Abstract
This paper is the first study of the sample path behavior of processes with stationary independent increments taking values in a nondiscrete, locally compact, metrizable, totally disconnected Abelian group. After some preparatory results of independent interest we give a general integral criterion for a deterministic function to be a local modulus of right-continuity for the paths of the process and then study the sets of “fast” and “slow” points where the local growth of the process is anomalously large or small. We establish the lim sup behavior for the sequence of first exit times from a collection of concentric balls for an arbitrary process and show that no deterministic function can act as an exact lower envelope. Under appropriate conditions similar results hold for the related sojourn time sequence. We consider various candidates for measuring the variation of the paths of the process, show that they exist and coincide in our situation, and then determine the common value for a general process. Using earlier results we calculate the Hausdorff and packing dimensions of the image of an interval, exhibit the correct Hausdorff measure for this set, and establish a dichotomy that classifies measure functions into those that lead to a zero packing measure for the image and those that lead to an infinite packing measure. Lastly, we prove some uniform dimension results, which bound the dimension of the image of a set in terms of the dimension of the set itself. These results hold almost surely for all sets simultaneously.
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Evans, S.N. Local properties of Lévy processes on a totally disconnected group. J Theor Probab 2, 209–259 (1989). https://doi.org/10.1007/BF01053411
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DOI: https://doi.org/10.1007/BF01053411