Fractal-Dimensional Properties of Subordinators

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing L\'evy processes). It was recently shown in [Savov, 2014] that almost surely $\lim_{\delta\to0}U(\delta)N(t,\delta) = t$, where $N(t,\delta)$ is the minimal number of boxes of size at most $\delta$ needed to cover a subordinator's range up to time $t$, and $U(\delta)$ is the subordinator's renewal function. Our main result is a central limit theorem (CLT) for $N(t,\delta)$, complementing and refining work in [Savov, 2014]. Box-counting dimension is defined in terms of $N(t,\delta)$, but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than $\delta$. This new process can be manipulated with remarkable ease in comparison to $N(t,\delta)$, and allows better understanding of the box-counting dimension of a subordinator's range in terms of its L\'evy measure, improving upon [Corollary 1, Savov, 2014]. Further, we shall prove corresponding CLT and almost sure convergence results for the new process.

A Lévy process is a stochastic process in R d which has stationary, independent increments, and starts at the origin. A subordinator X := (X t ) t≥0 is a non-decreasing real-valued Lévy process. The Laplace exponent Φ of a subordinator X is defined by the relation e −Φ(λ) = E[e −λX1 ] for λ ≥ 0. By the Lévy Khintchine formula [1, p72], Φ can always be expressed as where d is the linear drift, and Π is the Lévy measure, which determines the size and intensity of the jumps (discontinuities) of X, and satisfies the condition If the Lévy measure is infinite, then infinitesimally small jumps occur at an infinite rate, almost surely. We will not study processes with finite Lévy measure, as they have only finitely many jumps, and hence no fractal structure.
The box-counting dimension of a set in R d is lim δ→0 log(N (δ))/ log(1/δ), where N (δ) is the minimal number of d-dimensional boxes of side length δ needed to cover the set. The limsup and liminf respectively define the upper and lower box-counting dimensions. For further background reading, we refer to [1,2] for subordinators, [7,21,23] for Lévy processes, and [9,26] for fractals.
The paper is structured as follows: Section 2 outlines the statements of all of the main results; Section 3 contains the proof of the CLT result for N (t, δ) and the lemmas required for this proof; Section 4 contains the proofs of all of the main results on the new process L(t, δ); Section 5 extends this work to the graph of a subordinator, and considers the special case of a subordinator with regularly varying Laplace exponent.
We will complement and refine this work with a CLT on N (t, δ). When the subordinator has no drift, we require a mild condition on the Lévy measure: where I(u) := u 0 Π(x)dx, and Π(x) := Π((x, ∞)).
Remark 2.2. Condition (2) has many equivalent formulations, see [1,Ex. III.7] and [3, Section 2.1]. We emphasise that (2) is far less restrictive than regular variation (or even O-regular variation) of the Laplace exponent, and appears naturally in the context of the law of the iterated logarithm (see e.g. [1, p87]).

An Alternative Box-Counting Scheme, L(t, δ)
Definition 2.4. The process of δ-shortened jumps,X δ := (X δ t ) t≥0 , is obtained by shortening all jumps of X of size larger than δ to instead have size δ. That is, and Lévy measureΠ δ (dx) = Π(dx)½ {x<δ} + Π(δ)∆ δ , where ∆ δ denotes a unit point mass at δ, and Π is the Lévy measure of X. We will see in Theorem 2.7 that L(t, δ) can replace N (t, δ) in the definition lim δ→0 log(N (t, δ))/ log(1/δ) of the box-counting dimension of the range of X. Then we will prove almost sure convergence and CLT results for L(t, δ).
Remark 2.6. The log scale at which box-counting dimension is defined allows flexibility among functions to be taken in place of the optimal count. In particular, there is freedom between functions related by f ≍ g asymptotically, where the notation means that there exist positive constants A, B such that For more details, we refer to [9, p42].
Computing the moments of L(t, δ) is remarkably simple in comparison to the moments of N (t, δ), which are not well known. This is a key benefit of using L(t, δ) to study the box-counting dimension of the range of a subordinator.
3 Proof of Theorem 2.3

