Revisiting recurrence criteria of birth and death processes. Short proofs

The paper contains several new transparent proofs of criteria appearing in classification of birth and death processes (BDPs). They are almost purely probabilistic and differ from the classical techniques of three-term recurrence relations, continued fractions and orthogonal polynomials. Let ${T^\infty }$ be the passage time from zero to $\infty$. The regularity criterion says that ${T^\infty } < \infty$ if and only if $\mathbb {E}{T^\infty } < \infty$. It is heavily based on a result of Gong, Y., Mao, Y.-H. and Zhang, C. [J. Theoret. Probab. 25 (2012), no. 4, 950–980]. We obtain the latter expectation by using a two-term recurrence relation. We observe that the recurrence criterion is an immediate consequence of the well-known recurrence criterion for discrete-time BDPs and a result of Chung K. L. [Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967)]. We obtain the classical criterion of positive recurrence using technique of the common probability space. While doing so, we construct a monotone sequence of BDPs with finite state spaces converging to BDPs with an infinite state space.