In this paper we analyse the fractional Poisson process where the state probabilities pkνk(t), t ≥ 0, are governed by time-fractional equations of order 0 < νk ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of pkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νk differs from that constructed from the fractional state equations (in the case of νk = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

The skeleton of a (super-) critical Galton-Watson process with offspring mean 1 + r, r ≧ 0, and finite offspring variance, is considered. When r = 0 it is trivial. If r > 0 is small and the time unit is taken as α /r generations (α > 0) then the skeleton can be approximated by a Yule (linear pure birth) process of rate α. This approximation can be used to study the evolution of genetic types over a long period of time in an exponentially growing population.

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.

A non-homogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments. A connection with the interarrival times of the corresponding record counting process (which is pure birth) is also pointed out.