I'm working on a problem where I have $n^2$ real numbers $x_{11},...,x_{nn}$, all drawn i.i.d. from the same distribution $F$. I don't observe each $x_{ij}$, but I do observe the $n$ means:

$$\bar{x}_j = \frac{1}{n} \sum_{i=1}^n x_{ij}$$

As $n$ grows large, it is clear that I can get estimates for both the mean and variance of $F$. I can estimate the mean as

$$\hat{\mu} = \frac{1}{n} \sum_{j=1}^n \bar{x_j}$$

And I can use the central limit theorem to estimate variance as $\hat{\sigma^2} = \sum_{j=1}^n (\bar{x}_j - \hat{\mu})^2.$

If I could recover more moments (and assumed F was nice enough) I could recover the original distribution from $F$ using the moment generating functions (https://math.stackexchange.com/questions/353490/deducing-a-probability-distribution-from-its-moment-generating-function).

My intuition is that it is hard to recover more moments, since the observed $\bar{x_1},...,\bar{x_n}$ behave as if they were drawn from a normal distribution $N(\mu,\frac{\sigma^2}{n})$ when $n$ is large and $\mu,\sigma^2$ are the mean and variance of $F$.

My question is, is it possible to recover higher-order moments given just $\bar{x_1},...,\bar{x_n}$?