The bivariate maximum process and quasi-stationary structure of birth-death processes

AbstractLet N(t) be a birth-death process on {0,1,…} with state 0 reflecting and let qTK be the quasi-stationary distribution of the truncated process on {0,1,…, K} with λK > 0. It is shown that the sequence (qTK) increases stochastically with K. The bivariate Markov chain (M(t), N(t)) where M(t)=max0≤t′≤tN(t′) is studied as a stepping stone to the proof of the result