Summary
Let X(t) be a separable symmetric stable process of index α. Let P be a finite partition of [0,1], and ℘ a collection of partitions. The variation of a path X(t) is defined in three ways in terms of the sum \(\sum\limits_{t_i \in P} {|X(t_i ) - X(t_{i - 1} )|^\beta } \) collection ℘. Under certain conditions on ℘ and on the parameters α and Β, the distribution of the variation is shown to be a stable law. Under other conditions the distribution of the variational sum converges to a stable distribution.
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The author wishes to thank Prof. J. Chover for several helpful suggestions.
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Greenwood, P.E. The variation of a stable path is stable. Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 140–148 (1969). https://doi.org/10.1007/BF00537519
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DOI: https://doi.org/10.1007/BF00537519