Multifractal Analysis of Infinite Products of Stationary Jump Processes

Abstract

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.

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 ∫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 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: 𝐴(𝑡)=lim_𝑛→∞^𝑡₀^𝑛_𝑖=1Λ^(𝑖)(𝑠)d𝑠, (1.1) 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(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].

2. 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.

2.1. 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 byℎ^𝒞_𝑠(𝐹)≐lim_𝛿→0inf{_𝑖|𝑈_𝑖|^𝑠∶0<|𝑈_𝑖|≤𝛿,𝑈_𝑖∈𝒞∀𝑖;𝐹⊆_𝑖𝑈_𝑖}, (2.1) where the diameter of set 𝑈 is denoted by |𝑈|=sup{|𝑥−𝑦|∶𝑥,𝑦∈𝑈}. Then dim^𝒞(𝐹)≐sup_𝑠{𝑠≥0∶ℎ_𝑠(𝐹)=∞}=inf_𝑠{𝑠≥0∶ℎ_𝑠(𝐹)=0} (2.2) is called the Hausdorff dimension of set 𝐹 w.r.t. 𝒞. By definition, when choosing the powerset 2^ℝ𝑑, that is, the class of all subsets of ℝ^𝑑 as the class 𝒞 in (2.1), then these notions reduce to the usual 𝑠-dimensional Hausdorff measure ℎ_𝑠 and dimension dim:ℎ_𝑠(𝐹)=ℎ^2ℝd_𝑠(𝐹),dim(𝐹)=dim^2ℝd(𝐹). (2.3) Clearly, dim(𝐹)≤dim^𝒞(𝐹). The Hausdorff dimension w.r.t. 𝒞 is 𝜎-stable, that is, dim^𝒞(∪_𝑗𝐹_𝑗)=sup_𝑗dim^𝒞(𝐹_𝑗) 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 [𝑘₁𝑏^−𝑛,(𝑘₁+1)𝑏^−𝑛)×⋯×[𝑘_𝑑𝑏^−𝑛,(𝑘_𝑑+1)𝑏^−𝑛) (2.4) 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 {𝐶_𝑛(𝑥)}_ℕ∈𝒞 such that 𝑥∈𝐶_𝑛(𝑥)≠{𝑥} for all 𝑛, lim_𝑛→∞|𝐶_𝑛(𝑥)|=0,limsup_𝑛→∞log|𝐶_𝑛(𝑥)|log|𝐶_𝑛−1(𝑥)|≤1. (2.5) Then dim(𝐹)=dim^𝒞(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)=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 2.1.

Lemma 2.2 (see [45, Lemma 4.3.2.b]). Let 𝜈 be a Borel measure on ℝ^𝑑, 𝐹 a Borel subset, and 𝒞={𝐶^𝑛_𝑘∶𝑘=1,…,𝑁_𝑛,𝑛∈ℕ} a nested collection of cubes. Assume 𝜈(𝐹)>0, (i){𝐶^𝑛_𝑘}^𝑁𝑛_𝑘=1 cover 𝐹 for all 𝑛,(ii) there exists 𝑏>1 such that (1/𝑛)log|𝐶^𝑛_𝑘|→−log𝑏as 𝐶^𝑛_𝑘→{𝑥} for all 𝑥∈𝐹,(iii)log𝜈(𝐶^𝑛_𝑘)/log|𝐶^𝑛_𝑘|→𝑠 as 𝐶^𝑛_𝑘→{x} for all 𝑥∈𝐹. Then, dim(𝐹)=dim^𝒞(𝐹)≥𝑠.

2.2. 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 dim_loc𝜈(𝑥)=lim_𝑟→0log𝜈(𝐵(𝑥,𝑟))log𝑟 (2.6) if this limit exists. In multifractal analysis, we are usually interested in the properties like fractal dimension of the set {𝑥∶dim_loc𝜈(𝑥)=𝛼}.

A more approachable framework to study scaling properties is to consider a sequence of finer and finer partitions 𝐶^𝑛≐{𝐶^𝑛_𝑘∶𝑘=1,…,𝑁_𝑛} 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,…,2^𝑛.

Definition 2.3. Let 𝐹 be a Borel subset of ℝ^𝑑 and 𝐶^𝑛={𝐶^𝑛_𝑘∶𝑘=1,…,𝑁_𝑛} a partition of 𝐹 for each 𝑛=1,2,…. We call the 𝐶^𝑛nested partitions, if any 𝐶^𝑛_𝑘∈𝐶^𝑛 (𝑛≥2), 𝑘=1,…,𝑁_𝑛, is a subset of some 𝐶_𝑘^𝑛−1_′∈𝐶^𝑛−1, 𝑘^′∈{1,…,𝑁_𝑛−1}.

Let 𝛼^𝒞_𝑛(𝑥)≐log𝜈(𝐶^𝑛(𝑥))log|𝐶^𝑛,(𝑥)| (2.7) where 𝒞={𝐶^𝑛∶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 𝛼^𝒞(𝑥)≐lim_𝑛→∞𝛼^𝒞_𝑛(𝑥) (2.8) if this limit exists. One should think of 𝛼^𝒞(𝑥) as giving approximately the degree of Hölder regularity of 𝜈 at 𝑥, that is, an approximation to dim_loc𝜈(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 sets𝐸^𝒞_𝛼≐{𝑥∶lim_𝑛𝛼^𝒞_𝑛(𝑥)=𝛼},𝑉^𝒞_𝛼≐{𝑥∶liminf_𝑛𝛼^𝒞_𝑛(𝑥)≤𝛼} (2.9) characterizing the local scaling behaviour. The classical literature on multifractals studies typically the sets 𝐸^𝒞_𝛼, calling dim𝐸^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∶liminf_𝑛𝛼^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(𝑉^𝒞_𝛼). Trivially, 𝐸^𝒞_𝛼⊆𝑉^𝒞_𝛼 and dim(𝐸^𝒞_𝛼)≤dim(𝑉^𝒞_𝛼).

