulative processes (3.2)An(t)≐∫0tΛn(s)ds=∫0t∏i=0n-1Λ(i)(s)ds,n=1,2,…
can be associated with positive measures defined on the Borel sets ℬ of [0,1]: (3.3)μn(B)≐∫BΛn(s)ds,n=1,2,…,B∈ℬ.
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≐limnμn 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, themultifractal jump process (MJP) is the limit
(3.4)A(t)≐μ([0,t])≐limn→∞μn([0,t])=limn→∞An(t)a.s.
An MJP is called degenerate if A(1)=0 almost surely, and nondegenerate otherwise.It can be shown thatA is non-degenerate if and only if E[A(1)]=1 (see [39]). The convergence of An(1) in ℒp for some p>1 naturally implies nondegeneracy since then E[A(1)]=limnE[An(1)]=1. To provide criteria for convergence in ℒp is thus of central importance. As pointed out in [2], criteria for convergence in ℒ2 are particularly manageable and useful. In the following theorem, which is a straightforward adaption of [39, Corollary 3], the ℒ2 conditions are stated for MJPs.Theorem 3.2 (see [39, Corollary 3]).
Assume (A1), (A2), andE[M2]<∞. If there are positive γ1, γ2 and C with
(3.5)σ2e-γ1|x|≤Cov(Λ(0),Λ(x))=∑j=0∞(λx)jj!Cov(M0,Mj)e-λx≤C|x|-γ2,
then An(t) converges in ℒ2 if and only if b>1+σ2 where σ2=Var(M).Condition (3.5) allows a very large family of time series Mk. For example, a sufficient condition for convergence is(3.6)Cov(M0,Mj)≤Cj+1,j=0,1,2,…,
for some constant C>0.The majority of the following results are stated assuming a converging MJP in someℒp, 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 Mj are i.i.d. and the ℒ3 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 measureμ is ergodic with respect to the time shift operator. Ergodicity of μ, together with the positivity of the multipliers Mk, 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 measureμ on ℝ+ of an MJP is ergodic.Proof.
Consider the spacesΩi=(Xi,ℬi,Pi), i=0,1,…, and Ω=Πi=0∞Ωi=(X,ℬ,P), where Xi is the set of piece-wise continuous functions, ℬi its Borel-algebra, and Pi is the law of Λ(i). The shift operator is weakly mixing in Ωi and thus it is also weakly mixing in Ω0×⋯×Ωn-1 for any fixed n. Since the B0×⋯×Bn-1×Πi=n∞Ωi 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:(3.7)μ(n)(B)≐∫B∏i=0n-11Λ(i)(τ)dμ(τ),
that is, the measure where n first terms of the product are neglected. Thus(3.8)μ(n)(B)=distb-nμ(bnB).Proposition 3.4.
Assume thatA is a non-degenerate MJP. Then, with probability one, μ([0,t])>0 for all t∈(0,1].Proof.
First consider an arbitraryt>0 and make a change of variable to get
(3.9)μ([0,t])=∫0tΛn(s)dμ(n)(s)=distb-n∫0tbnΛn(sb-n)dμ̃(s),
where μ and μ̃ are identically distributed and μ̃ independent of Λn. Then by the positivity of Λn and ergodicity of μ, we have
(3.10)P(μ([0,t])=0)=P(b-n∫0tbnΛn(sb-n)dμ̃(s)=0)=P(b-nμ([0,tbn])=0)≤P(b-nμ([0,tbn])≤t2)→0asn→∞.
Thus μ([0,t])>0 a.s. and μ being a positive measure, trivially μ([0,s])>0 for all s≥t a.s.
Now denoteUt={ω:μ[0,t]=0}. Then
(3.11)P(μ([0,t])=0∀t∈(0,1])=P(⋃n=1∞U1/n)≤∑nP(U1/n)=0,
which completes the proof.


### 3.2. 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 Section2. In analogy with related multiplicative processes [15], we should expect moments of the multipliers to affect the multifractal properties through the structure function(3.12)β(q)≐q-1-logbE[Λq].
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]).
IfE[Λp]=E[Mp]<∞ and An(t) converges in ℒp for some p>1, then, for 0<q≤p,
(3.13)C̲tβ(q)+1≤E[A(t)q]≤C¯tβ(q)+1∀t∈[0,1],(3.14)T𝒟(q)=β(q)=q-1-logbE[Mq].Note that under the assumption (3.5) of Theorem 3.2, convergence of the MJP A(t) in ℒ2 is equivalent with β(2)>0.In order to analyze the MJP, it is natural to consider the nested sequence of partitionsJn induced by Λn. By assumptions (A1) and (A2), (3.15)Λ(i)(t)=Mk(i),Tk-1(i)≤t<Tk(i),
where {Tk(i)}k=1∞ forms a Poisson point process of intensity λbi and {Mk(i)}k=1∞ is a stationary positive time series independent of {Tk(i)}k. Consequently, the product process (3.16)Λn(t)=∏i=0n-1Λ(i)(t)
is constant on intervals Jkn≐[Tk-1n,Tkn), where the Tkn are defined by a Poisson point process of intensity λn=λ∑i=0n-1bi=O(bn). From the construction of {Tkn}k, it follows that the partitions Jn={Jkn:k=1,…} are nested. We denote the resulting collection of nested Poisson partitionsJn by 𝒫.Lemma 3.6.
Consider a nested Poisson partition𝒫. Let to be arbitrary. With probability one, as n→∞,
(3.17)log|Jn(to)|n→-logb,logmaxk|Jkn|n→-logb,logmink|Jkn|n→-2logb.Proof.
Without loss of generality, assumeλn=λbn and t0∈[0,1).
(1°): Clearly(3.18)P(Nopointsin[t0-x,t0+x])≤P(|Jn(t0)|>x)≤2P(Nopointsin[t0,t0+x/2]).
Thus for n large enough,
(3.19)P(|Jn(t0)|>b-nenε)≤2e(-1/2)λnb-nenε≤2e(-1/2)nελ,P(|Jn(t0)|<b-nenε)≤1-e-2λnb-ne-nε≤2λe-nε.
(2°): Since the maximal interval is always longer than a fixed interval, forn large enough,
(3.20)P(maxk|Jkn|<b-ne-nε)≤P(|Jn(t0)|<b-ne-nε)≤2λe-nε.
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
(3.21)P(maxk|Jkn|>b-nenε)≤∑i=1⌊2bne-nε⌋e(-1/2)λnb-nenε≤2e-n((λ/4)nε2-logb).
(3°): Let{Ukm}k=1m+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)P(mink|Ukm|<x)={1-(1-(m+1)x)m,0<x<11+m,1,x≥1m+1.
