On the Sequence of Events, Selected by a Counter From a Recurrent Process of Events
Lajos Takács · Theory of Probability and Its Applications · 1956
We consider a sequence of random variables $\{ t_n \} $, where $t_0 = 0$ and the differences $t_n - t_{n - 1} (n = 1,2, \cdots )$ are equidistributed, independent positive random variables, having the distribution function $F(x)$. From the sequence $\{ t_n \} $ a counter selects and records a sequence $\{ t'_n \} $ according to the following rule: 1) $t'_0 = t_0 = 0$; 2) a pulse starts in the counter at a time $t_0 $ , and lasts for a time $\chi _0 $; 3) if there is no pulse in the counter at time $t_n (n \geqq 1)$ (all previous pulses have ended), there is probability 1 that a new pulse will start at this time; 4) if there is a pulse in the counter at time $t_n (n \geqq 1)$, there is probability p that a new pulse will start; 5) the durations, $\chi _0 ,\chi _1 ,\chi _2 , \cdots $ of the pulses form a sequence of positive independent equidistributed random variables with distribution function $H(x)$ (which may be improper); 6) the sequence $\{ t_n \} $ consists of the starting times of those pulses which are not overlapped by previous pulses. For $p = 0$ this model represents a Geiger-Müller counter, and for $p = 1$ it represents an electronic amplifier. The sequence $\{ t'_n \} $ has properties analogous to those of $\{ t_n \} $. Consider $G(x)$, the unknown distribution function of the differences $t'_n - t'_{n - 1} ,(n, = 1,2,3, \cdots )$. Laplace transforms are used to investigate the dependence of $G(x)$ on $F(x)$ for the following two cases: a) when $p = 0$ or $p = 1$; b) when $0 \leqq p \leqq 1$ and $H(x)$ is an improper distribution function.