Between Strassen and Chung normalizations for Lévy's area process

Abstract

Let $\left\{L\right(t):t\ge 0\}$ be Lévy's, let $\gamma :R_{+}⤅R$, and let $\{Z_t:t\ge 3\}$ be the stochastic process defined by $Z_t\left(s\right)=L\left(\mathrm{ts}\right)/\left(2t\log \log t\right),0\le s\le 1$. Conditions on $\gamma$ are given such that the set of all limit points of $\left\{\gamma \right(t)Z_t:t\ge 3\}$ as $t\to \mathrm{\infty }$ is a.s. equal to the set of all continuous functions defined on $[0,1]$ which vanish at 0.