Let Nn denote the number of points of a Poisson process with density λbn located in [0,1]. Applying (3.22) gives
(3.23)P(mink|Jkn|<b-2ne-nε)=∑m=0∞P(mink=1,…,Nn+1|Jkn|<b-2ne-nε∣Nn=m)P(Nn=m)=∑m=0⌊b2nenε⌋-1[1-(1-(m+1)b-2ne-nε)m](λbn)mm!e-λbn+∑m=⌊b2nenε⌋∞(λbn)mm!e-λbn≤∑m=0∞(m2+m)b-2ne-nε(λbn)mm!e-λbn+(λbn)⌊b2nenε⌋⌊b2nenε⌋!≤e-nε(2λb-n+λ2)+e-nε,
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
(3.24)P(mink|Ikn|>b-2nenε)≤(e-λb-n(1+λb-nenε))b2ne-nε-1=exp(-λbn+λb-nenε+(b2ne-nε-1)log(1+λb-nenε))≤e-c1nε.
(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(·)=dim𝒫(·). In addition, by lemma 2.6 and theorem 2.8, the multifractal envelope T𝒫* provides an upper bound on the multifractal spectrum dim𝒫(Eα), almost surely.The connection to the measuresμ(n) defined in (3.7) is especially practical when considering sets Jn(t) because then(3.25)μ(Jn(t))=μ(n)(Jn(t))∏i=0n-1Λ(i)(t)=distb-nμ̃(bnJn(t))∏i=0n-1Λ(i)(t),
where μ̃ is distributed as μ and is independent of Jn and Λ(i), i=0,…,n-1. On the other hand, by corollary 3.5 and(3.8)(3.26)C1b-n(q-1-β(q))|I|1+β(q)≤E[μ(n)(I)q]≤C2b-n(q-1-β(q))|I|1+β(q)
for a fixed interval I. We find that T𝒫=T𝒟 under mild conditions.Proposition 3.7.
IfE[Mp]<∞ and An(t) converges in ℒp for some p>1, then
(3.27)T𝒫(q)=β(q)=q-1-logbE[Mq]for0<q≤p.Proof.
ConsiderE[∑k=1Nn|Jkn|1+ε] with ε>0. Denote by {Ukj:k=1,…,j+1} the partition resulting from j uniformly distributed points on [0,1]. Then
(3.28)E[|Ukj|α]=∫01jsα(1-s)j-1ds=j!Γ(1+α)Γ(j+1+α).
Using the above result and conditioning with respect to the number of intervals gives
(3.29)E[∑k=1Nn|Jkn|1+ε]=∑j=0∞E[∑k=1j+1|Ukj|1+ε]P(Nn=j+1)=∑j=0∞(j+1)j!Γ(2+ε)Γ(j+2+ε)(λn)je-λnj!≤Γ(2+ε)(λn)-ϵe-λn∑j=0∞(λn)j+εΓ(j+1+ε)=Γ(2+ε)(λn)-ϵ(1-Γ(ε,λn)Γ(ε))≤Cb-nε.
Now assumeγ=β(q)-ϵ with ϵ>0. Conditioning with respect to the partition Jn helps us to split the addends into three factors, that is,
(3.30)E[∑k=1Nnμ(Jkn)q|Jkn|-γ]=EE[∑k=1NnΛn(Tk-1n)q|Jkn|-γμ(n)(Jkn)q∣Jn].
Use the scaling law (3.26) together with the independence of μ(n) with respect to Jn and that
(3.31)E[Λn(Tkn)q∣Jn]=E[Λn(Tkn)q]=E[Mq]n=b-n(β(q)+1-q)
in order to estimate
(3.32)E[∑k=1Nnμ(Jkn)q|Jkn|-γ]≤Cb-n(q-1-β(q))E[Mq]nE[∑k=1Nn|Jkn|1+β(q)-γ]=CE[∑k=1Nn|Jkn|1+ε]≤C̃b-nε.
Next, assume thatγ=β(q)+ε. Condition again with respect to the partition Jn and apply the lower bound of (3.13) to get
(3.33)E[∑k=1Nnμ(Jkn)q|Jkn|-γ]≥CE[∑k=1Nn|Jkn|1-ε]≥CE[∑k=1Nn|Jkn|]=C.
Thus, we have found that∑n≥mE[Sn(q,γ)]→0 whenever γ<β(q) and ∑n≥mE[Sn(q,γ)]↛0 whenever γ>β(q).


### 3.3. Information Dimension

This section is devoted to analyzingEα1𝒫, the set of points with scaling exponent (3.34)α1𝒫≐α1≐β′(1)=1-E[ΛlogbΛ]=1-E[MlogbM]
which is the most relevant of all Eα𝒫.The upper bound of the multifractal spectrumdim𝒫(Eα𝒫)≤T𝒫* a.s. follows from lemma 2.6, lemma 3.6, and theorem 2.8. While T𝒫=T𝒟 under mild conditions (see above), we need to introduce the auxiliary set(3.35)Kα𝒫={x:limnαn𝒫(x)=α,limnlogb|Jn(x)|-n=1},
a subset of Eα𝒫 for which dim(Kα𝒫)=dim𝒫(Kα𝒫) 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 locationst, 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 dimKα𝒫, and thus dimEα𝒫, 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 (3.36)Q(φ)=E[∫01φ(t,ω)dμ(t)]
for all positive measurable functions φ(t,ω).Lemma 3.8.
Assume thatA(t) is a non-degenerate MJP. If (ω,T) is picked according to the Peyrière measure Q, then
(3.37)limn→∞log|Jn(T)|n=-logbQ-a.s.
Moreover, if An(t) converges in ℒq for some q>1 and bq-1>E[Λq], that is, β(q)>0, then
(3.38)logμ(n)(Jn(T))n→-logbQ-a.s.Proof.
Recall first thatE[μ(n)(Jkn)|Jn]=|Jkn| and that Λn(Tkn) is independent of Jn so that
(3.39)E[Λn(Tkn)∣Jn]=E[Λn]=1.
(1°) Condition with respect to the partition Jn in order to get
(3.40)PQ(|Jn(T)|<x)=E[∫011{|Jn(t)|<x}dμ(t)]=EE[∑k=1Nn1{|Jkn|<x}Λn(Tk-1n)μ(n)(Jkn)∣Jn]=E[∑k=1Nn1{|Jkn|<x}|Jkn|]=E[∫011{|Jn(t)|<x}dt]=∫01P(|Jn(t)|<x)dt≤C1xλn,
where C1 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:(3.41)PQ(|Jn(T)|>x)=∫01P(1{|Jn(t)|>x})dt≤C2e-λnx,
where C2 is independent of n. Here, the last inequality follows from estimating the probability that a sum of two exponential random variables exceeds x.