A Sufficient Condition for Theorem 2.3
We will first work towards proving the following sufficient condition: The proof of Lemma 3.1 relies upon the Berry-Esseen Theorem, a very useful result for proving central limit theorem results as it provides the speed of convergence, which is stated here in Lemma 3.2. See [10, p542] for more details. with the same distribution as Y , where Y has finite mean, finite absolute third moment, and finite non-zero variance, for all n ∈ N and x ∈ R, For brevity, we will only provide calculations for t = 1. The proofs for different values of t are essentially the same. Recall the definitions a(δ) := U (δ) −1 , σ 2 δ := Var(T δ ), and b(δ) := U (δ) − 3 2 σ δ . We shall aim to prove that for all x ∈ R, For each δ > 0, (7) provides an upper bound, and then under condition (2), we can prove that this bound converges to zero as δ → 0.
and since N (1, δ) only takes integer values, using the fact that T δ , it follows that It follows from Lemma 3.
Then, as δ → 0, the asymptotic behaviour of n is It follows, with x ′ depending on x and δ, that as δ → 0, Now, by the triangle inequality and symmetry of the normal distribution, combining (9) and (11), it follows that as δ → 0, for any x ∈ R, Recall that we wish to show that (12) converges to zero. By the Berry-Esseen Theorem and the fact that n ∼ U (δ) −1 , it follows that as δ → 0, Applying the triangle inequality, then Lemma 3.3 with m = 2 and m = 3 to Therefore if the condition (6) as in the statement of Lemma 3.1 holds, then the desired convergence in distribution (3) follows, as required.
Proof of Lemma 3.3. First, by the moments and tails lemma (see [15, p26]), By the definition of T δ , it follows that X u ≥ δ if and only if T δ ≤ u, and then which is finite and independent of δ. Therefore the lim sup is finite, as required.

Proof of Theorem 2.3
Theorem 2.3 is proven by a contradiction, using Lemma 3.4 to show that the sufficient condition in Lemma 3.6 holds.  (2) implies that for each η ∈ (0, 1), there exists a sufficiently large integer n such that Proof of Lemma 3.4. The integral condition (2) imposes that for some B > 1, Then, by effectively replacing 1/2 with 2 −n (so 1/2 is replaced by a smaller constant), we can replace B with B n , which can be made arbitrarily large by choice of n. This follows by splitting up the fraction, where we simply take n sufficiently large that B n > 1/η.
Using Lemma 3.4 for a contradiction is the step in the proof of Theorem 2.3 which requires the condition (2). In order to prove Theorem 2.3, we require the notation introduced in Definition 3.5. We refer to [14, p93] for more details.
Definition 3.5. Recalling from Remark 2.4 that the processX δ has Laplace One can ignore the drift d in Definition 3.5, since d = 0 throughout Section 3. The proof of Theorem 2.3 now requires the following lemma: then the desired convergence in distribution (3), as in Theorem 2.3, holds.
Proof of Theorem 2.3. Assume for a contradiction that there exists a sequence (δ m ) m≥1 converging to zero, such that lim m→∞ λ δm δ m = ∞. That is to say, assume that the sufficient condition in Lemma 3.6 doesn't hold. For brevity, we omit the dependence of δ m on m. Hence for all fixed η, n > 0, η ≥ e −λ δ 2 −n δ for all small enough δ > 0. By Fubini's theorem, Removing part of the first integral and noting 1 ≥ e −λ δ x for all x > 0, where α > 0 is fixed and chosen sufficiently large that x δ < where the last two inequalities respectively follow by Definition 2.4, Definition 3.5 (i) with d = 0, and the relation U (δ) −1 ≍ I(δ)/δ, see [1, p74]. So for a constant K > 0, for all sufficiently small δ > 0, we have shown ηI(δ) + I(2 −n δ) ≥ I(δ) (1+α)K .
Proof of Lemma 3.9. For all α > 0, recalling that E[T δ ] = U (δ), For the desired convergence in distribution (3) to hold, it is sufficient by Lemma 3.1 to show that lim δ→0 U (δ) Note that T δ ≥ t if and only ifX δ t ≤ δ since jumps of size larger than δ do not occur in either case, and so X t =X δ t when T δ ≥ t. It follows that (3) holds if Remark 3.10. The condition in Lemma 3.9 is not optimal. If for ε ∈ 0, 1 6 , the convergence in distribution (3) follows too. This stronger condition does not lead to any more generality than the condition (2) for driftless subordinators.

