Resting-state functional connectivity is a powerful tool for studying human functional brain networks. Temporal fluctuations in functional connectivity, i.e., dynamic functional connectivity (dFC), are thought to reflect dynamic changes in brain organization and non-stationary switching of discrete brain states. However, recent studies have suggested that dFC might be attributed to sampling variability of static FC. Despite this controversy, a detailed exposition of stationarity and statistical testing of dFC is lacking in the literature. This article seeks an in-depth exploration of these statistical issues at a level appealing to both neuroscientists and statisticians. We first review the statistical notion of stationarity, emphasizing its reliance on ensemble statistics. In contrast, all FC measures depend on sample statistics. An important consequence is that the space of stationary signals is much broader than expected, e.g., encompassing hidden markov models (HMM) widely used to extract discrete brain states. In other words, stationarity does not imply the absence of brain states. We then expound the assumptions underlying the statistical testing of dFC. It turns out that the two popular frameworks - phase randomization (PR) and autoregressive randomization (ARR) - generate stationary, linear, Gaussian null data. Therefore, statistical rejection can be due to non-stationarity, nonlinearity and/or non-Gaussianity. For example, the null hypothesis can be rejected for the stationary HMM due to nonlinearity and non-Gaussianity. Finally, we show that a common form of ARR (bivariate ARR) is susceptible to false positives compared with PR and an adapted version of ARR (multivariate ARR). Application of PR and multivariate ARR to Human Connectome Project data suggests that the stationary, linear, Gaussian null hypothesis cannot be rejected for most participants. However, failure to reject the null hypothesis does not imply that static FC can fully explain dFC. We find that first order AR models explain temporal FC fluctuations significantly better than static FC models. Since first order AR models encode both static FC and one-lag FC, this suggests the presence of dynamical information beyond static FC. Furthermore, even in subjects where the null hypothesis was rejected, AR models explain temporal FC fluctuations significantly better than a popular HMM, suggesting the lack of discrete states (as measured by resting-state fMRI). Overall, our results suggest that AR models are not only useful as a means for generating null data, but may be a powerful tool for exploring the dynamical properties of resting-state fMRI. Finally, we discuss how apparent contradictions in the growing dFC literature might be reconciled.