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Transactions of the American Mathematical Society
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
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

$\displaystyle \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\vert {{X_t}\, - \,{X_0}} \right\vert\, \leqslant \,\phi (t)\} $ has density 1 at the origin.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0536946-7
PII: S 0002-9947(1979)0536946-7
Keywords: Lebesgue density, approximate upper (lower) modulus, approximate continuity, Brownian motion, Gaussian process, scale-invariance
Article copyright: © Copyright 1979 American Mathematical Society