This study empirically investigated whether stock prices on the Nigerian stock exchange exhibit a random walk. Using monthly data from the Central Bank of Nigeria all share index from 1985-2011, the study employed a stepwise approach where the standard linear GARCH (1.1) is applied to capture randomness in terms of volatility clustering. The result proved that the Nigerian stock market is weakly stationary, meaning stock prices on the Nigerian stock market follows a random walk, which is an indication of weak-form efficiency. Therefore, the Nigerian stock market displays a random walk process. Nevertheless, the years 1987, 1991, 1995, 1997, 2001, 2002, 2008, and 2011 demonstrated negative skewness, which is a signification of non-randomness of the market for these years.