Sinceλn=O(bn), we have
(3.42)∑n=1∞PQ(|Jn(T)|<b-ne-nε)<∞,∑n=1∞PQ(|Jn(T)|>b-nenε)<∞.
(2°) Let ε>0. Write the Q-probability as a sum
(3.43)PQ(μ(n)(Jn(T))<b-ne-nε)=E[∑k=1Nn1{μ(n)(Jkn)<b-ne-nε}Λn(Tkn)μ(n)(Jkn)].
Scaling the μ(n)(Jkn) up by b-ne-nε and applying (3.39) lead to
(3.44)PQ(μ(n)(Jn(T))<b-ne-nε)≤b-ne-nεEE[∑k=0NnΛn(Tkn)∣Jn]≤b-ne-nεE[Nn].
This implies PQ(μ(n)(Jn(T)))≤Ce-nε, because E[Nn]=λn=O(bn).
On the other hand, by (3.26) and (3.29),
(3.45)EQ[μ(n)(Jn(T))q-1]=EE[∑k=1NnΛn(Tk-1n)μ(n)(Jkn)q∣Jn]≤C̃b-n(q-1-β(q))E[∑k=1Nn|Jkn|1+β(q)]≤Cb-n(q-1).
Then apply the Markov inequality to get
(3.46)PQ(μ(n)(Jn(T))≥b-nenε)≤Ce-nε(q-1).
(3°) Applying Borel-Cantelli to (1°) and (2°) completes the proof.To establish theQ-independence required for the LLN, we need the following result.Lemma 3.9.
Assume thatA(t) is a non-degenerate MJP. Then
(3.47)EQ[∏i=0n-1gi(Λ(i)(T))]=∏i=0n-1E[gi(Λ(i)(t))Λ(i)(t)]
for all n∈ℕ and positive Borel functions gi defined on ℝ+.Proof.
(Note: this is a replication of the proof appearing in [2]). Condition with respect to the partition Jn and use E[μn(Jkn)|Jn]=|Jkn|. In addition to this, utilize independence (w.r.t. Ω) and stationarity:
(3.48)EQ[∏i=0n-1gi(Λ(i)(T))]=E[∫01∏i=0n-1gi(Λ(i)(t))dμ(t)]=EE[∑k=0Nn∏i=0n-1gi(Λ(i)(Tk-1n))Λ(i)(Tk-1n)μ(n)(Ikn)∣Jn]=E[∑k=0Nn∏i=0n-1gi(Λ(i))(Tk-1n)Λ(i)(Tk-1n)|Jkn|]=E[∫01∏i=0n-1gi(Λ(i)(t))Λ(i)(t)dt]=∫01∏i=0n-1E[gi(Λ(i)(t))Λ(i)(t)]dt=∏i=0n-1E[gi(Λ(i)(t))Λ(i)(t)].Corollary 3.10.
Assume thatA(t) is a non-degenerate MJP. Then the random variables Λ(i)(T), i=0,1,…,n, are Q-independent.Proof.
Fixj and set gi(x)≡1 when i≠j, leaving gj arbitrary. Lemma 3.9 gives
(3.49)EQ[gj(Λ(j)(T))]=E[gj(Λ(j)(t))Λ(j)(t)].
Thus
(3.50)EQ[∏i=0ngi(Λ(i)(T))]=∏i=0nEQ[gi(Λ(i)(T))]
for all n∈ℕ and positive Borel functions gi defined on ℝ+.Corollary 3.11.
Assume thatA(t) is a non-degenerate MJP. Then EQ[logΛ(j)(T)]=E[ΛlogΛ] for all j∈ℕ.Proof.
Setgi(x)=1 if i≠j and gj(x)=logx and apply Lemma 3.9.We now show that the pathwiseν to be used in Lemma 2.2 is the realization of μ itself.Lemma 3.12.
Assume thatAn(t) converges in ℒq for some q>1. If β(q)>0, then the set Kα1𝒫 has full μ-measure Ω a.s., that is,
(3.51)μ(Kα1𝒫)=μ([0,1])Ω-a.s.Proof.
Notice first that(3.52)logμ(Jn(t))log|Jn(t)|=logΛ(0)(t)+⋯+logΛ(n-1)(t)log|Jn(t)|+logμ(n)(Jn(t))log|Jn(t)|=(1/n)∑i=0n-1logΛ(i)(t)(1/n)log|Jn(t)|+(1/n)logμ(n)(Jn(t))(1/n)log|Jn(t)|.
Then by Corollary 3.10, 3.11, and LLN,
(3.53)1n∑i=0n-1logΛ(i)(T)→EQ[Λ(T)]=E[ΛlogΛ]=E[MlogM].
On the other hand, by lemma 3.8,
(3.54)limn→∞1nlog(|Jn(T)|)=-logbQ-a.s.limn→∞1nlog(μ(n)(Jn(T)))=-logbQ-a.s.Lemma 3.13.
Assume thatAn(t) converges in ℒq for some q>1. If β(q)>0, then, with probability one, we have either μ([0,1])=0 or dim(Kα1𝒫)=dim𝒫(Kα1𝒫)≥α1.Proof.
Condition (i) of lemma2.2 is obvious; conditions (ii) and (iii) are satisfied by definition of Kα1. 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 thatAn(t) converges in ℒq for some q>1. If β(q)>0, then, with probability one,
(3.55)dim𝒫(Eα1𝒫)=dim(Eα1𝒞)=α1=1-E[MlogbM].For example, ifA(t) is an MJP satisfying (3.5) and β(2)>2, then dim𝒫(Eα1)=dim(Eα1)=α1.Proof.
First, note thatβ*(α1)=α1 since α1=β'(1); also β(q')>0 for 1<q'<q since β is convex; also, μ∈ℒq' for 1<q'<q. Second, by Lemmas 3.6 and 2.6, τ𝒫(q) is convex for q>0 a.s. and theorem 2.8 applies.
By Lemma3.13 we have
(3.56)α1≤dim𝒫(Kα1𝒫)=dim(Kα1𝒫)≤dim(Eα1𝒫)≤dim𝒫(Eα1)≤dim𝒫(Vα1𝒫)≤inf1<q′<q(q′α1-T𝒫(q′))=inf1<q′<q(q′α1-β(q′))=α1.


### 3.4. 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 computedimVα𝒫 for a range of exponents α.To this end, we fixq>0 for the remainder of the section. Introduce the auxiliary mother process (3.57)Λ¯(·)≐bβ(q)-q+1Λ(·)q=Λ(·)qE[Λq],
the structure function (3.58)β¯(p)≐p-1-logbE[Λ¯p]=β(pq)-pβ(q),
and the scaling exponent (3.59)αq≐β′(q).
