Why a Population Converges to Stability

A central theorem in mathematical demography tells us that the age distribution of a closed population with unchanging fertility and mortality behavior must converge to a fixed and stable form. Proofs rely on ready-made theorems borrowed from linear algebra or from asymptotic transform theory, notably the Perron-Frobenius and the Tauberian theorems. But while these are efficient and expedient, they give little insight into the mechanisms that forces the age distribution to converge. paper proposes a simple argument for convergence. An elementary device allows us to view the birth sequence as the product of an exponential sequence and a weighted smoothing process. Smoothing progressively damps out the peaks and hollows in the initial birth sequence; thus the birth sequence gradually becomes exponential, and this forces the age distribution to assume a fixed and final form.