Inequalities and principles of large deviations for the trajectories of processes with independent increments

We consider a homogeneous process $S(t)$ on $[0,\infty)$ with independent increments, establish the local and ordinary large deviation principles for the trajectories of the processes $s_T(t):=\frac1TS(tT)$, $t\in[0,1]$, as $T\to\infty$, and obtain a series of inequalities for the distributions of the trajectories of $S(t)$.