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)EQ¯[logbΛ]=E[Λ¯logbΛ]=1E[Λq]E[ΛqlogbΛ]=1-β′(q)=1-αq.
From this we find(3.61)EQ¯[logbΛ¯]=E[Λ¯logbΛ¯]=(β(q)-q+1)+qE[Λ¯logbΛ]=1-(qαq-β(q)).
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 ofμ¯ is qαq-β(q). We should also expect to find that with probability one, the measure μ assumes the scaling exponent α(t)=αq 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 Eαq of points with Hölder exponent αq almost surely has at least Hausdorff dimension qαq-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 ofVα𝒫, that is, the set (3.62)Wα𝒫≐{t:liminfnαn𝒫(t)≤α,limnlog|Jn(t)|n=-logb}
for α=αq. Combining this with an upper bound provided by inequality (2.24), we will arrive at a formula for the dimension of Vα𝒫.As a first step towards this goal, we mention a sufficient condition for convergence inℒp, 1≤p≤2, which follows directly from theorem 3.2, that is, A¯n(t) converges in ℒp if b>1+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 thatA¯n(t) converges in ℒp for some p>1. If β¯(p)>0 for some p>0, then Q¯-a.s.,
(3.63)limn→∞logb|Jn(T)|n=-1,limn→∞logbμ¯(Jn(T))n=-qαq+β(q),limn→∞logμ¯(Jn(T))log|Jn(t)|=1-E[Λ¯logbΛ¯]=qαq-β(q).To provide a counterpart to Lemma3.12, we need the following result concerning the local scaling of μ under the measure Q¯.Lemma 3.16.
Assume thatA(t) is a non-degenerate MJP and that A¯n(t) converges in ℒp for some p>1. Then
(3.64)liminfn→∞logbμ(n)(Jn(T))-n≤1Q¯-a.s.Proof.
(1°): Let us start with a simple observation on nested Poisson point processes. Given a sequence of independent Poisson point processes of densitiesb-kλ, their superposition forms a sequence of Poisson point processes of densities (1+⋯+b-n)λ. Denote In,[x,y]={Ikn,[x,y]} the partition of the interval [x,y] determined by a Poisson point process with density (1+⋯+b-n)λ. Adding extra points at the locations x=1,2,… makes the intervals even smaller. Thus the partition ∪i=1mI∞,[i-1,i] is finer than In,[0.m] for all n=0,1,… and m. In other words, for any m∈ℤ+,
(3.65)I∞,[i-1,i](t)⊆In,[0,m](t)∀n∈ℕ,∀i∈[1,m],∀t∈(i,i-1).
(2°): By a change of variables, and usingE[Λn¯(Tkn)∣Jn]=1 together with the independence of Λ¯n of Jn, μ(n), and μ¯(n), we get
(3.66)PQ¯(μ(n)(Jn(T))<b-n(1+ε))=E[∫011{μ(n)(Jn(t))<b-n(1+ε)}dμ¯(t)]=EE[∑k=1Nn1{μ(n)(Jkn)<b-n(1+ε)}Λ¯n(Tkn)μ¯(n)(Jkn)∣Jn]=E[b-n∑k=1Nn1{μ(bnJ̃kn)<b-nε}μ¯(bnJ̃kn)],
where J̃n is distributed as Jn and independent of μ and μ¯.
For each fixedn, construct a nested sequence of partitions Ii,[0,bn], i=0,1…, which are independent of μ and μ¯ and satisfy In,[0,bn]=bnJ̃n. Then write (3.66) as an integral split into parts of length 1 and apply (3.65) to get
(3.67)PQ¯(μ(n)(Jn(T))<b-n(1+ε))=E[b-n∫0bn1{μ(In,[0,bn](t))<b-nε}dμ¯(t)]≤E[b-n∑i=1⌊bn⌋∫i-1i1{μ(I∞,[i-1,i](t))<b-nε}dμ¯(t)]+E[b-nμ¯([⌊bn⌋,⌈bn⌉])]≤PQ¯(μ(I∞,[0,1](T))<b-nε)+b-n.
(3°): By proposition 3.4, μ(I)>0 a.s. for any fixed interval I. Thus,
(3.68)PQ¯(μ(I∞,[0,1](T))=0)=EE[∑k=1N1{μ(Ik∞,[0,1])=0}μ¯(Ik∞,[0,1])∣I∞,[0,1]]=0
and Q¯-a.s. μ(I∞,[0,1](T))>0. Applying this to (3.67) gives
(3.69)limn→∞PQ¯(logbμ(n)(Jn(T))-n>1+ε)=0.
It is easy to see by contradiction that the previous equation implies
(3.70)liminfnlogb(μ(n)(Jn(T)))-n≤1Q¯-a.s.
(4°): By the LLN, by the analogs of Corollaries3.10 and 3.11 for Q¯, and by (3.60), we have Q¯-a.s.
(3.71)1n∑i=0n-1logbΛ(i)(T)→EQ¯[Λ(T)]=1-αq.
By Corollary 3.15 we have (1/n)logb(|Jn(T)|)→-1Q¯-a.s. Applying these results and (3.70) to (3.52) completes the proof.Combining the previous results, we find the dimension of the setVαq𝒫.Theorem 3.17.
Assume thatA(t) is a non-degenerate MJP and that A¯n(t) converges in ℒp for some p>1. If β¯(p)>0, then, with probability one,
(3.72)dim𝒫(Vαq𝒫)=dim(Vαq𝒫)=qαq-β(q).Proof.
The proof follows the lines of lemmas3.12 and 3.13 and theorem 3.14. To get the lower bound, the set Kα1 is replaced by Wαq and Lemma 2.2 is applied to μ¯. The upper bound follows directly from inequality (2.24). In summary,
(3.73)dim𝒫(Vαq𝒫)=dim(Vαq𝒫)=f𝒫+(αq+)=T𝒫*(αq)=T𝒟*(αq)=β*(αq)=qαq-β(q).For example, ifA(t) is an MJP satisfying (3.5) and β(4)>2β(2)>0, then (3.72) holds for 0<q≤2.


## 3.1. Definition and Basic Properties

We start fromT-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: (A1)
Λ is stationary with Λ>0 and E[Λ]=1; there exists b>1 such that Λ(i)(·/bi) are i.i.d. processes, distributed as Λ;(A2)
Λ is a weakly mixing Markov process which is defined by Λ(t)≐Mk for Tk≤t<Tk+1, where T0=0 and {Tk+1-Tk}k are i.i.d. exponential random variables of mean 1/λ, and where the Mk form a stationary positive time series independent of {Tk}k satisfying E[Mk]=1 for all k.Recall that a real-valued stationary process is weakly mixing if for allB1,B2∈ℬ (the σ-algebra generated by the process) limT→∞T-1∫0T|P(B1∩StB2)-P(B1)P(B2)|dt=0 where St is the shift operator. Examples of interest include the cases where the sequence {Mk}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 be studied. For example, if Λ is a finite state Markov process, then analogous results can be derived.Next define the finite product processes(3.1)Λn(t)≐∏i=0n-1Λ(i)(t).
