In ordinary demographic projection, the starting age structure of a population ceases to matter after a time, whatever changes the birth and death rates undergo within bounds. The inhomogeneous or ‘weak’ ergodic theorem asserts this eventual independence of starting state. Does the same independence hold true for inverse projection? This paper proves that the answer is ‘yes’ provided the model life-table family satisfies certain reasonable conditions, but that the answer can be ‘no’, if more extreme life-table families are allowed. The inhomogeneous ergodic theorem for inverse projection proved here, and the counter-examples to its extension, have implications for the validation of aggregative reconstructions in historical demography and to the contemporary forecasting of sub-populations subject to constraints.