2.3. Pathwise Partition Function and Coarse Grain Spectra

In order to make the presentation simpler, we consider only the 1-dimensional case assuming that supp𝜈⊆[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 𝑆^𝒞_𝑛(𝑞,𝛾)≐𝑆_𝑛(𝑞,𝛾)=^𝑁𝑛_𝑘=1𝜈(𝐽^𝑛_𝑘)^𝑞|𝐽^𝑛_𝑘|^−𝛾, (2.10) 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 as𝜏_𝒞(𝑞)≐𝜏(𝑞)=sup{𝛾∶_𝑛𝑆^𝒞_𝑛(𝑞,𝛾)<∞}. (2.11) 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 𝜏^∗(𝛼)≐inf_𝑞(𝛼𝑞−𝜏(𝑞)). (2.12) 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 so-called “coarse grain” spectra. To this end, the following collections of intervals are of interest: 𝑁⁺_𝑛(𝛼)≐{𝑘∶𝐽^𝑛_𝑘∩[0,1]≠∅,log𝜈(𝐽^𝑛_𝑘)log|𝐽^𝑛_𝑘|<𝛼}. (2.13) 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 log𝜈(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: 𝑓⁺_𝒞(𝛼)≐𝑓⁺(𝛼)≐inf{𝛿>0∶_{𝑛≥𝑚𝑘∈𝑁+𝑛(𝛼)}|𝐽^𝑛_𝑘|^𝛿→0as𝑚→∞}. (2.14)

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 measure 𝜈, the following holds. If max_𝑘|𝐽^𝑛_𝑘|→0 as 𝑛→∞ then dim^𝒞(𝑉^𝒞_𝛼)≤𝑓⁺_𝒞(𝛼+)≐lim_𝜀↓0𝑓⁺_𝒞(𝛼+𝜀). (2.15)

Proof. Fix 𝛼 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∈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 𝑉^𝒞_𝛼⊆_{𝑛≥𝑚𝑘∈𝑁+𝑛(𝛼+𝜀)}𝐽^𝑛_𝑘. (2.16) This means that for every 𝑚 we have constructed a cover of 𝑉^𝒞_𝛼 with 𝐽^𝑛_𝑘 such that max_{𝑛≥𝑚,𝑘∈𝑁+𝑛(𝛼+𝜀)}|𝐽^𝑛_𝑘|→0,_{𝑛≥𝑚𝑘∈𝑁+𝑛(𝛼+𝜀)}|𝐽^𝑛_𝑘|^𝛾→0as𝑚→∞. (2.17) We conclude that ℎ^𝒞_𝛾(𝑉^𝒞_𝛼)=0 and dim^𝒞(𝑉^𝒞_𝛼)≤𝛾.

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 𝑆_𝑛(𝑞,𝛾)≥_{𝑁+𝑛(𝛼)}𝜈(𝐽^𝑛_𝑘)^𝑞|𝐽^𝑛_𝑘|^−𝛾≥_{𝑁+𝑛(𝛼)}|𝐽^𝑛_𝑘|^{𝑞𝛼−𝛾}. (2.18)
Fix a real 𝛼 and 𝑞≥0. Consider 𝛾<𝜏(𝑞). By definition of 𝜏, we find _{𝑛≥𝑚𝑁+𝑛(𝛼)}|𝐽^𝑛_𝑘|^{𝑞𝛼−𝛾}→0as𝑚→∞. (2.19) 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.

Lemma 2.6. Let 𝐽^𝑛 form nested partitions such that ∑_𝑛max_𝑘|𝐽^𝑛_𝑘|<∞. 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 𝑆_𝑛(𝑞,𝛾)≤(^𝑝_{𝑗=1𝑘∶log𝜈(𝐽𝑛𝑘)/log|𝐽𝑛𝑘|∈𝑈𝑗}+_{𝑘∶log𝜈(𝐽𝑛𝑘)/log|𝐽𝑛𝑘|>𝑀})𝜈(𝐽^𝑛_𝑘)^𝑞|𝐽^𝑛_𝑘|^−𝛾≤^𝑝_{𝑗=1𝑘∈𝑁+𝑛(𝑗𝜀)}|𝐽^𝑛_𝑘|^{𝑞((𝑗−1)𝜀)}|𝐽^𝑛_𝑘|^−𝛾+_{𝑘∶𝐽𝑛𝑘∩[0,1]≠∅}|𝐽^𝑛_𝑘|^𝑞𝑀|𝐽^𝑛_𝑘|^−𝛾. (2.20) By choice of 𝑀, and since the 𝐽^𝑛_𝑘 are disjoint subsets of [−1,2] for 𝑛 large enough, _{𝐽𝑛𝑘∩[0,1]≠∅}|𝐽^𝑛_𝑘|^𝑞𝑀|𝐽^𝑛_𝑘|^−𝛾≤_{𝐽𝑛𝑘∩[0,1]≠∅}|𝐽^𝑛_𝑘|²≤max_𝑘|𝐽^𝑛_𝑘|_{𝐽𝑛𝑘∩[0,1]≠∅}|𝐽^𝑛_𝑘|≤3max_𝑘|𝐽^𝑛_𝑘|. (2.21) 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 measure 𝜇 defined on [0,1] and a random nested collection 𝒞. The asymptotics of ensemble moments are given by 𝑇_𝒞(q)≐𝑇(𝑞)=sup{𝛾∶^∞_𝑛=1𝐄[𝑆^𝒞_𝑛(𝑞,𝛾)]<∞}. (2.22) 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, _𝑛≥𝑚𝐄[𝑆_𝑛(𝑞,𝛾)]=𝐄[_𝑛≥𝑚𝑆_𝑛(𝑞,𝛾)]<∞, (2.23) and thus ∑_𝑛≥𝑚𝑆_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, dim^𝒞(𝐸^𝒞_𝛼)≤dim^𝒞(𝑉^𝒞_𝛼)≤𝑓⁺_𝒞(𝛼+)≤inf_𝑞>0(𝑞𝛼−𝜏_𝒞(𝑞))≤inf_𝑞>0(𝑞𝛼−𝑇_𝒞(𝑞)). (2.24)

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].

