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Proceedings of the American Mathematical Society

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An upper asymptotic estimate of Brownian path variation


Author: Olaf P. Stackelberg
Journal: Proc. Amer. Math. Soc. 26 (1970), 168-173
MSC: Primary 60.62
DOI: https://doi.org/10.1090/S0002-9939-1970-0263160-2
MathSciNet review: 0263160
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Abstract: Let $ X(t,\omega )$ be standard Brownian motion, and denote by $ {\mathfrak{B}_n}$ the family of all partitions of $ [0, 1]$ with minimum distance between successive partition points $ \geqq 1/n$. Then

$\displaystyle P[ \lim \sup_{n \to \infty} \sup_{P \in \mathfrak{B}_{2n} \frac{\... ...g n}{\log^2 n} \sum\limits_{t_i \in P} (X(t_i) - X(t_{i - 1}))^2 \leqq K} ] = 1$

where $ K$ is an appropriate constant.

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DOI: https://doi.org/10.1090/S0002-9939-1970-0263160-2
Keywords: Brownian motion, sample path variation, asymptotic estimate
Article copyright: © Copyright 1970 American Mathematical Society

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