  Birth-death processes form a natural class where ideas and results on large
deviations can be tested. In this paper, we derive a large deviation principle
under the assumption that the rate of a jump down (death) is growing
asymptotically linearly with the population size, while the rate of a jump up
(birth) is growing sub-linearly. We establish a large deviation principle under
various forms of scaling of the underlying process and the corresponding
normalization of the logarithm of the large deviation probabilities. The
results show interesting features of dependence of the large deviation
functional upon the parameters of the process and the forms of scaling and
normalization.
