We present novel, robust methods for inference on market (non-)efficiency, volatility clustering, and nonlinear dependence in financial return series. In contrast to existing methodology, our proposed methods are robust against nonlinear dynamics and tail-heaviness of returns. Specifically, our methods only rely on return processes being stationary and weakly dependent (mixing) with finite moments of a suitable order. This includes robustness against power-law distributions associated with nonlinear dynamic models such as GARCH and stochastic volatility. The methods are easy to implement and perform well in realistic settings. We revisit a recent study by Baltussen, van Bekkum, and Da (2019, J. Financ. Econ., 132, 26–48) on autocorrelation in major stock indexes. Using our robust methods, we document that the evidence of the presence of negative autocorrelation is weaker, compared with the conclusions of the original study.