Copulas and Temporal Dependence

In this paper I identify a condition on the finite dimensional copulas of a univariate time series that ensures the series is weakly dependent in the sense of Doukhan and Louhichi (1999). This condition relates to the Kolmogorov-Smirnov distance between the joint copula of a group of variables in the past and a group of variables in the future, and the copula that would obtain if the past and future were independent. Interestingly, the implied form of weak dependence is not with respect to the class of Lipschitz functions, the class considered in most depth by Doukhan and Louhichi, but rather with respect to the class of absolutely continuous functions. I use the weak dependence property to prove a new strong law of large numbers and new invariance principles in which the only control on temporal dependence is expressed in terms of a condition on finite dimensional copulas.