In this paper we provide an introduction to statistical inference for the classical linear birth‒death process, focusing on computational aspects of the problem in the setting of discretely observed processes. The basic probabilistic properties are given in Section 2, focusing on computation of the transition functions. This is followed by a brief discussion of simulation methods in Section 3, and of frequentist methods in Section 4. Section 5 is devoted to Bayesian methods, from rejection sampling to Markov chain Monte Carlo and approximate Bayesian computation. In Section 6 we consider the time-inhomogeneous case. The paper ends with a brief discussion in Section 7.

The interpretation of the ‘standard’ Palm version of a stationary random measure ξ is that it behaves like ξ conditioned on containing the origin in its mass. The interpretation of the ‘modified’ Palm version is that it behaves like ξ seen from a typical location in its mass. In this paper we shall focus on the modified Palm version, comparing it with the standard version in the transparent case of mixed biased coin tosses, and then establishing a limit theorem that motivates the above interpretation in the case of random measures on locally compact second countable Abelian groups possessing Følner averaging sets.

We study the asymptotic behavior of the survival probability of a multi-type branching process in a random environment. In the one-dimensional situation, the class of processes considered corresponds to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.