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

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.