3. Multifractal Jump Processes

3.1. 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: (A1)Λ is stationary with Λ>0 and 𝐄[Λ]=1; there exists 𝑏>1 such that Λ^(𝑖)(⋅/𝑏^𝑖) are i.i.d. 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 𝑇₀=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.

Recall that a real-valued stationary process is weakly mixing if for all 𝐵₁,𝐵₂∈ℬ (the 𝜎-algebra generated by the process) lim_𝑇→∞𝑇⁻¹∫^𝑇₀|𝑃(𝐵₁∩𝑆^𝑡𝐵₂)−𝑃(𝐵₁)𝑃(𝐵₂)|d𝑡=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 be studied. For example, if Λ is a finite state Markov process, then analogous results can be derived.

Next define the finite product processes Λ_𝑛(𝑡)≐^𝑛−1_𝑖=0Λ^(𝑖)(𝑡). (3.1) For 𝑡∈[0,1], the cumulative processes 𝐴_𝑛(𝑡)≐^𝑡₀Λ_𝑛(𝑠)d𝑠=^𝑡₀^𝑛−1_𝑖=0Λ^(𝑖)(𝑠)d𝑠,𝑛=1,2,… (3.2) can be associated with positive measures defined on the Borel sets ℬ of [0,1]: 𝜇_𝑛(𝐵)≐_𝐵Λ_𝑛(𝑠)d𝑠,𝑛=1,2,…,𝐵∈ℬ. (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_𝑛𝜇_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(𝑡)≐𝜇([0,𝑡])≐lim_𝑛→∞𝜇_𝑛([0,𝑡])=lim_𝑛→∞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 𝐄[𝐴(1)]=lim_𝑛𝐄[𝐴_𝑛(1)]=1. To provide criteria for convergence in ℒ_𝑝 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), and 𝐄[𝑀²]<∞. If there are positive 𝛾₁, 𝛾₂ and 𝐶 with 𝜎²𝑒^{−𝛾1|𝑥|}≤Cov(Λ(0),Λ(𝑥))=^∞_𝑗=0(𝜆𝑥)^𝑗𝑗!Cov(𝑀₀,𝑀_𝑗)𝑒^−𝜆𝑥≤𝐶|𝑥|^−𝛾2, (3.5) then 𝐴_𝑛(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 𝑀_𝑘. For example, a sufficient condition for convergence isCov(𝑀₀,𝑀_𝑗𝐶)≤𝑗+1,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 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 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=0Ω_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 Ω₀×⋯×Ω_𝑛−1 for any fixed 𝑛. Since the 𝐵₀×⋯×𝐵_𝑛−1×Π^∞_𝑖=𝑛Ω_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:𝜇^(𝑛)(𝐵)≐_𝐵^𝑛−1_𝑖=01Λ^(𝑖)(𝜏)d𝜇(𝜏), (3.7) that is, the measure where 𝑛 first terms of the product are neglected. Thus𝜇^(𝑛)(𝐵)=^dist𝑏^−𝑛𝜇(𝑏^𝑛𝐵). (3.8)

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 𝜇([0,𝑡])=^𝑡₀Λ_𝑛(𝑠)d𝜇^(𝑛)(𝑠)=^dist𝑏^{−𝑛𝑡𝑏𝑛}₀Λ_𝑛(𝑠𝑏^−𝑛)d𝜇(𝑠), (3.9) 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 𝐏(𝜇([0,𝑡])=0)=𝐏(𝑏^{−𝑛𝑡𝑏𝑛}₀Λ_𝑛(𝑠𝑏^−𝑛)d𝜇(𝑠)=0)=𝐏(𝑏^−𝑛𝜇([0,𝑡𝑏^𝑛])=0)≤𝐏(𝑏^−𝑛𝜇([0,𝑡𝑏^𝑛𝑡])≤2)→0as𝑛→∞. (3.10) 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 denote 𝑈_𝑡={𝜔∶𝜇[0,𝑡]=0}. Then 𝐏(𝜇([0,𝑡])=0∀𝑡∈(0,1])=𝐏(^∞_𝑛=1𝑈_1/𝑛)≤_𝑛𝐏(𝑈_1/𝑛)=0, (3.11) 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 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 𝛽(𝑞)≐𝑞−1−log_𝑏𝐄[Λ^𝑞]. (3.12) 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)+1𝑇∀𝑡∈[0,1],_𝒟(𝑞)=𝛽(𝑞)=𝑞−1−log_𝑏𝐄[𝑀^𝑞]. (3.13)(3.14)

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 partitions 𝐽^𝑛 induced by Λ_𝑛. By assumptions (A1) and (A2), Λ^(𝑖)(𝑡)=𝑀_𝑘^(𝑖),𝑇^(𝑖)_𝑘−1≤𝑡<𝑇_𝑘^(𝑖), (3.15) where {𝑇_𝑘^(𝑖)}^∞_𝑘=1 forms a Poisson point process of intensity 𝜆𝑏^𝑖 and {𝑀_𝑘^(𝑖)}^∞_𝑘=1 is a stationary positive time series independent of {𝑇_𝑘^(𝑖)}_𝑘. Consequently, the product process Λ_𝑛(𝑡)=^𝑛−1_𝑖=0Λ^(𝑖)(𝑡) (3.16) is constant on intervals 𝐽^𝑛_𝑘≐[𝑇^𝑛_𝑘−1,𝑇^𝑛_𝑘), where the 𝑇^𝑛_𝑘 are defined by a Poisson point process of intensity 𝜆_𝑛=𝜆∑^𝑛−1_𝑖=0𝑏^𝑖=𝑂(𝑏^𝑛). 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.

