Geodesic
On the local behavior of processes with independent increments
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 507-512

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Let $\xi(t)$, $t\ge0$, $\xi(0)=0$, be a homogeneous process with independent increments. In [2] it was shown that $\lim\limits_{t\to0}(\xi(t)/t)$ exists and is finite if sample functions of $\xi(t)$ have a bounded variation. We prove that, in the opposite case, $$ \varlimsup_{t\to0}\frac{\xi(t)}t=\infty. $$ 
@article{TVP_1968_13_3_a11,
     author = {B. A. Rogozin},
     title = {On the local behavior of processes with independent increments},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {507--512},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_3_a11/}
}
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B. A. Rogozin. On the local behavior of processes with independent increments. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 507-512. http://geodesic.mathdoc.fr/item/TVP_1968_13_3_a11/