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
Q(θ) with birth rate λ< death rate μ, the
following results are proved, (1) there is an unique q-process
in random environment, ˉP(θ∗(0);t)=(ˉp(θ∗(0);t,i,j),i,j≥0), which is ergodic, that is,
limt→∞ˉp(θ∗(0);t,i,j)=ˉπ(θ∗(0);j)≥0 does not depend on i≥0
and ∑j≥0ˉπ(θ∗(0);j)=1, (2)
there is a bi-immigration birth and death process in random
environment (X∗={Xt,t≥0},ξ∗={ξt,t∈(−∞,∞)}) with random transition matrix ˉP(θ∗(0);t)
such that X∗ is a strictly stationary process.
