On the steady state of birth-death process with coefficients of moderate growth

The standard birth-death process with various forms of intensities (coefficients) is a source for obtaining natural skewed distributions which in turn are important in modeling different phenomena: the number of transcripts in eukaryotic cells, the number of words in a text, the number of citations of an author, etc. In these cases the coefficients of the respective birth-death process have moderate growth. The most general assumptions of growth for these cases are presented in this paper. These assumptions of moderate growth allow simplification of the well-known conditions for the existence of the steady state distribution of the process.