Having chosen the acceptance probabilities, we have guaranteed that if the \( {\cal P}_i^{(n)} \) has equilibrated, that is if it is equal to \( p_i \), it will remain equilibrated. Next we need to find the circumstances for convergence to equilibrium.

The dynamical equation can be written as $$ {\cal P}^{(n)}_i = \sum_j M_{ij}{\cal P}^{(n-1)}_j $$ with the matrix \( M \) given by $$ M_{ij} = \delta_{ij}\left [ 1 -\sum_k T_{i\rightarrow k} A_{i \rightarrow k} \right ] + T_{j\rightarrow i} A_{j\rightarrow i} \,. $$ Summing over \( i \) shows that \( \sum_i M_{ij} = 1 \), and since \( \sum_k T_{i\rightarrow k} = 1 \), and \( A_{i \rightarrow k} \leq 1 \), the elements of the matrix satisfy \( M_{ij} \geq 0 \). The matrix \( M \) is therefore a stochastic matrix.