Test of the Random Walke extension in view of stock returns'well-documented empirical departures from normality.10 Specifically,we consider the null hypothesis H* 1.For all t,E(e)=0,and E(e)=0 for any0. 2.e,}is o-mixing with coefficients (m)of size r/(2r-1)or is a-mixing with coefficients a(m)of size r/(r-1),where r>1,such that for all t and for anyT20,there exists some 6>0 for which E|ee,-,|2+<△<∞ (17) 3.1m2⑨-吃<∞ ng-oo nq t1 4.For all 1,E()=0 for any nonzero jand k wherejk. This null hypothesis assumes that X,possesses uncorrelated increments but allows for quite general forms of heteroscedasticity,including deter ministic changes in the variance (due,for example,to seasonal factors) and Engle's (1982)ARCH processes (in which the conditional variance depends on past information). Since M(g)still approaches zero under H*,we need only compute its asymptotic variance [call it 0()]to perform the standard inferences.We do this in two steps.First,recall that the following equality obtains asymp- totically: M,(q)g】 2(g-卫() (18) =1 q Second,note that under H*(condition 4)the autocorrelation coefficient estimators p(are asymptotically uncorrelated.12 If we can obtain asymp- totic variances (for each of the p()under H*,we may readily calculate the asymptotic variance 0(g)of M(g)as the weighted sum of the 6(p), 10 Of course,second moments are still assumed to be finite;otherwise,the variance ratio is no longer well defined.This rules out distributions with infinite varlance,such as those in the stable Pareto Levy family (with characteristic exponents that are less than 2)proposed by Mandelbrot (1963)and Fama (1965).We do,however,allow for many other forms of leptokurtosis,such as that generated by Engle's (1982) autoregressive conditionally heteroscedastic (ARCH)process. nCondition I is the essential property of the random walk that we wish to test.Conditions 2 and 3 are restrictions on the maximum degree of dependence and heterogeneity allowable while still permitting some form of the law of large numbers and the central limit theorem to obtain.See White (1984)for the precise definitions of and a-mixing random sequences.Condition 4 implies that the sample autocor. relations of e,are asymptotically uncorrelated;this condition may be weakened considerably at the expense of computational simplicity (see note 12). Although this restriction on the fourth cross-moments of e,may seem somewhat unintuitive,it is satisfied for any process with independent increments(regardless of heterogeneity)and also for linear gaussian ARCH processes.This assumption may be relaxed entirely,requiring the estimation of the asymptotic covariances of the autocorrelation estimators in order to estimate the limiting variance 0 of M(g)via relation (18).Although the resulting estimator of would be more complicated than Equation (20),it is conceptually straightforward and may readily be formed along the lines of Newey and West(1987).An even more general (and possibly more exact)sampling theory for the variance ratios may be obtained using the results of Dufour (1981)and Dufour and Roy (1985).Again,this would sacrifice much of the simplicity of our asymptotic results. 49