Eficient Capital markets tastes. Finally, lest I be accused of criticizing others too severely for am- biguity, lack of rigor and incorrect conclusions, By contrast, the stock market trader has a much more practical criterion for judging what constitutes important dependence in successive price changes. For his purposes the random walk model is valid as long as knowledge of the past behavior of the series of price changes cannot be used to increase expected gains. More specif ically, the independence assumption is an adequate description of reality as long 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 [10,p35 We know now, of course, that this last condition hardly requires a random walk. It will in fact be met by the submartingale model of(6) But one should not be too hard on the theoretical efforts of the early em pirical random walk literature. The arguments were usually appealing; where they fell short was in awareness of developments in the theory of stochastic processes. Moreover, we shall now see that most of the empirical evidence in the random walk literature can easily be interpreted as tests of more gene expected return or“ fair game” models 2. Tests of Market Efficiency in the Random Walk Literature as discussed earlier.,“ fair game” models imply the‘ impossibility"of various sorts of trading systems. Some of the random walk literature has been concerned with testing the profitability of such systems. More of the literature has, however, been concerned with tests of serial covariances of returns. We shall now show that. like a random walk the serial covariances of a fair game'are zero, so that these tests are also relevant for the expected return If xt is a"fair game, "its unconditional expectation is zero and its serial covariance can be written in general form as E(t+xf(x ) d where f indicates a density function. But if xt is a"fair game, E(X+1xt)=0 8. Our brief h review is meant only to provide perspective, and it is, of course, somewhat plete. For le, we have ignored the important contributions to the early random walk literature in stu arrant and other options by Sprenkle, kruizenga, Boness, and others. Much of this early work on options is summarized in [8] 9. More generally, if the sequence x, is a fair game with respect to the information sequen (4 ),(i. e, E(x++1l,)=0 for all 4,)i then x, is a fair game with respect to any 't that is a ibset of t (i.e, E(x++1lp' )=0 for all 't ). To show this, let t=('t "t).Then, using Stieltjes integrals and the symbol F to denote cumulative distinction functions, the conditional EG+1=∫x+m(+,1)=「[J广x+m+1)]rs