Geodesic
The limit theorems for some functionals of processes with independent increments
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 457-467

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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. 
@article{TVP_1973_18_3_a1,
     author = {Yu. A. Davydov},
     title = {The limit theorems for some functionals of processes with independent increments},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {457--467},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a1/}
}
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Yu. A. Davydov. The limit theorems for some functionals of processes with independent increments. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 457-467. http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a1/