RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Zametki, 1974, Volume 15, Issue 4, Pages 613–620 (Mi mz7385)  

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

A. T. Semenov

Novosibirsk State University

Abstract: Let $\xi_1,\xi_2,…$ be a sequence of independent identically distributed random variables with zero means. We consider the functional
$$ \eta_n=\sum_{k=0}^n\theta(S_k) $$
where $S_1=0$, $S_k=\sum_{i=1}^k\xi_i$ ($k\ge1$ and $\theta(x)=1$ for $x\ge0$, $\theta(x)=0$ for $x<0$. It is readily seen that $\eta_n$ is the time spent by the random walk $S_n$, $n\ge0$, on the positive semi-axis after $n$ steps.
For the simplest walk the asymptotics of the distribution $P(\eta_n=k)$ for $n\to\infty$ and $k\to\infty$, as well as for $k=O(n)$ and $k/n<1$, was studied in [1].
In this paper we obtain the asymptotic expansions in powers of $n^{-1}$ of the probabilities $P(\eta_n=nx)$ and $P(nx_1\le\eta_n\le nx_2)$ for $0<\delta_1\le x=k/n\le\delta_2<1$, $0<\delta_1\le x_1<x_2\le\delta_2<1$.

Full text: PDF file (467 kB)

English version:
Mathematical Notes, 1974, 15:4, 362–366

Bibliographic databases:

UDC: 519.2
Received: 27.02.1973

Citation: A. T. Semenov, “Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis”, Mat. Zametki, 15:4 (1974), 613–620; Math. Notes, 15:4 (1974), 362–366

Citation in format AMSBIB
\Bibitem{Sem74}
\by A.~T.~Semenov
\paper Asymptotic behavior of the dwell time distribution for a~random walk on a~positive semi-axis
\jour Mat. Zametki
\yr 1974
\vol 15
\issue 4
\pages 613--620
\mathnet{http://mi.mathnet.ru/mz7385}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=353457}
\zmath{https://zbmath.org/?q=an:0309.60041}
\transl
\jour Math. Notes
\yr 1974
\vol 15
\issue 4
\pages 362--366
\crossref{https://doi.org/10.1007/BF01095129}


Linking options:
  • http://mi.mathnet.ru/eng/mz7385
  • http://mi.mathnet.ru/eng/mz/v15/i4/p613

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математические заметки Mathematical Notes
    Number of views:
    This page:134
    Full text:62
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020