The time development of our initial PDF can now be represented through the action of the transition probability matrix applied \( n \) times. At a time \( t_n=n\epsilon \) our initial distribution has developed into

$$ \begin{equation*} w_i(t_n) = \sum_jW_{ij}(t_n)w_j(0), \end{equation*} $$

and defining

$$ \begin{equation*} W(il-jl,n\epsilon)=(W^n(\epsilon))_{ij} \end{equation*} $$

we obtain

$$ \begin{equation*} w_i(n\epsilon) = \sum_j(W^n(\epsilon))_{ij}w_j(0), \end{equation*} $$

or in matrix form

$$ \begin{equation} \tag{3} \boldsymbol{w}(n\epsilon) = \boldsymbol{W}^n(\epsilon)\boldsymbol{w}(0). \end{equation} $$