  The paper presents a systematic theory for asymptotic inference of
autocovariances of stationary processes. We consider nonparametric tests for
serial correlations based on the maximum (or ${\cal L}^\infty$) and the
quadratic (or ${\cal L}^2$) deviations. For these two 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 principle and propose a sufficient
condition for summability of joint cumulants of stationary processes. We adopt
a simulation-based block of blocks bootstrapping procedure that improves the
finite-sample performance.
