Central limit theorem: negative moments

Question

Let $X_{1},\ldots,X_{N}$ be i.i.d. random variables with mean 0 and variance 1. For simplicity, assume that $X_{1}$ has all finite moments. Let \begin{equation} S_{N} = \frac{1}{\sqrt{N}}\,\sum_{i=1}^{N}X_{i} \end{equation} The CLT says that $S_{N}$ converges in distribution to a normal random variable $X \sim N(0,1)$ as $N \to \infty$. It is also known that the following integer moments converge: For all $q \in \mathbb{N}$ we have \begin{equation} \lim_{N \to \infty}\mathbb{E}(|S_{N}|^{2q}) = \mathbb{E}(|X|^{2q}) \end{equation} My question is to what extent the above limit continues to be valid when $q$ is non-integer, especially when $q$ is slightly negative $-1/2 < q < 0$? Is there some uniform integrability that would prove this?