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Two well-documented "empirical regularities" of asset return series are

  1. Serial dependence: (i) Linear autocorrelation in $x$ (up to 20 minutes); (ii) Autocorrelation of non-linear functions of the series (e.g., $x^2$, $|x|$, $\cos(x)$; up to several weeks)
  2. Non-gaussianity: (i) Leptokurtosis; (ii) Positive skewness. A possible reference is Cont (2000).

I can see how (1.) (especially for $x^2$) can be evidence of volatility clustering, which makes the series non-stationary, but would this impact Gaussianity of the series across the entire sample? Are extreme values somehow made more likely by property (1.)? Is there a plausible mechanism that relates (1.) and (2.) or are they two distinct statistical features of the data?

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There are a lot of misconceptions in your post.

Take as an example a popular model for returns: a GARCH(1,1) model: $$ Z_t = \sqrt{h_t} e_t, \hspace{10mm} e_t \sim \text{IID}(0,1) \\ h_t = \alpha_0 + \alpha_1 Z_{t-1}^2 + \beta_1 h_{t-1}. $$ The parameters are $\alpha_0 > 0$, $\alpha_1, \beta_1 \ge 0$, and to ensure stationarity and causality $\alpha_1 + \beta_1 < 1$.

1.) If this model is true, the returns are stationary, even though you might not think so if you look at a time plot of $Z_t$ (there is volatility clustering).

2.) There is no autocorrelation at any lag $> 0$.

3.) The squared returns are autocorrelated. In fact, assuming $EZ_t^4 <\infty$, you can show that the squared returns are an ARMA process.

4.) The above is not evidence of volatility clustering. For example, you can look at an ARCH(1) model, and the squared returns will be autocorrelated (this is called the ARCH effect), however the conditional variances (volatilities) will not be.

5.) Volatility clustering does not make a process non-stationary. This is because $\text{Var}(Z_{t+1}|Z_{1:t}) \neq \text{Var}(Z_{t+1})$. Note that this is GARCH(1,1) model is a stationary model that does possess volatility clustering. The volatility $\text{Var}(Z_{t+1}|Z_{1:t}) = h_t$, which is positively autocorrelated. However, the marginal variance $\text{Var}(Z_t)$ is a function that is free of $t$. Also, the autocorrelation is trivial, and hence this is weakly stationary.

6.) None of the above results assume anything about any distribution. In fact, $e_t$ are commonly assumed to be $t-$distributed.

7.) I didn't say anything about asymmetric returns, but there are a lot of fancier GARCH models that handle this.

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