For t∈[0,1], the cumulative processes (3.2)An(t)≐∫0tΛn(s)ds=∫0t∏i=0n-1Λ(i)(s)ds,n=1,2,…
can be associated with positive measures defined on the Borel sets ℬ of [0,1]: (3.3)μn(B)≐∫BΛn(s)ds,n=1,2,…,B∈ℬ.
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≐limnμn 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, themultifractal jump process (MJP) is the limit
(3.4)A(t)≐μ([0,t])≐limn→∞μn([0,t])=limn→∞An(t)a.s.
An MJP is called degenerate if A(1)=0 almost surely, and nondegenerate otherwise.It can be shown thatA is non-degenerate if and only if E[A(1)]=1 (see [39]). The convergence of An(1) in ℒp for some p>1 naturally implies nondegeneracy since then E[A(1)]=limnE[An(1)]=1. To provide criteria for convergence in ℒp is thus of central importance. As pointed out in [2], criteria for convergence in ℒ2 are particularly manageable and useful. In the following theorem, which is a straightforward adaption of [39, Corollary 3], the ℒ2 conditions are stated for MJPs.Theorem 3.2 (see [39, Corollary 3]).
Assume (A1), (A2), andE[M2]<∞. If there are positive γ1, γ2 and C with
(3.5)σ2e-γ1|x|≤Cov(Λ(0),Λ(x))=∑j=0∞(λx)jj!Cov(M0,Mj)e-λx≤C|x|-γ2,
then An(t) converges in ℒ2 if and only if b>1+σ2 where σ2=Var(M).Condition (3.5) allows a very large family of time series Mk. For example, a sufficient condition for convergence is(3.6)Cov(M0,Mj)≤Cj+1,j=0,1,2,…,
for some constant C>0.The majority of the following results are stated assuming a converging MJP in someℒp, 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 Mj are i.i.d. and the ℒ3 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 measureμ is ergodic with respect to the time shift operator. Ergodicity of μ, together with the positivity of the multipliers Mk, 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 measureμ on ℝ+ of an MJP is ergodic.Proof.
Consider the spacesΩi=(Xi,ℬi,Pi), i=0,1,…, and Ω=Πi=0∞Ωi=(X,ℬ,P), where Xi is the set of piece-wise continuous functions, ℬi its Borel-algebra, and Pi is the law of Λ(i). The shift operator is weakly mixing in Ωi and thus it is also weakly mixing in Ω0×⋯×Ωn-1 for any fixed n. Since the B0×⋯×Bn-1×Πi=n∞Ωi 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:(3.7)μ(n)(B)≐∫B∏i=0n-11Λ(i)(τ)dμ(τ),
that is, the measure where n first terms of the product are neglected. Thus(3.8)μ(n)(B)=distb-nμ(bnB).Proposition 3.4.
Assume thatA is a non-degenerate MJP. Then, with probability one, μ([0,t])>0 for all t∈(0,1].Proof.
First consider an arbitraryt>0 and make a change of variable to get
(3.9)μ([0,t])=∫0tΛn(s)dμ(n)(s)=distb-n∫0tbnΛn(sb-n)dμ̃(s),
where μ and μ̃ are identically distributed and μ̃ independent of Λn. Then by the positivity of Λn and ergodicity of μ, we have
(3.10)P(μ([0,t])=0)=P(b-n∫0tbnΛn(sb-n)dμ̃(s)=0)=P(b-nμ([0,tbn])=0)≤P(b-nμ([0,tbn])≤t2)→0asn→∞.
Thus μ([0,t])>0 a.s. and μ being a positive measure, trivially μ([0,s])>0 for all s≥t a.s.
Now denoteUt={ω:μ[0,t]=0}. Then
(3.11)P(μ([0,t])=0∀t∈(0,1])=P(⋃n=1∞U1/n)≤∑nP(U1/n)=0,
which completes the proof.


## 3.2. 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 Section2. In analogy with related multiplicative processes [15], we should expect moments of the multipliers to affect the multifractal properties through the structure function(3.12)β(q)≐q-1-logbE[Λq].
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]).
IfE[Λp]=E[Mp]<∞ and An(t) converges in ℒp for some p>1, then, for 0<q≤p,
(3.13)C̲tβ(q)+1≤E[A(t)q]≤C¯tβ(q)+1∀t∈[0,1],(3.14)T𝒟(q)=β(q)=q-1-logbE[Mq].Note that under the assumption (3.5) of Theorem 3.2, convergence of the MJP A(t) in ℒ2 is equivalent with β(2)>0.In order to analyze the MJP, it is natural to consider the nested sequence of partitionsJn induced by Λn. By assumptions (A1) and (A2), (3.15)Λ(i)(t)=Mk(i),Tk-1(i)≤t<Tk(i),
where {Tk(i)}k=1∞ forms a Poisson point process of intensity λbi and {Mk(i)}k=1∞ is a stationary positive time series independent of {Tk(i)}k. Consequently, the product process (3.16)Λn(t)=∏i=0n-1Λ(i)(t)
is constant on intervals Jkn≐[Tk-1n,Tkn), where the Tkn are defined by a Poisson point process of intensity λn=λ∑i=0n-1bi=O(bn). From the construction of {Tkn}k, it follows that the partitions Jn={Jkn:k=1,…} are nested. We denote the resulting collection of nested Poisson partitionsJn by 𝒫.Lemma 3.6.
Consider a nested Poisson partition𝒫. Let to be arbitrary. With probability one, as n→∞,
(3.17)log|Jn(to)|n→-logb,logmaxk|Jkn|n→-logb,logmink|Jkn|n→-2logb.Proof.
Without loss of generality, assumeλn=λbn and t0∈[0,1).
(1°): Clearly(3.18)P(Nopointsin[t0-x,t0+x])≤P(|Jn(t0)|>x)≤2P(Nopointsin[t0,t0+x/2]).
Thus for n large enough,
(3.19)P(|Jn(t0)|>b-nenε)≤2e(-1/2)λnb-nenε≤2e(-1/2)nελ,P(|Jn(t0)|<b-nenε)≤1-e-2λnb-ne-nε≤2λe-nε.