Lemma 3.6. Consider a nested Poisson partition 𝒫. Let 𝑡_𝑜 be arbitrary. With probability one, as 𝑛→∞, log|𝐽^𝑛(𝑡_𝑜)|𝑛→−log𝑏,logmax_𝑘|𝐽^𝑛_𝑘|𝑛→−log𝑏,logmin_𝑘|𝐽^𝑛_𝑘|𝑛→−2log𝑏. (3.17)

Proof. Without loss of generality, assume 𝜆_𝑛=𝜆𝑏^𝑛 and 𝑡₀∈[0,1).
(1°): Clearly 𝐏(Nopointsin[𝑡₀−𝑥,𝑡₀+𝑥])≤𝐏(|𝐽^𝑛(𝑡₀)|>𝑥)≤2𝐏(Nopointsin[𝑡₀,𝑡₀+𝑥/2]). (3.18) Thus for 𝑛 large enough, 𝐏(|𝐽^𝑛(𝑡₀)|>𝑏^−𝑛𝑒^𝑛𝜀)≤2𝑒^{(−1/2)𝜆𝑛𝑏−𝑛𝑒𝑛𝜀}≤2𝑒^{(−1/2)𝑛𝜀𝜆},𝐏(|𝐽^𝑛(𝑡₀)|<𝑏^−𝑛𝑒^𝑛𝜀)≤1−𝑒^{−2𝜆𝑛𝑏−𝑛𝑒−𝑛𝜀}≤2𝜆𝑒^−𝑛𝜀. (3.19)
(2°): Since the maximal interval is always longer than a fixed interval, for 𝑛 large enough, 𝐏(max_𝑘|𝐽^𝑛_𝑘|<𝑏^−𝑛𝑒^−𝑛𝜀)≤𝐏(|𝐽^𝑛(𝑡₀)|<𝑏^−𝑛𝑒^−𝑛𝜀)≤2𝜆𝑒^−𝑛𝜀. (3.20) 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 𝐏(max_𝑘|𝐽^𝑛_𝑘|>𝑏^−𝑛𝑒^𝑛𝜀)≤^{⌊2𝑏𝑛𝑒−𝑛𝜀⌋}_𝑖=1𝑒^{(−1/2)𝜆𝑛𝑏−𝑛𝑒𝑛𝜀}≤2𝑒^{−𝑛((𝜆/4)𝑛𝜀2−log𝑏)}. (3.21)
(3°): Let {𝑈^𝑚_𝑘}^𝑚+1_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, that𝐏(min_𝑘|𝑈^𝑚_𝑘|<𝑥)=1−(1−(𝑚+1)𝑥)^𝑚1,0<𝑥<,11+𝑚1,𝑥≥.𝑚+1 (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 𝐏(min_𝑘|𝐽^𝑛_𝑘|<𝑏^−2𝑛𝑒^−𝑛𝜀)=^∞_𝑚=0𝐏(min_{𝑘=1,…,𝑁𝑛+1}|𝐽^𝑛_𝑘|<𝑏^−2𝑛𝑒^−𝑛𝜀∣𝑁_𝑛=𝑚)𝐏(𝑁_𝑛==𝑚)_{⌊𝑏2𝑛𝑒𝑛𝜀⌋−1𝑚=0}[1−(1−(𝑚+1)𝑏^−2𝑛𝑒^−𝑛𝜀)^𝑚](𝜆𝑏^𝑛)^𝑚𝑒𝑚!^{−𝜆𝑏𝑛}+^∞_{𝑚=⌊𝑏2𝑛𝑒𝑛𝜀⌋}(𝜆𝑏^𝑛)^𝑚𝑒𝑚!^{−𝜆𝑏𝑛}≤^∞_𝑚=0(𝑚²+𝑚)𝑏^−2𝑛𝑒^−𝑛𝜀(𝜆𝑏^𝑛)^𝑚𝑒𝑚!^{−𝜆𝑏𝑛}+(𝜆𝑏^𝑛)^{⌊𝑏2𝑛𝑒𝑛𝜀⌋}⌊𝑏^2𝑛𝑒^𝑛𝜀⌋!≤𝑒^−𝑛𝜀(2𝜆𝑏^−𝑛+𝜆²)+𝑒^−𝑛𝜀, (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 𝐏(min_𝑘|𝐼^𝑛_𝑘|>𝑏^−2𝑛𝑒^𝑛𝜀)≤(𝑒^{−𝜆𝑏−𝑛}(1+𝜆𝑏^−𝑛𝑒^𝑛𝜀))^{𝑏2𝑛𝑒−𝑛𝜀−1}=exp(−𝜆𝑏^𝑛+𝜆𝑏^−𝑛𝑒^𝑛𝜀+(𝑏^2𝑛𝑒^−𝑛𝜀−1)log(1+𝜆𝑏^−𝑛𝑒^𝑛𝜀))≤𝑒^{−𝑐1𝑛𝜀}. (3.24) (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 then𝜇(𝐽^𝑛(𝑡))=𝜇^(𝑛)(𝐽^𝑛(𝑡))^𝑛−1_𝑖=0Λ^(𝑖)(𝑡)=^dist𝑏^−𝑛𝜇(𝑏^𝑛𝐽^𝑛(𝑡))^𝑛−1_𝑖=0Λ^(𝑖)(𝑡), (3.25) 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)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) (3.26) 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 𝑇_𝒫(𝑞)=𝛽(𝑞)=𝑞−1−log_𝑏𝐄[𝑀^𝑞]for0<𝑞≤𝑝. (3.27)

