Fractal-Dimensional Properties of Subordinators

Abstract

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1-10, 2014) that almost surely lim δ→0 U (δ)N (t, δ) = t, where N (t, δ) is the minimal number of boxes of size at most δ needed to cover a subordinator's range up to time t, and U (δ) is the subordinator's renewal function. Our main result is a central limit theorem (CLT) for N (t, δ), complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N (t, δ), 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 δ. This new process can be manipulated with remarkable ease in comparison with N (t, δ), and allows better understanding of the box-counting dimension of a subordinator's range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process.

Transience and Recurrence of Markov Processes with Constrained Local Time

“…By (2), Π(g β (t)) ≤ t −1 , and so (2B) t −1 log(t) −mα(β−1)+mδ(β−1) Φ h y (t) as t → ∞. Finally, mα(1 − β) < −1, so choosing δ small enough, we conclude that for some ε > 0, (2B)…”

“…Proof for (a) Recall the notation (11), (7). By Lemma 6.1, for all h > 0, y > g(h), g h y (t)/ log(t) ≥ (1 − A −1 )g(t)/ log(t) > 1 as t → ∞, so by (60), Lemma 4.10, and Lemma Now, lim t→∞ tΠ(g h y (t)/ log(t)) ≥ lim t→∞ tΠ((1 − 1/A)g(t)/ log(t)) = 0 by (2), uniformly in h > 0, y > g(h) by Lemma 6.1. Applying Lemma 5.1 with H(t) = t −2 , as g h y (t) ≥ (1 − 1/A)g(t), uniformly in h > 0, y > g(h), as t → ∞, (2A) P X 0, g h y (t) log(t)…”

“…By (2), lim t→∞ ( * ) = 0 for each n > 1 and θ ∈ (0, 1), so in case (i), by (64), (J1) log(t) α+δ Π(g(t))t −(n−1) = o(Π(g(t)Φ(t)) as t → ∞ then θ → 1.…”

“…> h) = P(X (0,g(h)) h ≤ g(h))e −hΠ(g(h)) , and since lim h→∞ hΠ(g(h)) = 0 by (2), by Markov's inequality, as h → ∞, P (X h ≤ g(h))…”

“…So as h → ∞, g(h)(y −w −v) ≥ g(h−s)+x 0 ,so we can apply (3), and we apply (46) too. Now, g(h) −α L(g(h)) = Π(g(h)) for L slowly varying at ∞, so by(2), uniformly in y > K by the uniform convergence theorem[8, Theorem 1.2.1], as h → ∞,(h − s)u(g(h)(y − w − v))g(h)dy × P ∆ g(h) ds; X ∆ g(h) ∈ g(h)dw; S ∆ g(h) g(h)dv; O h s)L(g(h)(y − w − v)) g(h) α (y − w − v) 1+α dy × P ∆ g(h) ds; X ∆ g(h) ∈ g(h)dw; S ∆ g(h) s) g(h) α y 1+α L(g(h)y)dy × P ∆ g(h) ds; X ∆ g(h) ∈ g(h)dw; S ∆ g(h) g(h)dv; O h ds; X ∆ g(h) ∈ g(h)dw; S ∆ g(h) g(h)dv; O h ds; X ∆ g(h) ∈ g(h)dw; S ∆ g(h) g(h)dv; O h ≤ o(1) × y −1−α L(g(h)y) L(g(h)) P(O h )dy. (98)Upper Bound for σ 3 h (dy) Disintegrating on the values of ∆ g(h) 1…”