Small dead-time expansion in counting distributions and moments

Considering type I counters affected by a dead-time τ , we study the τ expansion of the probabilities and moments of the underlying stochastic renewal process. For the counting distributions and probabilities we extend results from the literature and analyse their approximation properties. Our results show, in particular, that for increasing counting numbers ever larger orders of the τ expansion are required for accurate approximations. Furthermore, the τ expansion for the first and second moments are obtained; their series is proved to coincide with the respective long time asymptotics. This asymptotics is demonstrated to converge exponentially fast to the exact quantities for growing time.