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Let $X_{1},\ldots,X_{N}$ be i.i.d. random variables with mean 0 and variance 1. Assume that $X_{i}$ are continuous random variables with all finite moments and a nice density function. 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?

Nick
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Ele
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  • You might start by wondering whether $\mathbb{E}(|X|^{2q})$ is finite for negative $q$ if $X \sim N(0,1)$ – Henry Dec 22 '23 at 12:24
  • I think it is provided -1/2 < q < 0 ? – Ele Dec 22 '23 at 12:25
  • I doubt it theoretically, and trying with $q=-0.4$ I do not see convergence empirically (there are occasionally very large values). – Henry Dec 22 '23 at 12:41
  • Doubt what? that $\mathbb{E}(|X|^{2q})$ is finite for -1/2 < q < 0 ? – Ele Dec 22 '23 at 12:42
  • Exactly that: if I take $\mathbb{E}(|X|^{-0.8})$ and simulate a very large number of times, I do not see it giving an obvious value. No do I expect it to: $|X|^{0.8}$ has a positive density near $0$ so its reciprocal would not have a positive expectation – Henry Dec 22 '23 at 12:45
  • I thought that $\mathbb{E}(|X|^{2q}) = c\int_{0}^{\infty}x^{2q}e^{-x^{2}/2}dx$. Now split the integral on some small interval $x \in [0,\epsilon]$. The Gaussian PDF is bounded by 1, so it's just integrating $\int_{0}^{\epsilon}x^{2q}dx$ which is finite if and only if $q > -1/2$. – Ele Dec 22 '23 at 12:48
  • That is an interesting point – Henry Dec 22 '23 at 12:50
  • So my original question...? – Ele Dec 22 '23 at 12:51
  • If $S_N$ takes the value $0$ with positive probability (for example when $N$ is even and $X_i$ take the values $1$ and $-1$ with positive probability) then the expectation is infinite. Hence some further assumptions are needed. – Davide Giraudo Dec 24 '23 at 11:27
  • Let's say the $X_{i}$ are continuous random variables with some well-defined continuous and bounded density function. – Ele Dec 24 '23 at 12:09

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