Let ${X(t) : t geq 0}$ be a real-valued stochastics process with stationary independent increments. In this paper, under the condition of stochastic compactness, we obtain appropriate function $alpha(t)$ and random function $eta(t)$ such that for some positive finite constant C, lim sup${X(t) - alpha(t)}/eta(t) = C$ a.s. both as t tends to zero and infinity.