We wish to derive the required properties of $T$ and $A$ such that ${\cal P}_i^{(n\rightarrow \infty)} \rightarrow p_i$ so that starting from any distribution, the method converges to the correct distribution. Note that the description here is for a discrete probability distribution. Replacing probabilities $p_i$ with expressions like $p(x_i)dx_i$ will take all of these over to the corresponding continuum expressions.