In econometric literature, a lot of kinds of stochastic processes are employed, of course, including an autoregression (AR)，a moving average (MA)，an ARMA, and an ARIMA processes, etc. Among those, a random walk process is very unique and has some remarkable features. One of the most possible reasons to take a certain stochastic process is the fact that no one knows the reality of the time series data to be analyzed. This brief essay focused on a random walk process that is one of the most popular stochastic processes to unveil its essence. Moreover, some non-parametric testing methodologies such as a run test, a rank test, and a mean square successive difference test are also reviewed and applied to the normal random series generated by Box- Muller transformation.