Independence Assumption

“By contrast the theory of random walks says that the future path of the price level of a security is no more predictable than the path of a series of cumulated random numbers. In statistical terms the theory says that successive price changesare independent,identically distributed random variables. Most simply this implies that the series of price changes has no memory, that is, the past cannot be used to predict the future in any meaningful way.” (Fama, 1965, p. 34) (pdf)

“In statistical terms independence means that the probability distribution for the price change during time period t is independent of the sequence of price changes during previous time periods. That is, knowledge of the sequence of price changes leading up to time period t is of no help in assessing the probability distribution for the price change during time period t.2” (Fama, 1965, p. 35) (pdf)

“in fact we can probably never hope to find a time series that is characterized by perfect independence. Thus, strictly speaking, the random walk theory cannot be a completely accurate description of reality. For practical purposes, however, we may be willing to accept the independence assumption of the model as long as the dependence in the series of successive price changes is not above some “minimum acceptable” level.” (Fama, 1965, p. 35) (pdf)

“More specifically, the independence assumption is an adequate description of reality as long as the actual degree of dependence in the series of price changes is not sufficient to allow the past history of the series to be used to predict the future in a way which makes expected profits greater than they would be under a naive buy-and-hold model.” (Fama, 1965, p. 35) (pdf)