A Bartlett-type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.

Weakly electric knifefish are exceptionally maneuverable swimmers. In prior work, we have shown that they are able to move their entire body omnidirectionally so that they can rapidly reach prey up to several centimeters away. Consequently, in addition to being a focus of efforts to understand the neural basis of sensory signal processing in vertebrates, knifefish are increasingly the subject of biomechanical analysis to understand the coupling of signal acquisition and biomechanics. Here, we focus on a key subset of the knifefish's omnidirectional mechanical abilities: hovering in place, and swimming forward at variable speed. Using high-speed video and a markerless motion capture system to capture fin position, we show that hovering is achieved by generating two traveling waves, one from the caudal edge of the fin and one from the rostral edge, moving toward each other. These two traveling waves overlap at a nodal point near the center of the fin, cancelling fore-aft propulsion. During forward swimming at low velocities, the caudal region of the fin continues to have counter-propagating waves, directly retarding forward movement. The gait transition from hovering to forward swimming is accompanied by a shift in the nodal point toward the caudal end of the fin. While frequency varies significantly to increase speed at low velocities, beyond approximately one body length per second, the frequency stays near 10 Hz, and amplitude modulation becomes more prominent. A coupled central pattern generator model is able to reproduce qualitative features of fin motion and suggest hypotheses regarding the fin's neural control.