Proof. Consider 𝐄[∑^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|^1+𝜀] with 𝜀>0. Denote by {𝑈^𝑗_𝑘∶𝑘=1,…,𝑗+1} the partition resulting from 𝑗 uniformly distributed points on [0,1]. Then 𝐄[|𝑈^𝑗_𝑘|^𝛼]=¹₀𝑗𝑠^𝛼(1−𝑠)^𝑗−1d𝑠=𝑗!Γ(1+𝛼).Γ(𝑗+1+𝛼) (3.28) Using the above result and conditioning with respect to the number of intervals gives 𝐄[^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|^1+𝜀]=^∞_𝑗=0𝐄[^𝑗+1_𝑘=1|𝑈^𝑗_𝑘|^1+𝜀]𝐏(𝑁_𝑛=𝑗+1)=^∞_𝑗=0(𝑗+1)𝑗!Γ(2+𝜀)Γ(𝑗+2+𝜀)(𝜆_𝑛)^𝑗𝑒^−𝜆𝑛𝑗!≤Γ(2+𝜀)(𝜆_𝑛)^−𝜖𝑒^{−𝜆𝑛∞}_𝑗=0(𝜆_𝑛)^𝑗+𝜀Γ(𝑗+1+𝜀)=Γ(2+𝜀)(𝜆_𝑛)^−𝜖(1−Γ(𝜀,𝜆_𝑛))Γ(𝜀)≤𝐶𝑏^−𝑛𝜀. (3.29)
Now assume 𝛾=𝛽(𝑞)−𝜖 with 𝜖>0. Conditioning with respect to the partition 𝐽^𝑛 helps us to split the addends into three factors, that is, 𝐄[^𝑁𝑛_𝑘=1𝜇(𝐽^𝑛_𝑘)^𝑞|𝐽^𝑛_𝑘|^−𝛾]=𝐄𝐄[^𝑁𝑛_𝑘=1Λ_𝑛(𝑇^𝑛_𝑘−1)^𝑞|𝐽^𝑛_𝑘|^−𝛾𝜇^(𝑛)(𝐽^𝑛_𝑘)^𝑞∣𝐽^𝑛]. (3.30) Use the scaling law (3.26) together with the independence of 𝜇^(𝑛) with respect to 𝐽^𝑛 and that 𝐄[Λ_𝑛(𝑇^𝑛_𝑘)^𝑞∣𝐽^𝑛]=𝐄[Λ_𝑛(𝑇^𝑛_𝑘)^𝑞]=𝐄[𝑀^𝑞]^𝑛=𝑏^{−𝑛(𝛽(𝑞)+1−𝑞)} (3.31) in order to estimate 𝐄[^𝑁𝑛_𝑘=1𝜇(𝐽^𝑛_𝑘)^𝑞|𝐽^𝑛_𝑘|^−𝛾]≤𝐶𝑏^{−𝑛(𝑞−1−𝛽(𝑞))}𝐄[𝑀^𝑞]^𝑛𝐄[^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|^{1+𝛽(𝑞)−𝛾}]=𝐶𝐄[^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|^1+𝜀]≤𝐶𝑏^−𝑛𝜀. (3.32)
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 𝐄[^𝑁𝑛_𝑘=1𝜇(𝐽^𝑛_𝑘)^𝑞|𝐽^𝑛_𝑘|^−𝛾]≥𝐶𝐄[^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|^1−𝜀]≥𝐶𝐄[^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|]=𝐶. (3.33)
Thus, we have found that ∑_𝑛≥𝑚𝐄[𝑆_𝑛(𝑞,𝛾)]→0 whenever 𝛾<𝛽(𝑞) and ∑_𝑛≥𝑚𝐄[𝑆_𝑛(𝑞,𝛾)]↛0 whenever 𝛾>𝛽(𝑞).

3.3. Information Dimension

This section is devoted to analyzing 𝐸^𝒫_𝛼1, the set of points with scaling exponent 𝛼^𝒫₁≐𝛼₁≐𝛽^′(1)=1−𝐄[Λlog_𝑏Λ]=1−𝐄[𝑀log_𝑏𝑀] (3.34) 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 set𝐾^𝒫_𝛼={𝑥∶lim_𝑛𝛼^𝒫_𝑛(𝑥)=𝛼,lim_𝑛log_𝑏|𝐽^𝑛(𝑥)|−𝑛=1}, (3.35) 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 dim𝐾^𝒫_𝛼, and thus dim𝐸^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 𝑄(𝜑)=𝐄[¹₀𝜑(𝑡,𝜔)d𝜇(𝑡)] (3.36) 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 lim_𝑛→∞log|𝐽^𝑛(𝑇)|𝑛=−log𝑏Q-a.s. (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 log𝜇^(𝑛)(𝐽^𝑛(𝑇))𝑛→−log𝑏Q-a.s. (3.38)

