The simplest form of the random walk is a function that has an equal probability of being +1 or -1. The expected outcome of this function is zero and the standard deviation is the square root of the number of periods.

A simple random walk in one dimension has two important properties. The first property is that it is recurrent. Over time, it crosses zero infinitely often. The second property is that it is unbounded – it can exceed any positive or negative threshold.

Our stock markets can be considered to be nearly normal (or Gaussian) random walks with a positive drift. This means that the stock market changes by an amount drawn from a normal distribution.

The implication is thus, once you deduct the equity premium and the risk free rate, the stock markets returns are supposed to be random a mean value of 0%. This is the efficient markets hypothesis –