The Distribution of the First Jump

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 $S_0=0$, $S_1$, $S_2$, $\dots$, $S_n=\sum\limits_{k=1}^n\xi_i$, $n\geqq 1$, where $\xi_1,\xi_2,\dots,\xi_n,\dots$ 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].