Proof. Recall first that 𝐄[𝜇^(𝑛)(𝐽^𝑛_𝑘)|𝐽^𝑛]=|𝐽^𝑛_𝑘| and that Λ_𝑛(𝑇^𝑛_𝑘) is independent of 𝐽^𝑛 so that 𝐄[Λ_𝑛(𝑇^𝑛_𝑘)∣𝐽^𝑛]=𝐄[Λ_𝑛]=1. (3.39) (1°) Condition with respect to the partition 𝐽^𝑛 in order to get 𝐏_𝑄(|𝐽^𝑛(𝑇)|<𝑥)=𝐄[¹₀𝟏{|𝐽^𝑛(𝑡)|<𝑥}d𝜇(𝑡)]=𝐄𝐄[^𝑁𝑛_𝑘=1𝟏{|𝐽^𝑛_𝑘|<𝑥}Λ_𝑛(𝑇^𝑛_𝑘−1)𝜇^(𝑛)(𝐽^𝑛_𝑘)∣𝐽^𝑛]=𝐄[^𝑁𝑛_𝑘=1𝟏{|𝐽^𝑛_𝑘|<𝑥}|𝐽^𝑛_𝑘|]=𝐄[¹₀𝟏{|𝐽^𝑛=(𝑡)|<𝑥}d𝑡]₁₀𝐏(|𝐽^𝑛(𝑡)|<𝑥)d𝑡≤𝐶₁𝑥𝜆_𝑛, (3.40) 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: 𝐏_𝑄(|𝐽^𝑛(𝑇)|>𝑥)=¹₀𝐏(𝟏{|𝐽^𝑛(𝑡)|>𝑥})d𝑡≤𝐶₂𝑒^{−𝜆𝑛𝑥}, (3.41) 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.
Since 𝜆_𝑛=𝑂(𝑏^𝑛), we have ^∞_𝑛=1𝐏_𝑄(|𝐽^𝑛(𝑇)|<𝑏^−𝑛𝑒^−𝑛𝜀)<∞,^∞_𝑛=1𝐏_𝑄(|𝐽^𝑛(𝑇)|>𝑏^−𝑛𝑒^𝑛𝜀)<∞. (3.42) (2°) Let 𝜀>0. Write the 𝑄-probability as a sum 𝐏_𝑄(𝜇^(𝑛)(𝐽^𝑛(𝑇))<𝑏^−𝑛𝑒^−𝑛𝜀)=𝐄[^𝑁𝑛_𝑘=1𝟏{𝜇^(𝑛)(𝐽^𝑛_𝑘)<𝑏^−𝑛𝑒^−𝑛𝜀}Λ_𝑛(𝑇^𝑛_𝑘)𝜇^(𝑛)(𝐽^𝑛_𝑘)]. (3.43) Scaling the 𝜇^(𝑛)(𝐽^𝑛_𝑘) up by 𝑏^−𝑛𝑒^−𝑛𝜀 and applying (3.39) lead to 𝐏_𝑄(𝜇^(𝑛)(𝐽^𝑛(𝑇))<𝑏^−𝑛𝑒^−𝑛𝜀)≤𝑏^−𝑛𝑒^−𝑛𝜀𝐄𝐄[^𝑁𝑛_𝑘=0Λ_𝑛(𝑇^𝑛_𝑘)∣𝐽^𝑛]≤𝑏^−𝑛𝑒^−𝑛𝜀𝐄[𝑁_𝑛]. (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), 𝐄_𝑄[𝜇^(𝑛)(𝐽^𝑛(𝑇))^𝑞−1]=𝐄𝐄[^𝑁𝑛_𝑘=1Λ_𝑛(𝑇^𝑛_𝑘−1)𝜇^(𝑛)(𝐽^𝑛_𝑘)^𝑞∣𝐽^𝑛]≤𝐶𝑏^{−𝑛(𝑞−1−𝛽(𝑞))}𝐄[^𝑁𝑛_𝑘=1|𝐽^𝑛_𝑘|^1+𝛽(𝑞)]≤𝐶𝑏^{−𝑛(𝑞−1)}. (3.45) Then apply the Markov inequality to get 𝐏_𝑄(𝜇^(𝑛)(𝐽^𝑛(𝑇))≥𝑏^−𝑛𝑒^𝑛𝜀)≤𝐶𝑒^{−𝑛𝜀(𝑞−1)}. (3.46) (3°) Applying Borel-Cantelli to (1°) and (2°) completes the proof.

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 𝐄_𝑄[^𝑛−1_𝑖=0𝑔_𝑖(Λ^(𝑖)(𝑇))]=^𝑛−1_𝑖=0𝐄[𝑔_𝑖(Λ^(𝑖)(𝑡))Λ^(𝑖)(𝑡)] (3.47) 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: 𝐄_𝑄[^𝑛−1_𝑖=0𝑔_𝑖(Λ^(𝑖)(𝑇))]=𝐄[¹₀^𝑛−1_𝑖=0𝑔_𝑖(Λ^(𝑖)(𝑡))d𝜇(𝑡)]=𝐄𝐄[^𝑁𝑛_𝑘=0^𝑛−1_𝑖=0𝑔_𝑖(Λ^(𝑖)(𝑇^𝑛_𝑘−1))Λ^(𝑖)(𝑇^𝑛_𝑘−1)𝜇^(𝑛)(𝐼^𝑛_𝑘)∣𝐽^𝑛]=𝐄[^𝑁𝑛_𝑘=0^𝑛−1_𝑖=0𝑔_𝑖(Λ^(𝑖))(𝑇^𝑛_𝑘−1)Λ^(𝑖)(𝑇^𝑛_𝑘−1)|𝐽^𝑛_𝑘|]=𝐄[¹₀^𝑛−1_𝑖=0𝑔_𝑖(Λ^(𝑖)(𝑡))Λ^(𝑖)=(𝑡)d𝑡]_{10𝑛−1𝑖=0}𝐄[𝑔_𝑖(Λ^(𝑖)(𝑡))Λ^(𝑖)(𝑡)]d𝑡=^𝑛−1_𝑖=0𝐄[𝑔_𝑖(Λ^(𝑖)(𝑡))Λ^(𝑖)(𝑡)]. (3.48)

Corollary 3.10. Assume that 𝐴(𝑡) is a non-degenerate MJP. Then the random variables Λ^(𝑖)(𝑇), 𝑖=0,1,…,𝑛, are 𝑄-independent.