(2°): Since the maximal interval is always longer than a fixed interval, forn large enough,
(3.20)P(maxk|Jkn|<b-ne-nε)≤P(|Jn(t0)|<b-ne-nε)≤2λe-nε.
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
(3.21)P(maxk|Jkn|>b-nenε)≤∑i=1⌊2bne-nε⌋e(-1/2)λnb-nenε≤2e-n((λ/4)nε2-logb).
(3°): Let{Ukm}k=1m+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)P(mink|Ukm|<x)={1-(1-(m+1)x)m,0<x<11+m,1,x≥1m+1.
Let Nn denote the number of points of a Poisson process with density λbn located in [0,1]. Applying (3.22) gives
(3.23)P(mink|Jkn|<b-2ne-nε)=∑m=0∞P(mink=1,…,Nn+1|Jkn|<b-2ne-nε∣Nn=m)P(Nn=m)=∑m=0⌊b2nenε⌋-1[1-(1-(m+1)b-2ne-nε)m](λbn)mm!e-λbn+∑m=⌊b2nenε⌋∞(λbn)mm!e-λbn≤∑m=0∞(m2+m)b-2ne-nε(λbn)mm!e-λbn+(λbn)⌊b2nenε⌋⌊b2nenε⌋!≤e-nε(2λb-n+λ2)+e-nε,
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
(3.24)P(mink|Ikn|>b-2nenε)≤(e-λb-n(1+λb-nenε))b2ne-nε-1=exp(-λbn+λb-nenε+(b2ne-nε-1)log(1+λb-nenε))≤e-c1nε.
(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(·)=dim𝒫(·). In addition, by lemma 2.6 and theorem 2.8, the multifractal envelope T𝒫* provides an upper bound on the multifractal spectrum dim𝒫(Eα), almost surely.The connection to the measuresμ(n) defined in (3.7) is especially practical when considering sets Jn(t) because then(3.25)μ(Jn(t))=μ(n)(Jn(t))∏i=0n-1Λ(i)(t)=distb-nμ̃(bnJn(t))∏i=0n-1Λ(i)(t),
where μ̃ is distributed as μ and is independent of Jn and Λ(i), i=0,…,n-1. On the other hand, by corollary 3.5 and(3.8)(3.26)C1b-n(q-1-β(q))|I|1+β(q)≤E[μ(n)(I)q]≤C2b-n(q-1-β(q))|I|1+β(q)
for a fixed interval I. We find that T𝒫=T𝒟 under mild conditions.Proposition 3.7.
IfE[Mp]<∞ and An(t) converges in ℒp for some p>1, then
(3.27)T𝒫(q)=β(q)=q-1-logbE[Mq]for0<q≤p.Proof.
ConsiderE[∑k=1Nn|Jkn|1+ε] with ε>0. Denote by {Ukj:k=1,…,j+1} the partition resulting from j uniformly distributed points on [0,1]. Then
(3.28)E[|Ukj|α]=∫01jsα(1-s)j-1ds=j!Γ(1+α)Γ(j+1+α).
Using the above result and conditioning with respect to the number of intervals gives
(3.29)E[∑k=1Nn|Jkn|1+ε]=∑j=0∞E[∑k=1j+1|Ukj|1+ε]P(Nn=j+1)=∑j=0∞(j+1)j!Γ(2+ε)Γ(j+2+ε)(λn)je-λnj!≤Γ(2+ε)(λn)-ϵe-λn∑j=0∞(λn)j+εΓ(j+1+ε)=Γ(2+ε)(λn)-ϵ(1-Γ(ε,λn)Γ(ε))≤Cb-nε.
Now assumeγ=β(q)-ϵ with ϵ>0. Conditioning with respect to the partition Jn helps us to split the addends into three factors, that is,
(3.30)E[∑k=1Nnμ(Jkn)q|Jkn|-γ]=EE[∑k=1NnΛn(Tk-1n)q|Jkn|-γμ(n)(Jkn)q∣Jn].
Use the scaling law (3.26) together with the independence of μ(n) with respect to Jn and that
(3.31)E[Λn(Tkn)q∣Jn]=E[Λn(Tkn)q]=E[Mq]n=b-n(β(q)+1-q)
in order to estimate
(3.32)E[∑k=1Nnμ(Jkn)q|Jkn|-γ]≤Cb-n(q-1-β(q))E[Mq]nE[∑k=1Nn|Jkn|1+β(q)-γ]=CE[∑k=1Nn|Jkn|1+ε]≤C̃b-nε.
Next, assume thatγ=β(q)+ε. Condition again with respect to the partition Jn and apply the lower bound of (3.13) to get
(3.33)E[∑k=1Nnμ(Jkn)q|Jkn|-γ]≥CE[∑k=1Nn|Jkn|1-ε]≥CE[∑k=1Nn|Jkn|]=C.
Thus, we have found that∑n≥mE[Sn(q,γ)]→0 whenever γ<β(q) and ∑n≥mE[Sn(q,γ)]↛0 whenever γ>β(q).


## 3.3. Information Dimension

This section is devoted to analyzingEα1𝒫, the set of points with scaling exponent (3.34)α1𝒫≐α1≐β′(1)=1-E[ΛlogbΛ]=1-E[MlogbM]
which is the most relevant of all Eα𝒫.The upper bound of the multifractal spectrumdim𝒫(Eα𝒫)≤T𝒫* a.s. follows from lemma 2.6, lemma 3.6, and theorem 2.8. While T𝒫=T𝒟 under mild conditions (see above), we need to introduce the auxiliary set(3.35)Kα𝒫={x:limnαn𝒫(x)=α,limnlogb|Jn(x)|-n=1},
a subset of Eα𝒫 for which dim(Kα𝒫)=dim𝒫(Kα𝒫) 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 locationst, 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 dimKα𝒫, and thus dimEα𝒫, 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 (3.36)Q(φ)=E[∫01φ(t,ω)dμ(t)]
for all positive measurable functions φ(t,ω).Lemma 3.8.
Assume thatA(t) is a non-degenerate MJP. If (ω,T) is picked according to the Peyrière measure Q, then
(3.37)limn→∞log|Jn(T)|n=-logbQ-a.s.
Moreover, if An(t) converges in ℒq for some q>1 and bq-1>E[Λq], that is, β(q)>0, then
(3.38)logμ(n)(Jn(T))n→-logbQ-a.s.Proof.
Recall first thatE[μ(n)(Jkn)|Jn]=|Jkn| and that Λn(Tkn) is independent of Jn so that
(3.39)E[Λn(Tkn)∣Jn]=E[Λn]=1.
