Another Version of Strassen's Log Log Law with an Application to Approximate Upper Functions of a Gaussian Process with a Positive Index

Abstract

Let $\{Y(t, \omega) = (X_1(t, \omega), \cdots, X_d(t, \omega)); 0 \leq t \leq 1\}$ be a $d$-dimensional Gaussian process whose components are independent copies of a Gaussian process with index $\alpha$; that is, $E\lbrack X(t, \omega)\rbrack = 0, X(0, \omega) = 0$, and $E\lbrack(X(t, \omega) - X(s, \omega))^2\rbrack = \sigma^2(|t - s|)$, where $\sigma(t) = t^\alpha, 0 < \alpha < 1$. Let $h(t)$ be a positive, non-increasing, continuous function and set $q = \sup\big\{r \geq 0; \int_{+0} e^{-rh^2(t)/2} dt/t = + \infty\big\}.$ Then, as an application of a version of Strassen's $\log \log$ law, we have \begin{equation*}\begin{split}\lim \sup_{t\downarrow 0} t^{-1}m(\{0 \leq s \leq t;\|Y(s, \omega)\| > \sigma(s)h(s)\}) \\ &= \sup_{x\in B}m(\{0 \leq s \leq 1;\|x(s)\| \geq \sigma(s)/ \sqrt{q}\}), \quad\text{a.s.},\end{split}\end{equation*} where $\| \|$ denotes the usual Euclidean norm, $m(\Gamma)$ denotes the Lebesgue measure of a linear set $\Gamma$, and $B$ is the unit ball of the direct sum of $d$ copies of the reproducing kernel Hilbert space with the kernel $R(s, t) = (\sigma^2(t) + \sigma^2(s) - \sigma^2(|t - s|))/2$. In case of the $d$-dimensional Brownian motion, Strassen [7] had proved that the right-hand side of the above formula is equal to $1 - \exp\{-4(q - 1)\}$ if $q \geq 1$, and 0 if $q \leq 1$. As a corollary, $\sigma(t)h(t)$ is an approximate upper function as introduced by D. German [2] if and only if $q \leq 1$. Especially, if $\lim_{t\downarrow0}h(t)/ \sqrt{2 \log \log 1/t} = c, \sigma(t)h(t)$ is an approximate upper function if and only if $c \geq 1$.