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ACTA WASAENSIA 39of returns would then not imply true variation in the distribution of t through time,but instead be a mere result of sampling error, as the law of large numbers would thenbe no longer applicable to 2nd nor any higher moments.The stable distribution hypothesis explains also leptokurtosis at varying time horizons,since application of the central limit theorem–used in the derivation of normallydistributed multiperiod returns–requires finite variance of the single period returns.Sums of independent increments with infinite variance, on the other hand, converge indistribtution to the non-normal members of the stable distribtution family.As such, Lévy stable distributions are the only possible limiting distributions of independenlyand identically distributed random variables 51 . Regarding long-term logreturnsas sums of iid short-term logreturns, this would imply that Lévy stable distributionsare the only candidates for describing long-term returns. DuMouchel (1973)showed, however, that the rate of convergence to the Lévy stable limit can be extremelyslow in the case of infinite variances, requiring numbers of observations of order 10 3before convergence to a stable limit could be observed. Furthermore, if returns arenot identically distributed, then every infinitely divisible distribution, that is everydistribution whose characteristic function ϕ may be expressed as the k’th power ofsome characteristic function ϕ k : ϕ(u) =[ϕ k (u)] k ,k ∈ N, is permissable as a limitdistribution for the sum of independent random variables 52 .Sums of iid stable random variables are themselves Lévy stable distributed with rescaledlocation and scale parameters, but identical skewness index β and characteristic exponentα S as the individual summands. Regarding long-term logreturns as sums of iidshort-term logreturns would then imply that long-term returns should have the sametail index α as their subperiod returns.Some researchers 53 accepted Mandelbrot’s infinite variance hypothesis merely upon indicationof α < 2 in their datasets without further testing of fit. Stability of the tail51 see Lévy (1925) and the discussions e.g. in Mandelbrot (1963) and Fama (1963).52 The result is due to Khintchine (1937). See also the discussion in Mantegna & Stanley (2000: page30—33).53 see e.g. the studies by Fama (1965); Teichmoeller (1971); Simkowitz & Beedles (1980).