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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 \[ P[ \lim \sup _{n \to \infty } \sup _{P \in \mathfrak {B}_{2n} \frac {\log \log n}{\log ^2 n} \sum \limits _{t_i \in P} (X(t_i) - X(t_{i - 1}))^2 \leqq K} ] = 1\] where $K$ is an appropriate constant.
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
- Priscilla E. Greenwood, An asymptotic estimate of Brownian path variation, Proc. Amer. Math. Soc. 21 (1969), 134–138. MR 235617, DOI https://doi.org/10.1090/S0002-9939-1969-0235617-3
- Paul Lévy, Le mouvement brownien plan, Amer. J. Math. 62 (1940), 487–550 (French). MR 2734, DOI https://doi.org/10.2307/2371467
- Otto Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Research Nat. Bur. Standards 45 (1950), 239–245. MR 0045863
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60.62
Retrieve articles in all journals with MSC: 60.62
Additional Information
Keywords:
Brownian motion,
sample path variation,
asymptotic estimate
Article copyright:
© Copyright 1970
American Mathematical Society