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.