The paper presents a systematic theory for asymptotic inferences based on autocovariances of stationary processes. We consider nonparametric tests for serial correlations using the maximum (or L ∞ ) and the quadratic (or L 2 ) deviations of sample autocovariances. For these cases, with proper centering and rescaling, the asymptotic distributions of the deviations are Gumbel and Gaussian, respectively. To establish such an asymptotic theory, as byproducts, we develop a normal comparison

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... and propose a sufficient condition for summability of joint cumulants of stationary processes. We adapt a blocks of blocks bootstrapping procedure proposed by Künsch (1989) and Liu and Singh (1992) to the L ∞ based tests to improve the finite-sample performance.