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

Abstract

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 $c > 0,0 0$.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\} $.