Stochastic Volatility and Fat Tail in Return Distribution

How are the two related?

Suppose volatility is constant and prices are lognormally distributed. Then the joint distribution of return across time is normal because sum of normally distributed variables is still normal.

If volatility is not constant yet still deterministic, then the same argument holds. In other words, the sum of normally distributed variables with different standard deviation is still normally distributed. In terms of stochastic calculus, the Ito integral of a deterministic function \int f dW is distributed as N(0,\int f^2 ds).

If volatility is stochastic, however, we will have fat-tailed return distribution instead of normal. Why is that? Shouldn't it still be normal because we're still summing up normally distributed variables? We have to think more rigorously in terms of stochastic calculus. For a stochastic function g, we cannot argue that \int g dW is distributed normally as we did previously for the case of deterministic volatility.