I'm looking at page 219 of the book "Time Series: Theory and Methods" by Brockwell and Davis, and I can't seem to get my head around one line. Observe that

\begin{align*} n\text{Var}(\bar{X}) &= \frac{1}{n} \sum_{i,j=1}^n \text{Cov}(X_i,X_j) \\ &\vdots \\ &= \sum_{|h|<n} \left(1 - \frac{|h|}{n} \right)\gamma(h)\\ &\le \sum_{|h|<n} | \gamma(h)|. \end{align*}

Using the inequality they say

If $\gamma(n) \to 0$ as $n \to \infty$, then $\lim_{n \to \infty}$ $n^{-1} \sum_{|h| < n}|\gamma(h)| = 2\lim_{n \to \infty}|\gamma(n)| =$ $0$...

Why are those limits equal? I was told it could be showed with Fejer's theorem because it was a Cesaro sum, but wouldn't the book mention this? I am wondering if it's because of something simpler.


\begin{align*} n^{-1} \sum_{|h| < n}|\gamma(h)| &= \frac{1}{n}\gamma(0) + \frac{2}{n}\sum_{h=1}^n|\gamma(h)|\\ &= \frac{1}{n}\gamma(0) + \frac{2}{n}\sum_{h=1}^n[h-(h-1)]|\gamma(h)|\\ &\to 0+2\lim_{h \to \infty}|\gamma(h)| \tag{*} \\ &=0. \end{align*} The line (*) follows from Cesaro's Lemma, i.e. if $b_n \to \infty$ and $v_n \to v_{\infty}$ then $b_n^{-1}\sum_{k=1}^n (b_k - b_{k-1})v_k \to v_{\infty}$.

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