Covariances Estimation for Long-Memory Processes

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

Covariances Estimation for Long-Memory Processes

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

Central limit theorem for solutions of random initialized differential equations: a simple proof

We study the central and noncentral limit theorems for the convolution of a certain kernel h with F(ξ(⋅)), where ξ is a stationary Gaussian process and F is a square integrable function with respect to the standard Gaussian measure. Our method consists in showing that in the weak dependence case, we can use the Lindeberg method, approaching the initial Gaussian process by an m-dependent process. We could say that only variance computations are needed to get the two types of limits. Then we apply the obtained results to the solutions of the certain differential equations.