The study of a population's evolution under the action of a genetic operator, or composition of operators, is more difficult when the population size is finite because the examination of the expected population at each generation does not necessarily yield the overall expected result. In certain circumstances, some interesting properties of a population change during evolution in such a way that a valid conclusion can be drawn by charting the change from expected population to expected population. We establish sufficient conditions that ensure that the evolution of a property of a population can be determined by examination of the expected populations only. An example of the application of this characterization is a proof that a finite size population under repeated crossover in the absence of selection or mutation converges, in the sense of expected outcome, to the population with maximum diversity. The proof extends the results already established by others for infinite populations.

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