A random walk is given as a sum of n iids, which has zero mean and constant variance. Based on this definition, the realization of a random walk at time index t is given by the sum *S = x _{1} + x_{2} + ... + x_{n}*. The following figure shows the random walk obtained from iids, which vary according to a normal distribution of zero mean and unit variance.

The random walk is important because if such behavior is found in a time series, it can be easily reduced to zero mean model by taking differences of the observations from two consecutive time indices as *S _{t} - S_{t-1} = x_{t}* is an iid with zero mean and constant variance.

Figure 1.13: Random walk ...