We investigate the short-term temporal evolution of the classical birth-and-death process with linear birth and death rates.

We prove weak convergence of its increments to those of the Skellam process in the Skorohod space.

Similar convergence results are established under a different set of assumptions on the model parameters and the time horizon.

We also discuss related short-term approximations in terms of Skellam distributions for the marginals of such birth-and-death processes which start from a growing number of particles.

For two special cases of the pure birth and pure death processes, our results yield those on weak convergence to the corresponding Poisson processes.

This is joint work with Prof. V. Vinogradov recently published in the Journal of Stochastic Analysis, 2022, 3(1).