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Short Communications
The Behavior of a Jump for Processes with Independent Increments
B. A. Rogozin Novosibirsk
Abstract:
Let
$\xi(t), t \geq 0$ be a process with independent increments,
$\xi(0)=0$,
$\tau_y=\inf\{t:\xi(t)\ge y\}$,
$\Gamma_y=\xi(\tau_y)-y$.
We prove that:
1)
$\mathbf{P}\{\Gamma_y>0\}=0$ for all
$y>0$ if and only if
$$
\ln\mathbf{M}e^{i\lambda\xi(t)}=t\biggl\{ia\lambda-\frac{\sigma^2\lambda^2}{2}+\int_{-\infty}^0 \biggl(e^{i\lambda x}-1-\frac{i\lambda x}{1+x^2}\biggr)ds(x)\biggr\};
$$
2) $\mathbf{P}\{\Gamma_y>0\}>0,\ y>0,\ y\ne kh,\ k=1,2,\dots,h>0$, and
$\mathbf{P}\{\Gamma_{kh}>0\}=0$,
$k=1,2,\dots$ if and only if
$$
\ln\mathbf{M}e^{i\lambda\xi(t)}=t\biggl\{p_1(e^{i\lambda h-1})+\sum_{k=-\infty}^{-1}p_k(e^{i\lambda kh}-1)\biggr\},
$$
$$
p_k\ge 0,\quad k=1,-1,-2,\dots,\quad \sum_{k=-\infty}^{-1}p_k<\infty,\quad p_1>0;
$$
3) in all other cases
$\mathbf{P}\{\Gamma_y>0\}>0$ for all
$y>0$.
Received: 26.11.1970