On Poisson Sequences of Chance Events
A. Ya. Khinchin · Theory of Probability and Its Applications · 1956
In the classical applications of the theory of chance events it is usually supposed that the probability of the occurrence of k events of the given type in a time interval of length t is given by Poisson’s formula \[ v_k (t) = e^{ - \lambda t} \frac{{(\lambda t)^t }}{{k1}}\quad (\lambda > 0{\text{ a constant}}).\] Indeed, as is known, it is easy to show that this is always the case if the given sequence of events is free from after-effects and stationary, and satisfies the condition \[ (1)\quad \psi _2 (t) = o(t)\quad (t \to 0), \] where \[ \psi _k (t) = \sum\limits_{r = k}^\infty {v_r (t)} .\] In the case of a nonstationary sequence $\Lambda (t)$ will always mean the average number of events occurring in the interval $(0,t)$ (so that in the stationary case $\Lambda (t) = \lambda t$). It is usual then to write for the probability $v_k (\alpha ,\beta )$ of the occurrence of k events in the time interval $(\alpha ,\beta )$\[ (2)\quad v_k (\alpha ,\beta ) = e^{ - [\Lambda (\beta ) - \Lambda (\alpha )]} \frac{{[\Lambda (\beta ) - \Lambda (\alpha )]^k }} {{k!}}, \] thus again postulating a Poisson distribution. The assumption of freedom from after-effects is obviously preserved and we reach the pertinent question: what requirements must take the place of (1), so that the Poisson formula (2) remains valid for the given sequence in this general case? The present paper aims at answering this question. We write in general \[ \sum\limits_{r = k}^\infty {v_r (\alpha ,\beta )} = \psi _k (\alpha ,\beta ),\] and call the given sequence ordinary, if for arbitrary $t > 0$ and $\varepsilon > 0$ there exists a constant $\delta > 0$ so that for $0 \leqq \alpha 0$.