The Ergodicity for Bi-Immigration Birth and Death Processes in Random Environment

The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in random environment Qθ with birth rate λ < death rate μ, the following results are proved, (1) there is an unique q-process in random environment, , which is ergodic, that is, does not depend on i ≥ 0 and , (2) there is a bi-immigration birth and death process in random environment (X* = {Xt, t ≥ 0}, ɛ* = {ɛt, t ∈ (-∞, ∞)}) with random transition matrix such that X* is a strictly stationary process.