Another view of sequential sampling in the birth process with immigration

We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. Letting S(a, b) be the number of families observed in (a, b), we study the expected sample variance and its asymptotics for p consecutive sequential samples [Formula: see text], for [Formula: see text]. By conditioning on the sizes of the samples, we provide a connection between [Formula: see text] and p sequential samples of sizes [Formula: see text], drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166-1194, 2023. https://doi.org/10.3150/22-BEJ1494 ). We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of [Formula: see text], where the lengths of intervals are logarithmically equal, we revisit Fisher's 1943 multi-sampling problem and give another explanation of what Fisher's model could have meant in the world of sequential samples drawn from a BI process.