By using the context of the theory of Markov processes’ cycle (or circuit) representation, we study the circuit and weight representation of discrete-time birth-death processes, that is, birth-death chains. This will give us the possibility to investigate suitable criterions regarding transience and recurrence for the corresponding “adjoint” Markov chains describing uniquely the discrete-time birth-death chains by directed circuits and weights in fixed environments.

cycle representation theory, birth-death chain, transience, recurrence.