Let Sn be a random variable for the step size. N denotes an instance. The chosen step size is independent from previous instances. Sn is distributed according to f(s). Such a process is called white noise provided the step size is gaussian distributed and step sizes are not correlated.
The step sizes from subsequent steps could be summed up to result in a path with realizations X1 until Xn denoting the position of the random walker at an given instance n.
The sum of all steps is again a random variable D X describing the random walk having its . This leads us to the Brownian motion or also called Wiener process.