Small dead-time expansion in counting distributions and moments

Considering type I counters affected by a dead-time ττ, we study the ττ 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 ττ expansion are required for accurate approximations. Furthermore, the ττ 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.