A new variant of the functional law of the iterated logarithm

The full description of the set of limit points of the sequence (3) is given, where $W(t)$ is a $d$-dimensional Brownian motion consisting of $d$ independent Brownian motions and $\varphi( \cdot )$ is arbitrary function such that $\varphi(t)\uparrow\infty$ ($t\uparrow\infty$). We show that with probability one this set coincides with the set $K_{R(\varphi)}$ specified in theorems 1–3. The sequences of the form (18) are also considered. The result of V. Strassen is a special case when $\varphi(t)=\sqrt{2\ln\ln t}$. The generalization of Hartman–Wintner's theorem is obtained. Theorems 4, 5 are valid for all sequences satisfying the almost sure invariance principles (martingale-differences, sequences with mixing etc.).