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 𝐄_𝑄[^𝑛_𝑖=0𝑔_𝑖(Λ^(𝑖)(𝑇))]=^𝑛_𝑖=0𝐄_𝑄[𝑔_𝑖(Λ^(𝑖)(𝑇))] (3.50) 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[logΛ^(𝑗)(𝑇)]=𝐄[ΛlogΛ] for all 𝑗∈ℕ.

Proof. Set 𝑔_𝑖(𝑥)=1 if 𝑖≠𝑗 and 𝑔_𝑗(𝑥)=log𝑥 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_𝛼1 has full 𝜇-measure Ω a.s., that is, 𝜇(𝐾^𝒫_𝛼1)=𝜇([0,1])Ω-a.s. (3.51)

Proof. Notice first that log𝜇(𝐽^𝑛(𝑡))log|𝐽^𝑛=(𝑡)|logΛ⁽⁰⁾(𝑡)+⋯+logΛ^(𝑛−1)(𝑡)log|𝐽^𝑛+(𝑡)|log𝜇^(𝑛)(𝐽^𝑛(𝑡))log|𝐽^𝑛=(𝑡)|(1/𝑛)∑^𝑛−1_𝑖=0logΛ^(𝑖)(𝑡)(1/𝑛)log|𝐽^𝑛+(𝑡)|(1/𝑛)log𝜇^(𝑛)(𝐽^𝑛(𝑡))(1/𝑛)log|𝐽^𝑛.(𝑡)| (3.52) Then by Corollary 3.10, 3.11, and LLN, 1𝑛^𝑛−1_𝑖=0logΛ^(𝑖)(𝑇)→𝐄_𝑄[Λ(𝑇)]=𝐄[ΛlogΛ]=𝐄[𝑀log𝑀]. (3.53) On the other hand, by lemma 3.8, lim_𝑛→∞1𝑛log(|𝐽^𝑛(𝑇)|)=−log𝑏Q-a.s.lim_𝑛→∞1𝑛log(𝜇^(𝑛)(𝐽^𝑛(𝑇)))=−log𝑏Q-a.s. (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^𝒫_𝛼1)=dim^𝒫(𝐾^𝒫_𝛼1)≥𝛼₁.

Proof. Condition (i) of lemma 2.2 is obvious; conditions (ii) and (iii) are satisfied by definition of 𝐾_𝛼1. 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, dim^𝒫(𝐸^𝒫_𝛼1)=dim(𝐸^𝒞_𝛼1)=𝛼₁=1−𝐄[𝑀log_𝑏𝑀]. (3.55)

For example, if 𝐴(𝑡) is an MJP satisfying (3.5) and 𝛽(2)>2, then dim^𝒫(𝐸_𝛼1)=dim(𝐸_𝛼1)=𝛼₁.

Proof. First, note that 𝛽^∗(𝛼₁)=𝛼₁ since 𝛼₁=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 𝛼₁≤dim^𝒫(𝐾^𝒫_𝛼1)=dim(𝐾^𝒫_𝛼1)≤dim(𝐸^𝒫_𝛼1)≤dim^𝒫(𝐸_𝛼1)≤dim^𝒫(𝑉^𝒫_𝛼1)≤inf_{1<𝑞′<𝑞}(𝑞^′𝛼₁−𝑇_𝒫(𝑞^′))=inf_{1<𝑞′<𝑞}(𝑞^′𝛼₁−𝛽(𝑞^′))=𝛼₁. (3.56)

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 compute dim𝑉^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 Λ(⋅)≐𝑏^{𝛽(𝑞)−𝑞+1}Λ(⋅)^𝑞=Λ(⋅)^𝑞𝐄[Λ^𝑞], (3.57) the structure function 𝛽(𝑝)≐𝑝−1−log_𝑏𝐄[Λ^𝑝]=𝛽(𝑝𝑞)−𝑝𝛽(𝑞), (3.58) 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 𝛽𝐄_𝑄[log_𝑏Λ]=𝐄[Λlog_𝑏1Λ]=𝐄[Λ^𝑞]𝐄[Λ^𝑞log_𝑏Λ]=1−𝛽^′(𝑞)=1−𝛼_𝑞. (3.60) From this we find𝐄_𝑄[log_𝑏Λ]=𝐄[Λlog_𝑏Λ]=(𝛽(𝑞)−𝑞+1)+𝑞𝐄[Λlog_𝑏Λ]=1−(𝑞𝛼_𝑞−𝛽(𝑞)). (3.61) 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_𝛼𝑞 of points with Hölder 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 𝑊^𝒫_𝛼≐{𝑡∶liminf_𝑛𝛼^𝒫_𝑛(𝑡)≤𝛼,lim_𝑛log|𝐽^𝑛(𝑡)|𝑛=−log𝑏} (3.62) 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, 𝐴_𝑛(𝑡) converges in ℒ_𝑝 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 that 𝐴_𝑛(𝑡) converges in ℒ_𝑝 for some 𝑝>1. If 𝛽(𝑝)>0 for some 𝑝>0, then 𝑄-a.s., lim_𝑛→∞log_𝑏|𝐽^𝑛(𝑇)|𝑛=−1,lim_𝑛→∞log_𝑏𝜇(𝐽^𝑛(𝑇))𝑛=−𝑞𝛼_𝑞+𝛽(𝑞),lim_𝑛→∞log𝜇(𝐽^𝑛(𝑇))log|𝐽^𝑛(𝑡)|=1−𝐄[Λlog_𝑏Λ]=𝑞𝛼_𝑞−𝛽(𝑞). (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.

