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This article investigates the problem of testing for a unit root in the case that the error, { u t}, of the model is a strictly stationary, mixing process with just barely infinite variance. Such errors have the property that for every δ such that 0 ≤ δ < 2, the moments E| u t| δ are finite. Under some additional restrictions on the rate of decay of the mixing rates, these errors belong to the domain of the non-normal attraction of the normal law and obey the invariance principle. This in turn implies that there might be conditions under which the usual Phillips-type test statistics for unit roots may still converge to the corresponding Dickey–Fuller distributions. In such a case, the unit-root hypothesis can be tested within an infinite-variance framework without any modifications to either the tests themselves or the critical values employed. This article derives a necessary and sufficient condition for convergence of the standard test statistics to the Dickey–Fuller distributions. By means of Monte Carlo simulations, the article also shows that this condition is likely to hold in the case that { u t} is a serially correlated, integrated generalized autoregressive conditionally heteroskedastic (IGARCH) process and the standard unit-root tests work well. [ABSTRACT FROM AUTHOR]