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Let $\{x_i\}$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $\frac{1}{n}\sum_{i=1}^n x_i$ is asymptotically normal.

What do we know about the mean and variance of this asymptotic Gaussian? I know that if $x_i \sim \mathcal{N}(\mu,\sigma)$ that the sample mean has distribution $\mathcal{N}(\mu, \sqrt{\sigma}/n)$. What would the approach be if $x_i$ comes from some arbitrary distribution (say, exponential)?

And in the multivariate case (i.e. $x_i \in \mathbb{R}^n$) or more generally the continuous case (i.e. $x_i(t)$ is a function over a continuous-valued $t$), we would also need to find covariance.

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Let's assume that $x$ is distributed according to some distribution, not necessarily Gaussian, with mean $\mu$ and standard deviation $\sigma$. Then, the distribution for the mean of a sample of $N$ iid ${x_i}$ converges to a Gaussian of mean $\mu$ and standard deviation $\sigma/\sqrt{N}$ if the following conditions hold:

  • The ${x_i}$ are independent, although the convergence will still occur if the correlation is not too strong
  • The convergence to a Gaussian still occurs if the ${x_i}$ have different pdf's, but their variances are of a similar order of magnitude
  • The variances of the pdf's of the $x_i$ must be finite
  • The theorem applies for $N\to \inf$. For finite $N$, the shape of the pdf of the sample mean will still be approximately Gaussian in its center (the tails will differ, though)

If each $x_i$ is a vector of dimension $n$, the situation is the same. The theorem can be applied to each dimension separately unless there are correlations between each vector's components.

In the continuous case, you say that $t$ is continuous, but $x$ is not necessarily continuous. Let's consider that $x(t)$ is piece-wise constant, that is, uniform over some intervals of $t$, but the value can differ from interval to interval. The width of each interval $dt_i$ can be in general a random variable too, where $i$ is the number of the interval. In this case, I expect that if the $\{dt_i\}$ fulfill the conditions mentioned above, and if there is no correlation between the $dt_i$ interval and the (uniform) value that $x$ takes in that interval, then the CLT should also apply.

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  • $\begingroup$ Fantastic, thank you! Could you share a source for the bullet points you gave? These are exactly what I'm looking for, and I'd like to know more. $\endgroup$ – 900edges Feb 24 at 14:59
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    $\begingroup$ You can find more details about those bullet points and some other minor ones in e.g. "Critical Phenomena in Natural Sciences" by Didier Sornette $\endgroup$ – kastellane Feb 24 at 15:01

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