Lemma 3.16. Assume that 𝐴(𝑡) is a non-degenerate MJP and that 𝐴_𝑛(𝑡) converges in ℒ_𝑝 for some 𝑝>1. Then liminf_𝑛→∞log_𝑏𝜇^(𝑛)(𝐽^𝑛(𝑇))−𝑛≤1Q-a.s. (3.64)

Proof. (1°): Let us start with a simple observation on nested Poisson point processes. Given a sequence of independent Poisson point processes of densities 𝑏^−𝑘𝜆, their superposition forms a sequence of Poisson point processes of densities (1+⋯+𝑏^−𝑛)𝜆. Denote 𝐼^{𝑛,[𝑥,y]}={𝐼_𝑘^{𝑛,[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 ∪^𝑚_𝑖=1𝐼^{∞,[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 𝐏_𝑄(𝜇^(𝑛)(𝐽^𝑛(𝑇))<𝑏^{−𝑛(1+𝜀)})=𝐄[¹₀𝟏{𝜇^(𝑛)(𝐽^𝑛(𝑡))<𝑏^{−𝑛(1+𝜀)}}d𝜇(𝑡)]=𝐄𝐄[^𝑁𝑛_𝑘=1𝟏{𝜇^(𝑛)(𝐽^𝑛_𝑘)<𝑏^{−𝑛(1+𝜀)}}Λ_𝑛(𝑇^𝑛_𝑘)𝜇^(𝑛)(𝐽^𝑛_𝑘)∣𝐽^𝑛]=𝐄[𝑏^{𝑁−𝑛𝑛}_𝑘=1𝟏{𝜇(𝑏^𝑛𝐽^𝑛_𝑘̃)<𝑏^−𝑛𝜀}𝜇(𝑏^𝑛𝐽^𝑛_𝑘̃)], (3.66) where 𝐽^𝑛̃ is distributed as 𝐽^𝑛 and independent of 𝜇 and 𝜇.
For each fixed 𝑛, construct a nested sequence of partitions 𝐼^{𝑖,[0,𝑏𝑛]}, 𝑖=0,1…, which are independent of 𝜇 and 𝜇 and satisfy 𝐼^{𝑛,[0,𝑏𝑛]}=𝑏^𝑛𝐽^𝑛̃. Then write (3.66) as an integral split into parts of length 1 and apply (3.65) to get 𝐏_𝑄(𝜇^(𝑛)(𝐽^𝑛(𝑇))<𝑏^{−𝑛(1+𝜀)})=𝐄[𝑏^{−𝑛𝑏𝑛}₀𝟏{𝜇(𝐼^{𝑛,[0,𝑏𝑛]}(𝑡))<𝑏^−𝑛𝜀}𝑑𝜇(𝑡)]≤𝐄[𝑏^{−𝑛⌊𝑏𝑛⌋}_𝑖=1^𝑖_𝑖−1𝟏{𝜇(𝐼^{∞,[𝑖−1,𝑖]}(𝑡))<𝑏^−𝑛𝜀}𝑑𝜇(𝑡)]+𝐄[𝑏^−𝑛𝜇([⌊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))=0)=𝐄𝐄[^𝑁_𝑘=1𝟏{𝜇(𝐼_𝑘^∞,[0,1])=0}𝜇(𝐼_𝑘^∞,[0,1])∣𝐼^∞,[0,1]]=0 (3.68) and 𝑄-a.s. 𝜇(𝐼^∞,[0,1](𝑇))>0. Applying this to (3.67) gives lim_𝑛→∞𝐏_𝑄(log_𝑏𝜇^(𝑛)(𝐽^𝑛(𝑇))−𝑛>1+𝜀)=0. (3.69) It is easy to see by contradiction that the previous equation implies liminf_𝑛log_𝑏(𝜇^(𝑛)(𝐽^𝑛(𝑇)))−𝑛≤1𝑄-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. 1𝑛^𝑛−1_𝑖=0log_𝑏Λ^(𝑖)(𝑇)→𝐄_𝑄[Λ(𝑇)]=1−𝛼_𝑞. (3.71) By Corollary 3.15 we have (1/𝑛)log_𝑏(|𝐽^𝑛(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 set 𝑉^𝒫_𝛼𝑞.

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_𝛼𝑞)=dim(𝑉^𝒫_𝛼𝑞)=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_𝛼1 is replaced by 𝑊_𝛼𝑞 and Lemma 2.2 is applied to 𝜇. The upper bound follows directly from inequality (2.24). In summary, dim^𝒫(𝑉^𝒫_𝛼𝑞)=dim(𝑉^𝒫_𝛼𝑞)=𝑓⁺_𝒫(𝛼⁺_𝑞)=𝑇^∗_𝒫(𝛼_𝑞)=𝑇^∗_𝒟(𝛼_𝑞)=𝛽^∗(𝛼_𝑞)=𝑞𝛼_𝑞−𝛽(𝑞). (3.73)

For example, if 𝐴(𝑡) is an MJP satisfying (3.5) and 𝛽(4)>2𝛽(2)>0, then (3.72) holds for 0<𝑞≤2.