Transient analysis of the subordinated chain of a state dependent pure birth process

Consider a pure birth process with intensities {\lambda}(k) =1/(1+k), with k=0,1,2,…, we show that the subordinated chain is assimilable to a Bernoullian scheme with dependent successes probabilities, also we show a direct link with a degenerate Polya urn replacement scheme \cite{Janson}. We compute explicitly transition probabilities, by generating function method, of the subordinated chain and give some interesting bounds using the “centering sequence” approach proposed by MacDiarmid \cite{McDiarmidcentering}.