(1°) Condition with respect to the partition Jn in order to get
(3.40)PQ(|Jn(T)|<x)=E[∫011{|Jn(t)|<x}dμ(t)]=EE[∑k=1Nn1{|Jkn|<x}Λn(Tk-1n)μ(n)(Jkn)∣Jn]=E[∑k=1Nn1{|Jkn|<x}|Jkn|]=E[∫011{|Jn(t)|<x}dt]=∫01P(|Jn(t)|<x)dt≤C1xλn,
where C1 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:(3.41)PQ(|Jn(T)|>x)=∫01P(1{|Jn(t)|>x})dt≤C2e-λnx,
where C2 is independent of n. Here, the last inequality follows from estimating the probability that a sum of two exponential random variables exceeds x.
Sinceλn=O(bn), we have
(3.42)∑n=1∞PQ(|Jn(T)|<b-ne-nε)<∞,∑n=1∞PQ(|Jn(T)|>b-nenε)<∞.
(2°) Let ε>0. Write the Q-probability as a sum
(3.43)PQ(μ(n)(Jn(T))<b-ne-nε)=E[∑k=1Nn1{μ(n)(Jkn)<b-ne-nε}Λn(Tkn)μ(n)(Jkn)].
Scaling the μ(n)(Jkn) up by b-ne-nε and applying (3.39) lead to
(3.44)PQ(μ(n)(Jn(T))<b-ne-nε)≤b-ne-nεEE[∑k=0NnΛn(Tkn)∣Jn]≤b-ne-nεE[Nn].
This implies PQ(μ(n)(Jn(T)))≤Ce-nε, because E[Nn]=λn=O(bn).
On the other hand, by (3.26) and (3.29),
(3.45)EQ[μ(n)(Jn(T))q-1]=EE[∑k=1NnΛn(Tk-1n)μ(n)(Jkn)q∣Jn]≤C̃b-n(q-1-β(q))E[∑k=1Nn|Jkn|1+β(q)]≤Cb-n(q-1).
Then apply the Markov inequality to get
(3.46)PQ(μ(n)(Jn(T))≥b-nenε)≤Ce-nε(q-1).
(3°) Applying Borel-Cantelli to (1°) and (2°) completes the proof.To establish theQ-independence required for the LLN, we need the following result.Lemma 3.9.
Assume thatA(t) is a non-degenerate MJP. Then
(3.47)EQ[∏i=0n-1gi(Λ(i)(T))]=∏i=0n-1E[gi(Λ(i)(t))Λ(i)(t)]
for all n∈ℕ and positive Borel functions gi defined on ℝ+.Proof.
(Note: this is a replication of the proof appearing in [2]). Condition with respect to the partition Jn and use E[μn(Jkn)|Jn]=|Jkn|. In addition to this, utilize independence (w.r.t. Ω) and stationarity:
(3.48)EQ[∏i=0n-1gi(Λ(i)(T))]=E[∫01∏i=0n-1gi(Λ(i)(t))dμ(t)]=EE[∑k=0Nn∏i=0n-1gi(Λ(i)(Tk-1n))Λ(i)(Tk-1n)μ(n)(Ikn)∣Jn]=E[∑k=0Nn∏i=0n-1gi(Λ(i))(Tk-1n)Λ(i)(Tk-1n)|Jkn|]=E[∫01∏i=0n-1gi(Λ(i)(t))Λ(i)(t)dt]=∫01∏i=0n-1E[gi(Λ(i)(t))Λ(i)(t)]dt=∏i=0n-1E[gi(Λ(i)(t))Λ(i)(t)].Corollary 3.10.
Assume thatA(t) is a non-degenerate MJP. Then the random variables Λ(i)(T), i=0,1,…,n, are Q-independent.Proof.
Fixj and set gi(x)≡1 when i≠j, leaving gj arbitrary. Lemma 3.9 gives
(3.49)EQ[gj(Λ(j)(T))]=E[gj(Λ(j)(t))Λ(j)(t)].
Thus
(3.50)EQ[∏i=0ngi(Λ(i)(T))]=∏i=0nEQ[gi(Λ(i)(T))]
for all n∈ℕ and positive Borel functions gi defined on ℝ+.Corollary 3.11.
Assume thatA(t) is a non-degenerate MJP. Then EQ[logΛ(j)(T)]=E[ΛlogΛ] for all j∈ℕ.Proof.
Setgi(x)=1 if i≠j and gj(x)=logx and apply Lemma 3.9.We now show that the pathwiseν to be used in Lemma 2.2 is the realization of μ itself.Lemma 3.12.
Assume thatAn(t) converges in ℒq for some q>1. If β(q)>0, then the set Kα1𝒫 has full μ-measure Ω a.s., that is,
(3.51)μ(Kα1𝒫)=μ([0,1])Ω-a.s.Proof.
Notice first that(3.52)logμ(Jn(t))log|Jn(t)|=logΛ(0)(t)+⋯+logΛ(n-1)(t)log|Jn(t)|+logμ(n)(Jn(t))log|Jn(t)|=(1/n)∑i=0n-1logΛ(i)(t)(1/n)log|Jn(t)|+(1/n)logμ(n)(Jn(t))(1/n)log|Jn(t)|.
Then by Corollary 3.10, 3.11, and LLN,
(3.53)1n∑i=0n-1logΛ(i)(T)→EQ[Λ(T)]=E[ΛlogΛ]=E[MlogM].
On the other hand, by lemma 3.8,
(3.54)limn→∞1nlog(|Jn(T)|)=-logbQ-a.s.limn→∞1nlog(μ(n)(Jn(T)))=-logbQ-a.s.Lemma 3.13.
Assume thatAn(t) converges in ℒq for some q>1. If β(q)>0, then, with probability one, we have either μ([0,1])=0 or dim(Kα1𝒫)=dim𝒫(Kα1𝒫)≥α1.Proof.
Condition (i) of lemma2.2 is obvious; conditions (ii) and (iii) are satisfied by definition of Kα1. 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 thatAn(t) converges in ℒq for some q>1. If β(q)>0, then, with probability one,
(3.55)dim𝒫(Eα1𝒫)=dim(Eα1𝒞)=α1=1-E[MlogbM].For example, ifA(t) is an MJP satisfying (3.5) and β(2)>2, then dim𝒫(Eα1)=dim(Eα1)=α1.Proof.
First, note thatβ*(α1)=α1 since α1=β'(1); also β(q')>0 for 1<q'<q since β is convex; also, μ∈ℒq' for 1<q'<q. Second, by Lemmas 3.6 and 2.6, τ𝒫(q) is convex for q>0 a.s. and theorem 2.8 applies.
By Lemma3.13 we have
(3.56)α1≤dim𝒫(Kα1𝒫)=dim(Kα1𝒫)≤dim(Eα1𝒫)≤dim𝒫(Eα1)≤dim𝒫(Vα1𝒫)≤inf1<q′<q(q′α1-T𝒫(q′))=inf1<q′<q(q′α1-β(q′))=α1.


