The evolutionary algorithm stochastic process is well-known to be Markovian. These have been under investigation in much of the theoretical evolutionary computing research. When the mutation rate is positive, the Markov chain modeling of an evolutionary algorithm is irreducible and, therefore, has a unique stationary distribution. Rather little is known about the stationary distribution. In fact, the only quantitative facts established so far tell us that the stationary distributions of Markov chains modeling evolutionary algorithms concentrate on uniform populations (i.e., those populations consisting of a repeated copy of the same individual). At the same time, knowing the stationary distribution may provide some information about the expected time it takes for the algorithm to reach a certain solution, assessment of the biases due to recombination and selection , and is of importance in population genetics to assess what is called a “genetic load” (see the introduction for more details). In the recent joint works of the first author, some bounds have been established on the rates at which the stationary distribution concentrates on the uniform populations. The primary tool used in these papers is the “quotient construction” method. It turns out that the quotient construction method can be exploited to derive much more informative bounds on ratios of the stationary distribution values of various subsets of the state space. In fact, some of the bounds obtained in the current work are expressed in terms of the parameters involved in all the three main stages of an evolutionary algorithm: namely, selection, recombination, and mutation.

The frequency with which various elements of the search space of a given evolutionary algorithm are sampled is affected by the family of recombination (reproduction) operators. The original Geiringer theorem tells us the limiting frequency of occurrence of a given individual under repeated application of crossover alone for the classical genetic algorithm. Recently, Geiringer's theorem has been generalized to include the case of linear GP with homologous crossover (which can also be thought of as a variable length GA). In the current paper we prove a general theorem which tells us that under rather mild conditions on a given evolutionary algorithm, call it A , the stationary distribution of a certain Markov chain of populations in the absence of selection is unique and uniform. This theorem not only implies the already existing versions of Geiringer's theorem, but also provides a recipe of how to obtain similar facts for a rather wide class of evolutionary algorithms. The techniques which are used to prove this theorem involve a classical fact about random walks on a group and may allow us to compute and/or estimate the eigenvalues of the corresponding Markov transition matrix which is directly related to the rate of convergence towards the unique limiting distribution.

This paper addresses the relationship between schemata and crossover operators. In Appendix A a general mathematical framework is developed which reveals an interesting correspondence between the families of reproduction transformations and the corresponding collections of invariant subsets of the search space. On the basis of this mathematical apparatus it is proved that the family of masked crossovers is, for all practical purposes, the largest family of transformations whose corresponding collection of invariant subsets is the family of Antonisse's schemata. In the process, a number of other interesting facts are shown. It is proved that the full dynastic span of a given subset of the search space under either one of the traditional families of crossover transformations (one-point crossovers or masked crossovers) is obtained after [log 2 n ] iterations where n is the dimension of the search space. The generalized notion of invariance introduced in the current paper unifies Radcliffe's notions of firespectfl and figene transmissionfl. Besides providing basic tools for the theoretical analysis carried out in the current paper, the general facts established in Appendix A provide a way to extend Radcliffe's notion of figenetic representation functionfl to compare various evolutionary computation techniques via their representation.