extension in view of stock returns'well-documented empirical departure from normality. 10 Specifically, we consider the null hypothesis H*. 1, For all t, E(e )=0, and E(EE-r)=0 for any T 2.(6 )is o-mixing with coefficients p(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 any T 20, there exists some 8>0 for which E|∈-,|m<△<∞ (17) E(e2)=o2 4. For all t,E(∈1-天1-)=0 for any nonzero j and k where≠k but allows for onesis assumes that X, possesses uncorrelated increments 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 8(q)] to perform the standard inferences. We do this in two steps. First, recall that the following equality obtains asymp M(g∑2a-Dn cond. note that under H*(condition 4) the autocorrelation coefficient imators p(j) are asymptotically uncorrelated. 12 If we can obtain asymp totic variances 8()) for each of the p() under H*, we may readily calculate the asymptotic variance e(q)of M, q) as the weighted sum of the 8() do,however, allow for many other forms of leptokurtosis, such as that generated by Engle's(1982) autoregressive conditionally heteroscedastic (ARCH) process Condition 1 is the essential property of the random walk that we wish to test. Conditions 2 and 3 are the law of bers and the central lim condition 4 tions of are asymptotically uncorrelated; this condition may be weakened considerably at th of computational simplicity (see note 12) Although this restriction on the fourth cross-moments of e, may seem somewhat unintuitive, it of M(a via sing the results of Dufour (1981) and Dufour and Roy (1985). Again, this would sacrifice much of the icity of our asymptotic results