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I am working on time series. Theorem A5 of this document http://www-stat.wharton.upenn.edu/~stine/stat910/lectures/11_clt.pdf claims proof of asymptotic normality for autoregressive process of a special type (idependent white noise) of the sample mean. Also, the Wold decomposition says that it is almost possible to represent a time series as an autoregressive model with uncorrelated white noise and some deterministic term (See Shumway and Stoffer section B4 of the appendix). Which makes me wonder if there is any theoretical justification to believe that

  • A sum of elements of stationary time series is asymptotically normal?

By the above, it seems almost true. Is it true, or can we make it true by adding some weak conditions, such as $\gamma(h) \to 0$ as $h\to \infty$ or that there exist $M \in \mathrm{N}$ such that $\mathrm{Cov}(X_i,X_j)=0$ for all $|i-j|>M$?

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  • $\begingroup$ This is the common fallacy that the distribution of $X_1+X_2+\cdots + X_n$ approaches a normal distribution (with genuflections in the direction of the Central Limit Theorem). An equally common fallacy is that the distribution of $\frac 1n(X_1+X_2+\cdots + X_n)$ approaches a normal distribution (with genuflections in the direction of the Central Limit Theorem). Neither is true and neither is actually supported by the Central Limit Theorem. $\endgroup$ – Dilip Sarwate Sep 23 '17 at 19:45
  • $\begingroup$ Ok. I am interested in any number of results which could prove normality for the sum under stationary time series with conditions on $\gamma(h)$ if necessary. I am not interested in the standard CLT because of dependence obviously $\endgroup$ – Mikkel Rev Sep 23 '17 at 19:52
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What you are searching for appears to include the CLT for Ergodic Stationary Martingale Differences,

Billingsley, P. (1961). The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society, 12(5), 788-792.

See also the CLT for mixing processes, e.g. here

White, H., & Domowitz, I. (1984). Nonlinear regression with dependent observations. Econometrica: Journal of the Econometric Society, 143-161.

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