A law of iterated logarithm for stationary Gaussian processes

In this article the following results are established. Theorem A. Let $ \{ X(t):0 \leqslant t &lt; \infty \} $ $ E[X(t)] \equiv 0$ satisfies the following conditions. (a) $ r(t) = 1 - \vert t{\vert^\alpha }H(t) + o(\vert t{\vert^\alpha }H(t))$as $ t \to 0$ $ 0 &lt; \alpha \leqslant 2$ varies slowly at zero, and (b) $ r(t) = O(1/\log t)$as $ t \to \infty $ defined on $ [a,\infty )$ \phi (t)$ --> $ \phi (\infty ) = \infty ,P[X(t) > \phi (t)$ $ {t_n} \to \infty ] = 0or1$ $ I(\phi ) = \int_a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt} $ $ g(x) = 1/_\sigma ^{ \sim - 1}(1/x)$ and $ _\sigma ^{ \sim 2}(t) = 2\vert t{\vert^\alpha }H(t)$. Theorem C. Let $ \{ {X_n}:n \geqslant 1\} $ 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty$ --> $ \gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty $ $ \{ \phi (n):n \geqslant 1\} $ $ {\lim _{n \to \infty }}\phi (n) = \infty $ $ \Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty $ <DIV ALIGN="CENTER" CLASS="mathdisplay"> $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sum\limits_{1 \leq k \leq n} {{I_k}} /\sum\limits_{1 \leq k \leq n} {E[{I_k}] = 1\quad a.s.,} $ where $ {I_k}$ <IMG WIDTH="117" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img27.gif" ALT="$ \{ {X_k} > \phi (k)\} $">.