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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 3, Pages 498–515 (Mi tvp395)

The Distribution of the First Jump

B. A. Rogozin

Novosibirsk

Abstract: The method of factorization [3], [4] is used for determining the distributions of the first positive value $\chi^ +$ and the maximum $\eta^+$ in the sequence $S_0=0$, $S_1$, $S_2$, …, $S_n=\sum\limits_{k=1}^n\xi_i$, $n\geqq 1$, where $\xi_1,\xi_2,…,\xi_n,…$ is a sequence of independent identically distributed random variables. The distribution of the first jump $\chi_x^+$ over a level $x>0$ and its limit as $x\to\infty$ are expressed in terms of the distribution of $\chi^+$. Formulas are given for evaluating these distributions by means of the distribution of $\xi_1$. The results presented supplement those in [8], [9] and [7].

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English version:
Theory of Probability and its Applications, 1964, 9:3, 450–465

Bibliographic databases:

Citation: B. A. Rogozin, “The Distribution of the First Jump”, Teor. Veroyatnost. i Primenen., 9:3 (1964), 498–515; Theory Probab. Appl., 9:3 (1964), 450–465

Citation in format AMSBIB
\Bibitem{Rog64} \by B.~A.~Rogozin \paper The Distribution of the First Jump \jour Teor. Veroyatnost. i Primenen. \yr 1964 \vol 9 \issue 3 \pages 498--515 \mathnet{http://mi.mathnet.ru/tvp395} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=169298} \zmath{https://zbmath.org/?q=an:0139.35501} \transl \jour Theory Probab. Appl. \yr 1964 \vol 9 \issue 3 \pages 450--465 \crossref{https://doi.org/10.1137/1109060} 

• http://mi.mathnet.ru/eng/tvp395
• http://mi.mathnet.ru/eng/tvp/v9/i3/p498

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This publication is cited in the following articles:
1. E. L. Presman, “Factorization methods and boundary problems for sums of random variables given on Markov chains”, Math. USSR-Izv., 3:4 (1969), 815–852
2. Sergey V. Nagaev, “Asymptotic formulas for probabilities of large deviations of ladder heights”, Theory Stoch. Process., 14(30):1 (2008), 100–116
3. R. Aliev, T. Khaniev, B. Gever, “Weak convergence theorem for ergodic distribution of stochastic process with a discrete interference of change and generalized reflecting barrier”, Theory Probab. Appl., 60:3 (2016), 502–513
4. R. T. Aliev, T. A. Khaniev, “On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk with Two Boundaries”, Math. Notes, 102:4 (2017), 444–454