The Annals of Probability

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

Norio Kono

Full-text: Open access

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 $c \geq 1$.

Article information

Source
Ann. Probab., Volume 10, Number 2 (1982), 303-319.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993859

Digital Object Identifier
doi:10.1214/aop/1176993859

Mathematical Reviews number (MathSciNet)
MR647506

Zentralblatt MATH identifier
0478.60047

JSTOR
links.jstor.org

Subjects
Primary: 60G17: Sample path properties
Secondary: 60G15: Gaussian processes 60F15: Strong theorems

Keywords
Approximate upper lower modulus Gaussian process sample path property Strassen's log log law

Citation

Kono, Norio. Another Version of Strassen's Log Log Law with an Application to Approximate Upper Functions of a Gaussian Process with a Positive Index. Ann. Probab. 10 (1982), no. 2, 303--319. doi:10.1214/aop/1176993859. https://projecteuclid.org/euclid.aop/1176993859


Export citation