Hausdorff Dimension of Operator Stable Sample Paths

Dimension results for sample paths of operator stable Lévy processes

“…Theorem 3.1 removes the condition on the density of X ðtÞ in Theorem 2.2 of Becker-Kern et al [1] and extends their results to X ðEÞ: This solves the problems in Remarks 3.8 and 3.9 of their paper.…”

“…The following lemma is a slight variant of Lemmas 3.3 and 3.4 in Becker-Kern et al [1] which can also be proven directly using Corollary 2.2.5 in Meerschaert and Scheffler [18]. …”

“…The second method is more analytic and is based on (1.3) and a result of Khoshnevisan and Xiao [13]. Compared to the arguments in Becker-Kern et al [1], our methods in this paper make use of other characteristics of an operator stable Le´vy process than its transition densities and hence they are more general. In particular, the covering method allows us to obtain a formula for dim H X ðEÞ for every Borel set E & R þ :…”

“…In this paper, by using different methods, we show that the restrictions on the transition densities of the processes in Becker-Kern et al [1] can be removed and thus verify their conjectures on the Hausdorff and packing dimensions of X ð½0; 1Þ: More specifically, we apply two methods to calculate the Hausdorff dimension of the range of an operator stable Le´vy process in R d : The first method is based on the covering argument for determining the Hausdorff dimension and is closely related to the arguments of Pruitt and Taylor [22] and Hendricks [9]. The second method is more analytic and is based on (1.3) and a result of Khoshnevisan and Xiao [13].…”

“…This type of Le´vy processes is sometimes useful in constructing counterexamples (see [18]) and has been studied by several authors. Examples of operator stable Le´vy process with dependent components can be found in Shieh [25] and Becker-Kern et al [1]. For systematic information about operator stable laws and operator stable Le´vy processes, we refer to Meerschaert and Scheffler [18].…”