Ergodicity of Age Structure in Populations with Markovian Vital Rates, I: Countable States

This paper establishes new ergodic theorems for population age structure. Let A 1, A 2, … be denumerable Leslie matrices, and for ℓ = 1,2, m(ℓ)(0) be age structures (vectors with elements mi (ℓ)(0)), satisfying the assumptions of the Coale-Lopez theorem. Let {A(t, ωℓ)}∞ t − 1 be sample paths of a discrete-time Markov chain with sample space {A k }∞ k − 1, and m(ℓ)(t) = A(t, ωℓ)A(t − 1, ωℓ) … A(1, ωℓ)m(ℓ)(0). Then for b = 1, 2, … (weak stochastic ergodicity) lim t→∞ (E(mi (1)(t)/mi (1)(t)) b − E(mi (2)(t)/mi (2)(t)) b ) = 0 if the chain is finite and weakly ergodic (see [4]) or denumerable and weakly ergodic (see [13]). The limit holds and (strong stochastic ergodicity) {m(ℓ)(t)}∞ t − 1 converge in distribution, if the chain is homogeneous, aperiodic, positive recurrent, and uniformly geometrically ergodic.