The limit theorems for some functionals of processes with independent increments

This paper deals with asymptotical behaviour of the distribution of the functional $\frac1{D_T}\int_0^Th(S_t)\,dt$, where $S_t$ is a stochastic process with independent and stationary increments, $h(x)$ is a bounded function such that
$$ \frac1{T^\beta}\int_0^Th(x)\,dx\to p,\quad\frac1{T^\beta}\int_{-T}^0h(x)\,dx\ge q,\quad T\to\infty,\quad0\le\beta\le1 $$
and $D_T$ is a normalizing factor.