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

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