Assume that time series $(X_t)$ is given by: \begin{equation} X_t = \sum_{i=0}^{\infty} c_i \varepsilon_{t - i}, \end{equation} where $(\varepsilon_t)$ is a weak white noise $\text{WN}(0, \sigma^2)$ and sequence $(c_n)$ converges in $l^2$.
How to prove, that the autocovariance function: \begin{equation} \gamma_X(h) = Cov(X_{t+h}, X_t) \end{equation} converges to zero, as $h \rightarrow \infty$?
I found the solution on my own, so I share it with you.
We know that \begin{equation} \gamma_X(h) = \sigma^2\sum_{i = 0}^\infty c_ic_{i+h}. \end{equation} To test the convergence consider \begin{equation} \frac{1}{\sigma^2}|\gamma_X(h)| \leq \sum_{i=0}^\infty|c_i|\cdot|c_{i+h}| \leq \sqrt{\sum_{i=0}^\infty|c_i|^2}\cdot\sqrt{\sum_{i=0}^\infty|c_{i+h}|^2} = ||c_i||_{\mathcal{l}^2}\cdot||c_{i+h}||_{\mathcal{l}^2}. \end{equation} As $(c_i)$ is a member of the $\mathcal{l}^2$ space, so $||c_i||<\infty$. What remains is to check $\lim_{h\rightarrow\infty}||c_{i+h}||_{\mathcal{l}^2}$. \begin{equation} ||c_{i+h}||_{\mathcal{l}^2}^2 = \sum_{i=0}^\infty|c_{i+h}|^2 = \sum_{i=h}^\infty|c_i|^2 \rightarrow 0,\;\;\text{as}\;\;h\rightarrow\infty. \end{equation} Which holds because the $\mathcal{l}^2$ space is complete and tails of convergent series converge to zero.
Hence \begin{equation} |\gamma_X(h)| \leq \sigma^2 ||c_i||_{\mathcal{l}^2}\cdot||c_{i+h}||_{\mathcal{l}^2} \rightarrow 0,\;\;\text{as}\;\;h\rightarrow\infty. \end{equation}