## 3.4. 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 computedimVα𝒫 for a range of exponents α.To this end, we fixq>0 for the remainder of the section. Introduce the auxiliary mother process (3.57)Λ¯(·)≐bβ(q)-q+1Λ(·)q=Λ(·)qE[Λq],
the structure function (3.58)β¯(p)≐p-1-logbE[Λ¯p]=β(pq)-pβ(q),
and the scaling exponent (3.59)αq≐β′(q).
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)EQ¯[logbΛ]=E[Λ¯logbΛ]=1E[Λq]E[ΛqlogbΛ]=1-β′(q)=1-αq.
From this we find(3.61)EQ¯[logbΛ¯]=E[Λ¯logbΛ¯]=(β(q)-q+1)+qE[Λ¯logbΛ]=1-(qαq-β(q)).
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 ofμ¯ is qαq-β(q). We should also expect to find that with probability one, the measure μ assumes the scaling exponent α(t)=αq 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 Eαq of points with Hölder exponent αq almost surely has at least Hausdorff dimension qαq-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 ofVα𝒫, that is, the set (3.62)Wα𝒫≐{t:liminfnαn𝒫(t)≤α,limnlog|Jn(t)|n=-logb}
for α=αq. Combining this with an upper bound provided by inequality (2.24), we will arrive at a formula for the dimension of Vα𝒫.As a first step towards this goal, we mention a sufficient condition for convergence inℒp, 1≤p≤2, which follows directly from theorem 3.2, that is, A¯n(t) converges in ℒp if b>1+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 thatA¯n(t) converges in ℒp for some p>1. If β¯(p)>0 for some p>0, then Q¯-a.s.,
(3.63)limn→∞logb|Jn(T)|n=-1,limn→∞logbμ¯(Jn(T))n=-qαq+β(q),limn→∞logμ¯(Jn(T))log|Jn(t)|=1-E[Λ¯logbΛ¯]=qαq-β(q).To provide a counterpart to Lemma3.12, we need the following result concerning the local scaling of μ under the measure Q¯.Lemma 3.16.
Assume thatA(t) is a non-degenerate MJP and that A¯n(t) converges in ℒp for some p>1. Then
(3.64)liminfn→∞logbμ(n)(Jn(T))-n≤1Q¯-a.s.Proof.
(1°): Let us start with a simple observation on nested Poisson point processes. Given a sequence of independent Poisson point processes of densitiesb-kλ, their superposition forms a sequence of Poisson point processes of densities (1+⋯+b-n)λ. Denote In,[x,y]={Ikn,[x,y]} the partition of the interval [x,y] determined by a Poisson point process with density (1+⋯+b-n)λ. Adding extra points at the locations x=1,2,… makes the intervals even smaller. Thus the partition ∪i=1mI∞,[i-1,i] is finer than In,[0.m] for all n=0,1,… and m. In other words, for any m∈ℤ+,
(3.65)I∞,[i-1,i](t)⊆In,[0,m](t)∀n∈ℕ,∀i∈[1,m],∀t∈(i,i-1).
(2°): By a change of variables, and usingE[Λn¯(Tkn)∣Jn]=1 together with the independence of Λ¯n of Jn, μ(n), and μ¯(n), we get
(3.66)PQ¯(μ(n)(Jn(T))<b-n(1+ε))=E[∫011{μ(n)(Jn(t))<b-n(1+ε)}dμ¯(t)]=EE[∑k=1Nn1{μ(n)(Jkn)<b-n(1+ε)}Λ¯n(Tkn)μ¯(n)(Jkn)∣Jn]=E[b-n∑k=1Nn1{μ(bnJ̃kn)<b-nε}μ¯(bnJ̃kn)],
where J̃n is distributed as Jn and independent of μ and μ¯.
For each fixedn, construct a nested sequence of partitions Ii,[0,bn], i=0,1…, which are independent of μ and μ¯ and satisfy In,[0,bn]=bnJ̃n. Then write (3.66) as an integral split into parts of length 1 and apply (3.65) to get
(3.67)PQ¯(μ(n)(Jn(T))<b-n(1+ε))=E[b-n∫0bn1{μ(In,[0,bn](t))<b-nε}dμ¯(t)]≤E[b-n∑i=1⌊bn⌋∫i-1i1{μ(I∞,[i-1,i](t))<b-nε}dμ¯(t)]+E[b-nμ¯([⌊bn⌋,⌈bn⌉])]≤PQ¯(μ(I∞,[0,1](T))<b-nε)+b-n.
(3°): By proposition 3.4, μ(I)>0 a.s. for any fixed interval I. Thus,
(3.68)PQ¯(μ(I∞,[0,1](T))=0)=EE[∑k=1N1{μ(Ik∞,[0,1])=0}μ¯(Ik∞,[0,1])∣I∞,[0,1]]=0
and Q¯-a.s. μ(I∞,[0,1](T))>0. Applying this to (3.67) gives
(3.69)limn→∞PQ¯(logbμ(n)(Jn(T))-n>1+ε)=0.
It is easy to see by contradiction that the previous equation implies
(3.70)liminfnlogb(μ(n)(Jn(T)))-n≤1Q¯-a.s.
(4°): By the LLN, by the analogs of Corollaries3.10 and 3.11 for Q¯, and by (3.60), we have Q¯-a.s.
(3.71)1n∑i=0n-1logbΛ(i)(T)→EQ¯[Λ(T)]=1-αq.
By Corollary 3.15 we have (1/n)logb(|Jn(T)|)→-1Q¯-a.s. Applying these results and (3.70) to (3.52) completes the proof.Combining the previous results, we find the dimension of the setVαq𝒫.Theorem 3.17.
Assume thatA(t) is a non-degenerate MJP and that A¯n(t) converges in ℒp for some p>1. If β¯(p)>0, then, with probability one,
(3.72)dim𝒫(Vαq𝒫)=dim(Vαq𝒫)=qαq-β(q).Proof.
The proof follows the lines of lemmas3.12 and 3.13 and theorem 3.14. To get the lower bound, the set Kα1 is replaced by Wαq and Lemma 2.2 is applied to μ¯. The upper bound follows directly from inequality (2.24). In summary,
(3.73)dim𝒫(Vαq𝒫)=dim(Vαq𝒫)=f𝒫+(αq+)=T𝒫*(αq)=T𝒟*(αq)=β*(αq)=qαq-β(q).For example, ifA(t) is an MJP satisfying (3.5) and β(4)>2β(2)>0, then (3.72) holds for 0<q≤2.



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