Nonlinearity and Temporal Dependence
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Nonlinearities in the drift and diﬀusion coeﬀicients influence temporal dependence in diﬀusion models. We study this link using three measures of temporal dependence: rho-mixing, beta-mixing and alpha-mixing. Stationary diﬀusions that are rho-mixing have mixing coeﬀicients that decay exponentially to zero. When they fail to be rho-mixing, they are still beta-mixing and alpha-mixing; but coeﬀicient decay is slower than exponential. For such processes we ﬁnd transformations of the Markov states that have ﬁnite variances but inﬁnite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state-dependent, Poisson sampling alters the temporal dependence.
Chen, Xiaohong; Hansen, Lars P.; and Carrasco, Marine, "Nonlinearity and Temporal Dependence" (2008). Cowles Foundation Discussion Papers. 1956.