We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original...

We deal with real weakly stationary processes $\{X_t,t\in \mathbb{Z}\}$ with non-positive autocorrelations $\left\{r_k\right\}$, i. e. it is assumed that $r_k\le 0$ for all $k=1,2,\cdots$. We show that such processes have some special interesting properties. In particular, it is shown that each such a process can be represented as a linear process. Sufficient conditions under which the resulting process satisfies $r_k\le 0$ for all $k=1,2,\cdots$ are provided as well.

In the paper, a heteroskedastic autoregressive process of the first order is considered where the autoregressive parameter is random and errors are allowed to be non-identically distributed. Wild bootstrap procedure to approximate the distribution of the least-squares estimator of the mean of the random parameter is proposed as an alternative to the approximation based on asymptotic normality, and consistency of this procedure is established.