We test for departures from normal and independent and identically distributed (NIID) log returns, for log returns under the alternative hypothesis that are self-affine and either long-range dependent, or drawn randomly from an L-stable distribution with infinite higher-order moments. The finite sample performance of estimators of the two forms of self-affinity is explored in a simulation study. In contrast to rescaled range analysis and other conventional estimation methods, the variant of fluctuation analysis that considers finite sample moments only is able to identify both forms of self-affinity. When log returns are self-affine and long-range dependent under the alternative hypothesis, however, rescaled range analysis has higher power than fluctuation analysis. The techniques are illustrated by means of an analysis of the daily log returns for the indices of 11 stock markets of developed countries. Several of the smaller stock markets by capitalization exhibit evidence of long-range dependence in log returns.