## 2 A Genetic Algorithm Markov Chain and Related Preliminaries
The proportional selection operation selects two parents with replacement (i.e., two population members from \(Q^{0}\)), wherein higher-fitness population members are more likely to be chosen as parents. Once these two parents are selected, a random crossover point is selected \(\chi\in[1,\)\(z-1]\), and then, with a certain crossover probability \(p_{c}\), a child is created by exchanging the parents' genes at crossover point \(\chi\). As an example, assume a crossover point \(\chi\in[1,\)\(n-1]\) and two selected parents, \(P_{a}=[\theta_{a1},\ldots,\theta_{az}]\) and \(P_{b}=[\theta_{b1},\ldots,\theta_{bz}]\), with \(a,b\in[1,\)\(\Omega]\); then, the genes of the child are created by swapping genes of parents at the crossover point, which will be \(\left[\theta_{a1},\ldots,\theta_{a\chi},\ \theta_{b(\chi+1)},\ldots,\theta_{bz}\right]\). The mutation operation probabilistically changes a gene of a population member using a low mutation probability \(p_{m}\). If a gene is selected for mutation, its value is replaced by generating a random number using uniform distribution in a closed interval \([-1,1]\). If \(T^{s}\), \(T^{c}\), and \(T^{m}\) are represented as selection, crossover, and mutation operations, then the next generation population \(Q^{g+1}\) is generated from the current generation population \(Q^{g}\), as follows:  
\[Q^{g+1}=T^{m}\left(T^{C}\left(T^{S}(Q^{g})\right)\right) \tag{7}\]  
The matrix \(Q^{g}\) in Equation (6) may be written as \(Q^{g}=\left[\theta_{1}^{T},\ldots,\theta_{\Omega}^{T}\right]^{T}\) and it is used to represent one state in a discrete Markov Chain. Each generation \"\(g\)\" is a discrete event. Given that genes or individual components of \(\theta_{s}\) take continuous values in the interval \([-1,\)\(1]\), the interval is discretized into discrete categories using a minimal interval width \(\delta\)\(>0\). The discretized interval will contain categories in a set \(\{-1,\)\(-1+\delta,\)\(-1+2\delta,\)\(\ldots,\)\(0,\)\(\ldots,\)\(1-\delta,\)\(1\}\). By choosing a value of \(\delta\) so that zero is not eliminated from the set, there will be \(\eta\)\(=(2/\delta)\) total categories in the interval. [PERSON] (1965) showed that the number of unique states, \(U\), of the population will be given by the following expression:  
\[U= \begin{pmatrix}\eta^{z}+\Omega-1\\ \Omega\end{pmatrix}  \tag{8}\]  
The state of any population at any generation \"\(g\)\" can be represented by a vector \(\pi^{g}\) of dimension \(U\), taking values in the interval \([0,1]\) so that the sum of the components of the vector always sum to \(1\). As an example, \(\pi^{0}=[0,\)\(0,\)\(1,\)\(\ldots\)\(0]^{\intercal}\) means that the initial random population in generation 0 is in the third state of all possible unique states \(U\). Assuming that the initial state of the population is given by the state \(\pi^{0}\) and the steady-state distribution (after several generations) is given by \(\pi^{\star}\), the expression in Equation (7) is a Markov Chain, and iterative operations in Equation (7) can be represented as follows:  
\[\left[\pi^{\star}\right]^{T}=\left[\pi^{0}\right]^{T}R \tag{9}\]  
where \(R\) is the Markov Chain transition matrix of dimension \(U\times U\). Let \(r_{pq}\) be the individual elements of matrix \(R\); then, as long as \(p_{m}\
eq 0\), \(r_{pq}>0\), because each population state can be reached by any other population state since a mutation can randomly change any gene of any or all population members with a non-zero low probability. Also, \(r_{pq}\
eq 1\), because \(p_{m}\
eq 1\) and because there is always some uncertainty that the population from state \(p\) may not go to state \(q\) in the next step. Furthermore, since each \(r_{pq}>0\) and \(r_{pq}\
eq 1\), \(R\) is a stochastic matrix with the magnitude of its maximum eigenvalue equal to one and its second-largest eigenvalue less than one. The Markov Chain is ergodic and irreducible, with a unique stationary distribution that can be reached with any initial random population.