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

We consider a homogeneous process S(t) on [0,∞) with independent increments, establish the local and ordinary large deviation principles for the trajectories of the processes $s_T (t): = tfrac{1} {T}S(tT) $ , t ∈ [0, 1], as T → ∞, and obtain a series of inequalities for the distributions of the trajectories of S(t).