Since both \( \boldsymbol{W} \) and \( \boldsymbol{w} \) represent probabilities, they have to be normalized, i.e., we require that at each time step we have

$$ \begin{equation*} \sum_i w_i(t) = 1, \end{equation*} $$

and

$$ \begin{equation*} \sum_j W(j\rightarrow i) = 1, \end{equation*} $$

which applies for all \( j \)-values. The further constraints are \( 0 \le W_{ij} \le 1 \) and \( 0 \le w_{j} \le 1 \). Note that the probability for remaining at the same place is in general not necessarily equal zero.