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.