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Teoriya Veroyatnostei i ee Primeneniya, 1972, Volume 17, Issue 1, Pages 143–147 (Mi tvp4195)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Communications

The Behavior of a Jump for Processes with Independent Increments

B. A. Rogozin

Novosibirsk
Full-text PDF (614 kB) Citations (1)
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
English version:
Theory of Probability and its Applications, 1972, Volume 17, Issue 1, Pages 146–149
DOI: https://doi.org/10.1137/1117011
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: B. A. Rogozin, “The Behavior of a Jump for Processes with Independent Increments”, Teor. Veroyatnost. i Primenen., 17:1 (1972), 143–147; Theory Probab. Appl., 17:1 (1972), 146–149
Citation in format AMSBIB
\Bibitem{Rog72}
\by B.~A.~Rogozin
\paper The Behavior of a Jump for Processes with Independent Increments
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 1
\pages 143--147
\mathnet{http://mi.mathnet.ru/tvp4195}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=315794}
\zmath{https://zbmath.org/?q=an:0293.60068}
\transl
\jour Theory Probab. Appl.
\yr 1972
\vol 17
\issue 1
\pages 146--149
\crossref{https://doi.org/10.1137/1117011}
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  • https://www.mathnet.ru/eng/tvp/v17/i1/p143
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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