Nonlinearities in the drift and diﬀusion coeﬀicients influence temporal dependence in scalar diﬀusion models. We study this link using two notions of temporal dependence: beta-mixing and rho-mixing. We show that beta-mixing and rho-mixing with exponential decay are essentially equivalent concepts for scalar diﬀusions. For stationary diﬀusions that fail to be rho-mixing, we show that they are still beta-mixing except that the decay rates are slower than exponential. For such processes we ﬁnd transformations of the Markov states that have ﬁnite variances but inﬁnite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner similar to that of stochastic processes with long memory. Finally we show how nonlinear, state-dependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence.
Chen, Xiaohong; Hansen, Lars P.; and Carrasco, Marine, "Nonlinearity and Temporal Dependence" (2008). Cowles Foundation Discussion Papers. 1955.