Between Strassen and Chung Normalizations for Lévy's Area Process

Let {L(t): t ≥ 0} be Lévy's area process, let$\gamma \colon {\bf R}_{+}\mapsto {\bf R}$, and let {Zt: t≥ 3} be the stochastic process defined by Zt(s)=L(ts)/(2t log log t), 0 ≤ s ≤ 1. Conditions on γ are given such that the set of all limit points of {γ (t)Zt: t≥ 3} as t → ∞ is a.s. equal to the set of all continuous functions defined on [0, 1] which vanish at 0.