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Passage of Lévy processes across power law boundaries at small times. (English) Zbl 1140.60025

The paper completes and extends the research initiated by R. M. Blumenthal and R. K. Getoor [J. Math. Mech. 10, 492–516 (1961; Zbl 0097.33703)]. Namely, let \(X=(X_t, t\geq 0)\) be a Lévy process with characteristic triplet \((\gamma,\sigma,\Pi)\) where \(\gamma \in\mathbb{R}\), \(\sigma^2 = 0\) and the Lévy measure \(\Pi\) is restricted to \([-1,1]\) (\(\int(x^2 \wedge 1) \Pi(dx)\) is finite). The authors study when \(\limsup_{t\downarrow 0}| X_t| /t^{\kappa}\), \(\limsup_{t\downarrow 0}X_t/t^{\kappa}\) and/or \(\liminf_{t\downarrow 0}X_t/t^{\kappa}\) are a.s. finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of \(\kappa >0\). In general, the process crosses one- or two-sided \(t^{\kappa}\) boundaries for small \(t\) in quite different ways, but surprisingly this is not so for the case \(\kappa=1/2\). In the last case an integral test is provided to distinguish the possibilities. The Lévy – Itô decomposition for \(X\) plays an important role in the proofs. The introduction of the paper clarifies the relationships between the problems under consideration and some classical results as, e.g., the LIL by Khinchin and theorems by E. S. Shtatland [Theory Probab. Appl. 10, 317–322 (1965)] and B. A. Rogozin [Theor. Probab. Appl. 13, 482-486 (1968; Zbl 0177.21305)].

MSC:

60G51 Processes with independent increments; Lévy processes
60F15 Strong limit theorems
60G17 Sample path properties
60J65 Brownian motion
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References:

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