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.