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

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