Suppose we have a probability distribution p(x), and we wish to analyze its convolution to itself n times as n approaches infinity.
https://en.wikipedia.org/wiki/Convolution_power and several other sources state that its shape will approximate to a Gaussian but under the assumption that its mean is zero ($\int_{-\infty}^{\infty} xp(x) dx = 0$), and its standard deviation is 1 ($\int_{-\infty}^{\infty} x^2p(x) dx = 1$).
However, suppose I am working with a probability distribution with mean equal to 0, but the integral for its standard deviation does not converge on $\{-\infty, \infty\}$. Does this mean that its convolution power (as n grows large) cannot be described through a normal distribution? Are there any other approximations/directions that can be done?
You are correct - it may not be a Gaussian if the finite variance assumption is not satisfied. But the shape can still be characterized assuming certain tail behavior, see e.g.: https://en.wikipedia.org/wiki/Stable_distribution#A_generalized_central_limit_theorem