I am working on the following problem arising in time series analysis.
Let us assume that $\sum_{h \in \mathbb{Z}} |\gamma(h)|<\infty$. I would like to prove that \begin{equation*} 1) \; \; \; \lim_{n\to +\infty} \sum_{h >n } \gamma(h) = 0 \end{equation*} and the stronger result \begin{equation*} 2) \; \; \; \lim_{n\to +\infty} \sqrt{n}\sum_{h >n } \gamma(h) = 0 \end{equation*}
I was able to prove 1) using the limit of the partial sum $S_n = \sum_{h=-\infty}^n\gamma(h)$, and then writing $\sum_{h > n} \gamma(h) = \sum_{h \in \mathbb{Z}} \gamma(h) - \sum_{h = -\infty}^{n} \gamma(h) \ $ and taking the limit. However, for the second question I think I need additional assumptions on the sequence $(\gamma(h))$ in order to prove it.
The example given by Paul Sinclair give a good path, but one has to be sure that $a_{n-1}-a_n$ gives indeed the covariance function of a stationary sequence.
To do so, take $X_t=\sum_{i\in\mathbb Z}a_i\varepsilon_{t-i}$, where $(\varepsilon_k)$ is i.i.d., centered and has unit variance, $a_i> 0$, $\sum_{i\in\mathbb Z}a_i<\infty$ and $\sqrt{N}\sum_{i>N}a_i\geqslant 1$. Then $$ \gamma(h)=\mathbb E[X_0X_h]=\sum_{i\in\mathbb Z}a_{h+i}a_i $$ hence $\sum_{h\in\mathbb Z}\gamma(h)<\infty$ and $\gamma(h)\geqslant a_0a_h$.