Proofs of Results on L(t, δ)
Firstly, we prove Theorem 2.7, which confirms that L(t, δ) can replace N (t, δ) in the definition of the box-counting dimension of the range. This is done by showing that L(t, δ) ≍ N (t, δ), which is known to be sufficient by Remark 2.6.
Proof of Theorem 2.7. The jumps of the original subordinator X and the process with shortened jumpsX δ are all the same size, other than jumps bigger than size δ. The optimal number of intervals to cover the range, N (X, t, δ), always increases by 1 at each jump bigger than size δ, regardless of its size, so it follows that N (X, t, δ) = N (X δ , t, δ), with the obvious notation.
Instead of counting the number N (X, t, δ) of boxes needed to cover the range of X, consider those needed for the range of the subordinator X (0,δ) with Lévy measure Π(dx)½ {x<δ} (so all jumps of size larger than δ are removed), and adding Y δ t , which counts the number of jumps larger than size δ of X. It follows that N (X, t, δ) ≤ N (X (0,δ) , t, δ) + Y δ t ≤ 2N (X, t, δ).
Next we will prove the CLT result for L(t, δ), working with t = 1 for brevity. The proof is essentially the same for other values of t > 0. We will show convergence of the Laplace transform of 1 v(δ) (L(1, δ) − µ(δ)) to that of the standard normal distribution. Recall that Z ∼ N (0, 1) has Laplace transform E[e −λZ ] = e λ 2 /2 .
Proof of Theorem 2.10. By Remark 2.4 and (1), δL(t, δ) =X δ t is a subordinator with Laplace exponentΦ δ , and it follows that for any λ ≥ 0, Recalling the definition µ(δ) = 1 δ (d + I(δ)), where I(δ) := δ 0 xΠ δ (dx), and writingΦ δ in the Lévy Khintchine representation as in (1), it follows that Then applying the fact that y 2 2 − y 3 6 ≤ y − 1 + e −y ≤ y 2 2 for all y > 0, By the definition of v(δ), it follows that v(δ) 2 = 1 It is then sufficient, in order to show that (19) converges to λ 2 2 , to prove that Again by the definition of v(δ), for (20) to hold we require both Squaring the expression in (21), since x ≤ δ within each integral, it follows that By the binomial expansion, (a + b) 3 ≥ 3a 2 b for a, b > 0, and then since the Lévy measure is infinite. For (22), simply observe that Next we will prove the almost sure convergence result for L(t, δ). If there is a drift and the Lévy measure is finite, then the result is trivial. So we need only consider cases with infinite Lévy measure, and begin with the zero drift case. Using a Borel-Cantelli argument (see [15, p32] for details), we shall prove that lim inf δ→0 L(t, δ)/µ(δ) = lim sup δ→0 L(t, δ)/µ(δ) = t almost surely.
First, we will prove the almost sure convergence to t along a subsequence δ n converging to zero. Then, by monotonicity of µ(δ) and L(t, δ), we will deduce that for all δ between δ n and δ n+1 , L(t, δ)/µ(δ) also tends to t as δ n → 0.
When there is no drift, L(t, δ) is given by changing the original subordinator's jump sizes from y to 1 δ (y ∧ δ). By monotonicity of this map, it follows that for a fixed sample path of the original subordinator, each individual jump of the process L(t, δ n+1 ) is at least as big as the corresponding jump of the process L(t, δ n ). So L(t, δ) is non-decreasing as δ decreases, and so for all δ n+1 ≤ δ ≤ δ n , Then by our choice of the subsequence δ n , it follows that for all δ n+1 ≤ δ ≤ δ n , and since lim n→∞ L(t, δ n )/µ(δ n ) = t, it follows that Taking limits as r → 1, it follows that lim δ→0 L(t, δ)/µ(δ) = t almost surely.
For a process with a positive drift d > 0 and infinite Lévy measure, denote the scaling term obtained by removing the drift asμ(δ) := µ(δ) − d/δ. Then the above Borel-Cantelli argument forμ yields the almost sure limit along a subsequenceδ n as in (23). Then since the functions µ(δ) and L(t, δ) are again monotone in δ when there is a drift, the argument applies as in (24).

Extensions: Box-Counting Dimension of the Graph
The graph of a subordinator X up to time t is the set {(s, X s ) : 0 ≤ s ≤ t}.
The box-counting dimensions of the range and graph are closely related. This is evident when we consider the mesh box counting schemes M G (t, δ), M R (t, δ), denoting graph and range respectively. The mesh box-counting scheme counts the number of boxes in a lattice of side length δ to intersect with a set. Remark 5.2. It follows that the graph of every subordinator X has the same box-counting dimension as the range of X ′ t := t + X t , the original process plus a unit drift. and write N R (t, δ) as the optimal number of boxes needed to cover the range.
If d ≥ 1, then we have T 1 = T (δ,∞) because X δ ≥ dδ. It follows that each time N G (t, δ) increases by 1, so does N R (t, δ), and vice versa, so N G (t, δ) = N R (t, δ), and the box-counting dimension of the range and graph are equal when d ≥ 1.
For d ∈ (0, 1), a similar argument applies with a covering of δ d × δ rectangles rather than δ × δ squares. Starting with [0, , and so on. The number of these boxes is again N R (δ, t), since X δ d ≥ δ. By Remark 2.6 , this covering of rectangles can still be used to define the box-counting dimension of the range, since for k := 1 d , with N G (t, δ) and N ′ G (t, δ) as the number of squares and of rectangles respectively, N ′ G (t, δ) ≤ N G (t, δ) ≤ k N ′ G (t, δ/k).

Special Cases: Regular Variation of the Laplace Exponent
Corollary 5.5 is analogous to [24,Corollary 2], with L(t, δ) in place of N (t, δ). This allows very fine comparisons, not visible at the log-scale, to be made between subordinators whose Laplace exponents are regularly varying with the same index.