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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Variational sums for additive processes

Authors: William N. Hudson and J. David Mason
Journal: Proc. Amer. Math. Soc. 55 (1976), 395-399
MSC: Primary 60J30
MathSciNet review: 0405593
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Abstract: Let $ X(t),0 \leqq t \leqq T$, be an additive process, and let $ {X_{nk}}$ be the $ k$th increment of $ X(t)$ associated with the partition $ {\Pi _n}$ of $ [0,T]$. Assume $ \vert\vert{\Pi _n}\vert\vert \to 0$. Let $ \beta $ be the Blumenthal-Getoor index of $ X(T)$ and let $ 2 \geqq \gamma > \beta $. When the partitions are nested, $ \sum\nolimits_k {\vert{X_{nk}}{\vert^\gamma }} $ converges a.s. to $ \sum {\{ \vert J(s){\vert^\gamma }:0 \leqq s \leqq T\} } $, where $ J(s)$ is the jump of $ X(t)$ at $ s$. This convergence also holds when the partitions are not nested provided either $ X(t)$ has stationary increments or $ 1 \geqq \gamma > \beta $. This extends a result of P. W. Millar and completes a result of S. M. Berman.

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Keywords: Variational sums, infinitely divisible distributions, stochastic processes with independent increments
Article copyright: © Copyright 1976 American Mathematical Society