Misunderstanding of time series autocovariance I'm reading the "Time Series: Theory and Methods (2nd ed.)" by P.J.Brockwell and R.A.Davis. I've stopped at the one moment at pp.218-219 (Chapter 7 "Estimation of the mean and the Autocovariance function"). In the proof of theorem 7.1.1 if 
$\gamma(n) \rightarrow 0$ as $n \rightarrow +\infty$
then $$lim_{n \rightarrow +\infty} n^{-1} \sum_{|h| < n} \left(|\gamma(h)| \right) = 2 \lim_{n \rightarrow +\infty} \left( |\gamma(n)| \right) = 0$$. 
Could anyone explain me the first equality in this part of the proof, pls? I spend much time, but suppose, I'm not so intelligent for self-understanding...((((
Hope, you'll help me.
 A: It's not obvious, so let's approach this from first principles.
We only know, from the definition of $\gamma(n)\to 0,$ that for every $\epsilon\gt 0$ there exists an $N(\epsilon)\ge 0$ for which $n\ge N(\epsilon)$ implies $|\gamma(n)|\lt \epsilon.$
Let's exploit this to estimate the sum.  Pick any $\epsilon$ as in the definition, set $N=N(\epsilon/2)$, and for $m \gt N$ break the sum into two parts where every term in the latter part is bounded by $\epsilon/2,$ and (over)estimate the second sum:
$$s_m = \frac{1}{m}\sum_{h=0}^{m-1} |\gamma(h)|=\frac{1}{m}s_N + \frac{1}{m}\sum_{h=N}^{m-1} |\gamma(h)| \lt \frac{1}{m}s_N + \frac{1}{m}(m-N)\epsilon/2 \le \frac{1}{m}s_N + \epsilon/2.$$
By choosing $m$ to be greater than either $N$ or $2s_N/\epsilon$ you can guarantee $s_N/m \lt \epsilon/2.$  In such cases, $s_N/m + \epsilon_/2 \lt \epsilon.$  This shows that for any $\epsilon\gt 0,$ there is an $m$ for which $0 \le s_n \lt \epsilon$ for all $n\ge m.$  That's the very definition of having $0$ as a limit.

Notice that the quoted statement cannot be taken literally as a sequence of substitutions: it is not necessarily the case that for any $n,$ $s_n$ equals (or even is bounded above) by $2|\gamma(n)|.$  After all, it is possible that all the $\gamma(n)$ are eventually zero, but provided any single $\gamma(h)$ is nonzero, $s_n$ is always strictly positive.  I suspect the authors are implicitly quoting some theorem about summing convergent series.  Indeed, this is an elementary result in Cesàro summation--but the factor of $2$ is superfluous.
