The Upper and Lower Class Functions for Processes With Independents

In this paper, we derive the upper and lower class functions for prosses with independent increments.Let X=(X_t;t≥0)be a process with independent increments,EX_t= 0,< X >t< +∞,and lim_t→∞<X>_t=+∞. If exists a poisitive constant K such that\left|\Delta X_t\right| \tilde\equiv\left|X_t-X_t-1\right| \leq \frack < x_t^1 / 2\left(L l < x>_t\right)_3 / 2,then we have the following.Theorem 1. For any eventually no-decreasing function φ : l,∞)→(0, ∞) \mathrmP\left(X_t>< X_t^1 / 2 \phi\left(< X>_t\right)\right. i.o. )=0 or 1 according as I(\phi)=\int_1^\infty \frac\phi^2(t)t \exp \left\-\frac\phi^2(t)2\right\< \infty or =\infty,Theorem 2. Let φ be asabove, then \mathrmP\left(\sup _0 \leq s \leq t\left|X_s\right| \leq\left\langle X>_t^1 / 2 / \phi\left(< X>_t\right)\right.\right., i.o. )=0 or 1,according as J(\phi)=\int_1^\infty \frac\phi^2(t)t \exp \left\-\frac\pi^2 \phi^2(t)8\right\< \infty or =\infty.