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The fourth moment is $\mathbb{E}[X^4]$. Its finiteness is equivalent to square integrability of $X^2$. For centered variables, dividing it by the squared variance gives kurtosis.

A Gaussian scale mixture is a random variable $X=RZ$, where $Z$ has a standard normal distribution and is independent of a positive random scale $R$. Its probability density function is the average of the conditional normal densities. Random scales can produce larger kurtosis than a single normal law.

For finite fourth moment and positive variance, kurtosis is $\mathbb{E}[(X-\mathbb{E}X{)}^4]/\mathrm{Var}(X{)}^2$. The normal distribution has kurtosis $3$. A random scale can increase kurtosis even when the conditional distribution is normal.

Two useful features of autoregressive conditional heteroscedasticity are persistent changes in conditional scale and excess unconditional kurtosis. Its time-varying conditional variance can explain volatility clustering, where large absolute returns occur in groups even when signed returns have little autocorrelation. A homoscedastic autoregressive moving-average model has a fixed innovation variance.

Also, a conditional normal distribution with a random scale is a Gaussian scale mixture. Its unconditional kurtosis can exceed $3$, or its fourth moment can be infinite. A Gaussian autoregressive moving-average model remains jointly Gaussian and cannot reproduce this effect. In the particular lag-two ARCH process, persistence of the squared scale occurs within each parity subsequence; the two parity subsequences are independent.