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On the Distribution of the First Positive Sum for a Sequence of Independent Random Variables

Theory of probability and its applications, 1957-01, Vol.2 (1), p.122-129 [Peer Reviewed Journal]

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Title:
On the Distribution of the First Positive Sum for a Sequence of Independent Random Variables
Author: Sinai, Ya. G.
Subjects: Neighborhoods ; Probability ; Random variables
Is Part Of: Theory of probability and its applications, 1957-01, Vol.2 (1), p.122-129
Description: The following problem is considered in this paper. Let $x_1 ,x_2 , \cdots ,x_n $ be mutually independent identically distributed random variables whose characteristic function is \[ \varphi (t) = {\bf M}_{e^{i\xi t} } = e^{ - c| t |^\alpha } \left( {1 + i\beta \frac{t} {{| t |}}\omega (t,\alpha )} \right), \] where $c > 0,0 < \alpha < 2, - 1 < \beta < 1$ and \[ \omega (t,\alpha ) = \left\{ \begin{gathered} \tan \frac{\pi } {2}\alpha ,\quad \alpha \ne 1, \hfill \\ \log \left| t \right|,\quad \alpha = 1. \hfill \\ \end{gathered} \right. \] Let $s_n = x_1 + \cdots + x_n $, and $\nu $ be the first index for which $S_n > 0$. It is proved that the distribution function of random variables $S_\nu $ belongs to the domain of attraction of the stable law with the parameters \[ a' = \alpha ( {1 - F(0)} ),\quad \beta = - 1, \] where $F(0) = P\{ x_i < 0\} $.
Publisher: Philadelphia: Society for Industrial and Applied Mathematics
Language: English
Identifier: ISSN: 0040-585X
EISSN: 1095-7219
DOI: 10.1137/1102009
Source: ProQuest Central

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