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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Dispersion points for linear sets and approximate moduli for some stochastic processes


Author: Donald Geman
Journal: Trans. Amer. Math. Soc. 253 (1979), 257-272
MSC: Primary 28A10; Secondary 26A15, 60G15, 60G17
DOI: https://doi.org/10.1090/S0002-9947-1979-0536946-7
MathSciNet review: 536946
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Abstract: Let $\Gamma \in [0, 1]$ be Lebesgue measurable; then $\Gamma$ has Lebesgue density 0 at the origin if and only if \[ \int _\Gamma {{t^{ - 1}}\Psi ({t^{ - 1}} {\text {meas}}} \{ \Gamma \cap (0, t)\} ) dt < \infty \] for some continuous, strictly increasing function $\Psi (t) (0 \leqslant t \leqslant 1)$ with $\Psi (0) = 0$. This result is applied to the local growth of certain Gaussian (and other) proceses $\{ {X_t}, t \geqslant 0\}$ as follows: we find continuous, increasing functions $\phi (t)$ and $\eta (t) (t \geqslant 0)$ such that, with probability one, the set $\{ t:\eta (t) \leqslant \left | {{X_t} - {X_0}} \right | \leqslant \phi (t)\}$ has density 1 at the origin.


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Keywords: Lebesgue density, approximate upper (lower) modulus, approximate continuity, Brownian motion, Gaussian process, scale-invariance
Article copyright: © Copyright 1979 American Mathematical Society